TITLE: Would dimples help a ping pong ball travel faster?
QUESTION [2 upvotes]: Dimples are used on golf balls to promote the formation of a turbulent boundary layer that stays with the ball longer as it travels through the air, therefore greatly decreasing the pressure drag experienced by the ball. The increased skin friction drag created by the dimples on the ball's surface is very small compared to the pressure drag reduction that it benefits from; therefore, the overall drag decreases as a result of the dimples. This is useful because a golf ball needs to travel far distances with a potentially large amount of air resistance. Conversely, an airplane does not need dimples on its wings or body because it is already aerodynamic in shape (instead of blunt-faced like the golf ball) and therefore the pressure drag is already reduced such that adding dimples would actually increase the skin friction more than it would decrease the pressure drag, and the overall drag on the airplane would likely increase as a result.
My question is, based on this knowledge, would ping pong balls travel faster as a result of adding dimples? Would they benefit from dimples at all? They have a similar size and shape but much smaller mass than golf balls, which is where my uncertainty comes from.

REPLY [3 votes]: You are mostly correct that the dimples on a golf ball reduce its drag by promoting transition from laminar to turbulent flow, which reduces the size of the separated wake behind the ball. But that doesn't mean that a different ball would benefit in a similar way.
The Reynolds number (Density * Speed * Diameter / Viscosity) has to be within a certain range for the benefit to occur. At smaller values, the dimples (or other roughness) wouldn't be enough of a disturbance to cause transition, and at higher values the flow would be turbulent without them.
To see whether ping pong balls would benefit from dimples, we need to compare their speeds and diameters with those of golf balls. A quick Internet search found that golf balls travel at something like 80.5 m/s and have a diameter of about 42.6mm, whereas the numbers for ping pong balls are about 30 m/s and 40mm. Comparing the Reynolds numbers gives about 240,000 for a golf ball and 80,000 for a ping pong ball.
The figure below (taken from Fluid Dynamic Drag by S. Hoerner) plots experimental drag measurements for spheres as functions of Reynolds number, and shows a dramatic change in shape at Reynolds numbers in the transitional region between 100,000 and 1,000,000.
The next figure (from the same source) zooms in more closely to our region of interest, and shows that the ping pong ball's Reynolds number of about 80,000 is just below  the beginning of the transitional region, thus indicating that it would not be likely to benefit from adding dimples.


text:
\begin{document}

\maketitle
\begin{abstract}
  We study a class of elliptic problems with homogeneous
  Dirichlet boundary condition and a nonlinear
  reaction term $f$ which is nonlocal depending on the $L^{p}$-norm of the unknown function.
  The nonlinearity $f$ can make the problem degenerate since it may even
  have multiple singularities  in the nonlocal variable.

  We use  fixed point arguments for an appropriately defined solution map,
  to produce multiplicity of classical positive solutions with ordered norms.
\end{abstract}

\bigskip

\maketitle
\begin{center}
\begin{minipage}{12cm}
\tableofcontents
\end{minipage}
\end{center}

\bigskip

\section{Introduction}

  In this paper we consider the existence of multiple positive solutions for the following
class of nonlocal problems:
\begin{equation}\label{P1}\tag{$P_{I}$}
\left \{ \begin{array}{ll}
  -\Delta u = f\left(u, \displaystyle\int_{\Omega} | u|^{p}dx \right) & \mbox{in $\Omega$,}\\
  u=0 & \mbox{on $\partial\Omega$,}
\end{array}\right.
\end{equation}

  where $p\geq1$, $\Omega$ is a bounded domain in $\R^{N}$ with smooth boundary
  and  $f$ is a continuous function with a  suitable  behaviour in the second variable.

\medskip

  In the last decades partial differential equations involving nonlocal terms have attracted a great deal of attention
  of the mathematical community for different reasons. Indeed, equations involving nonlocal terms are usually more realistic
  to model different situations in nature, see Furter-Grinfeld \cite{FG} for a comparison between local and nonlocal models in population dynamics,
  or Kirchhoff \cite{Kir} for an improvement (based on the insertion of a nonlocal term) of the classical D'Alembert's wave equation
  in string deformation theory. From a mathematical point of view
  nonlocal equations are challenging since, in general, the presence of a nonlocal term
  makes the equation much more complicated. In many cases the known techniques
  cannot be applied in a straightforward way, so the development of alternative approaches is required.

Regarding stationary partial differential equations in very recent years two classes of problems involving nonlocal terms in the diffusion operator have been a quite active research field, namely the Kirchhoff and the generalised Carrier problems. A basic prototype of the first one is
\begin{equation}\label{P3}\tag{$KP$}
\left \{ \begin{array}{ll}
  -a\left(\displaystyle\int_{\Omega}|\nabla u|^{2}dx\right)\Delta u = f(u) & \mbox{in $\Omega$,}  \medskip \\
  u=0 & \mbox{on $\partial\Omega$,}
\end{array}\right.
\end{equation}
  where $a$ is usually a continuous real function bounded away from zero.
  Problem \eqref{P3} is the stationary version of a hyperbolic model to small transversal vibrations
  of elastic membranes, see Kirchhoff \cite{Kir} for details.
  On the other hand, generalised Carrier problems are of type
\begin{equation}\label{P4}\tag{$GCP$}
\left \{ \begin{array}{ll}
  -a\left(\displaystyle\int_{\Omega}|u|^{p}dx\right)\Delta u = f(u) & \mbox{in $\Omega$,} \medskip \\
  u=0 & \mbox{on $\partial\Omega$,}
\end{array}\right.
\end{equation}
  where $a$ is again a continuous real function. When $p=2$, problem \eqref{P4} is the stationary case
  of another hyperbolic model of transversal vibrations (not necessarily small)
  of elastic membranes which was introduced by Carrier in \cite{Car}, in the unidimensional case.
  As $p=1$, \eqref{P4} also appears as the stationary version of parabolic models
  in the study of the dispersion of biological populations where the diffusion rate
  of the specie takes into account the whole population. See, for instance, Chipot-Rodrigues \cite{CR}.
  For papers involving operators as in \eqref{P4} with $p>1$ arbitrary,
  we refer the reader to Figueiredo-Sousa-Morales-Rodrigo-Su\'arez \cite{FMS} and references therein.

 \medskip

  Recently, the above problems have been studied also in the degenerate case, that is,
  when $a$ vanishes in some points.
  Concerning \eqref{P3}, this line of research started with the papers of
  Ambrosetti-Arcoya \cite{AA1,AA2}.
  See also Santos J\'unior-Siciliano \cite{SS} who establish a multiplicity result of positive solutions
  depending on the number of degeneration points of the function $a$.
  For problem \eqref{P4},
  some interesting results can be found in the recent papers of
  Delgado-Morales-Rodrigo-Santos J\'unior-Su\'arez \cite{DMSS} and
  Gasi\'nski-Santos J\'unior \cite{GS1,GS2}.

\medskip

  Starting with the above considerations, our aim here is to study existence of positive
  solutions for (possibly) degenerate problems for which problem
  \eqref{P4} is just a particular case.
  The main goal of this paper is to prove that when the reaction term involved in the equation depends on the
  $L^{p}$-norm of the unknown function and it allows singularities in some points of its domain,
  the same sort of multiplicity result (as those observed in degenerate Kirchhoff
  and Carrier problems, see
  Santos J\'unior-Siciliano \cite{SS} and
  Gasi\'nski-Santos J\'unior \cite{GS1}) holds.

  It is worth to point out that such kind of results are much more general than those obtained for degenerate

  Carrier problems. In fact, if we choose, the nonlinearity $f$

  in \eqref{P1} of the form
$$
  f\left(u, \int_{\Omega} |u|^{p}dx \right )=\frac{g(u)}{a\left(\displaystyle\int_{\Omega} |u|^{p}dx\right)},
$$
  with $a$ being a continuous real function which vanishes in some points,
  then $f(u,\cdot)$ exhibits a diverging behaviour and we fall in the degenerate
  Carrier problems \eqref{P4}.

\subsection{Statement of the main result - Theorem \ref{main1}} \label{sec:Prelim}

  First let us introduce some notations.
  Along the paper,
  \begin{itemize}
  \item $\|\cdot\|$ denotes the $H_{0}^{1}(\Omega)$-norm,
  \item   $|\cdot|_{r}$ is the $L^{r}(\Omega)$-norm,
  \item   $\lambda_{1}$ is the first eigenvalue of the minus Laplacian operator in $\Omega$
  with zero Dirichlet boundary condition,
  \item   $\varphi_{1}$ is the positive eigenfunction associated to $\lambda_{1}$ normalized in the $H_{0}^{1}(\Omega)$-norm,
  \item $e_{1}$ is the positive eigenfunction associated to $\lambda_{1}$ normalized in the $L^{\infty}(\Omega)$-norm,
  \item $C_{1}>0$ stands for best constant of the Sobolev embedding of $H_{0}^{1}(\Omega)$ into $L^{1}(\Omega)$;
\item  $|\Omega|$ is the Lebesgue measure of $\Omega$.
  \end{itemize}

  \medskip

We assume that $f$ is a function having the following behaviour:

\begin{enumerate}[label=($f_{\arabic*}$),ref=$f_{\arabic*}$,start=0]
\item\label{f_{0}} there exist positive numbers $0=:t_{0}<t_{1}<t_{2}<\ldots<t_{K}$ ($K\geq 1$) such that
     $f:\R\times \mathcal{A}\to\R$ is continuous, where $\mathcal{A}=(0,\infty)\backslash\{t_{1}, \ldots, t_{K}\}$.
     For each $k\in\{1, \ldots, K\}$ and fixed $\alpha\in (t_{k-1}, t_{k})$,
     $f(\cdot,\alpha)\in C^1(\R)$,
     there exists $s_\alpha>0$ such that $f(s_{\alpha}, \alpha)=0$ and
     $f(s, \alpha)>0$ for $s\in (0, s_{\alpha})$.
  Moreover $\sup\limits_{\alpha\in Z}s_{\alpha}<\infty$, for each compact set $Z\subset (t_{k-1}, t_{k})$ and $k\in\{1, \ldots, K\}$;
 \item\label{f_{1}} for each $k\in\{1, \ldots, K\}$ and any $s\in (0,s_{\alpha})$, $\lim\limits_{\alpha\to t_{k}}f(s, \alpha)\in[\lambda_{1},+\infty]$;
 \item\label{f_{2}} for each $k\in\{1, \ldots, K\}$ and fixed $\alpha\in (t_{k-1}, t_{k})$, the map $(0, s_{\alpha})\ni s\mapsto f(s, \alpha)/s$ is decreasing;
 \item\label{f_{3}}$\inf\limits_{\alpha\in \mathcal{A}}\gamma_{\alpha}>\lambda_1$, where $\gamma_\alpha:=\lim\limits_{s\to 0^{+}}f(s, \alpha)/s$.
\end{enumerate}

\begin{remark} \label{rem_hyp123}\rm
  \textbf{(a)} Hypothesis \eqref{f_{1}} makes the problems \eqref{P1}
   singular in the nonlocal term
  whenever $\lim\limits_{\alpha\to t_{k}}f(s, \alpha)=+\infty$.\\
  \textbf{(b)} Hypotheses  \eqref{f_{2}} and \eqref{f_{3}} describe the behaviour of the function
  $\psi_{\alpha}\colon (0, s_{\alpha})\to (0, \gamma_{\alpha})$
  defined by
\begin{equation}\label{eq_psi}
  \psi_{\alpha}(s)=\frac{f(s,\alpha)}{s},\ \forall s\in (0, s_{\alpha}).
\end{equation}
  Hypothesis \eqref{f_{2}} guarantees its monotonicity, which will be crucial
  in the proof of the uniqueness of solutions for the auxiliary problem \eqref{AP}
  (see Proposition \ref{pro1}) and for other technical issues (see Lemmas \ref{odd}, \ref{odt}, and \ref{behhhh}).
  Hypothesis \eqref{f_{3}} says that the limits of the functions $\psi_{\alpha}$ at zero are separated from $\lambda_1$ (uniformly in $\alpha$).
\end{remark}

  Moreover we assume that:
  \medskip

\begin{enumerate}[label=($f_{\arabic*}$),ref=$f_{\arabic*}$,start=4]
 \item\label{f_{4}} $t_{K}<\big(\inf\limits_{\alpha\in\mathcal{A}}s_{\alpha}\big)^{p}\displaystyle\int_{\Omega}e_{1}^{p}dx$;
 \item\label{f_{5}} $\inf\limits_{\alpha\in (t_{k-1}, t_{k})}\max\limits_{s\in[0, s_{\alpha}]}f(s,\alpha)(s_{\alpha}^{p-1}/\alpha)<\lambda_{1}^{1/2}/C_{1}|\Omega|^{1/2}$,
             for all $k\in\{1, \ldots, K\}$.
\end{enumerate}

\medskip

\begin{remark}\label{rem_hyp45}\rm
  \textbf{(a)} Hypotheses \eqref{f_{4}}
  gives some bounds on the ``edges'' of the
  curvilinear trapezoid $\{s_{\alpha}:\ \alpha\in\ \mathcal{A}\}\times \mathcal{A}$,
  in which the interesting domain of $f$ is contained.\\
  \textbf{(b)} Hypotheses \eqref{f_{5}}
  is quite technical and  says that
  for some particular choice of $\alpha_0\in(t_{k-1}, t_{k})$ the maximum of the quotient $f(s,\alpha_0)/\alpha_0$
  over $s\in[0,s_{\alpha}]$ is bounded by some constant.
\end{remark}

  Our main result is the following.

\begin{theorem}\label{main1}
  If conditions \eqref{f_{0}}-\eqref{f_{5}} hold, then problem \eqref{P1} has at least $2K$ classical positive solutions with ordered $L^{p}$-norms, namely
$$
  0<\int_{\Omega}u_{1, 1}^{p}dx<\int_{\Omega}u_{1, 2}^{p}dx<t_{1}<\ldots<t_{K-1}<\int_{\Omega}u_{K, 1}^{p}dx<\int_{\Omega}u_{K, 2}^{p}dx<t_{K}.
$$
\end{theorem}

  The proof of the above theorem will follow the following general steps:
  \begin{itemize}
  \item[Step 1.] We first introduce an auxiliary problem with a truncated nonlinearity and fixed nonlocal term;
  \item[Step 2.] By variational methods we show that the auxiliary problem has a unique solution;
  \item[Step 3.] We define a suitable map which gives the $L^{p}$-norm

  of the solution;
  \item[Step 4.] Finally, we show that this map has fixed points, which are indeed solutions of the considered problem.
  \end{itemize}

  The structure of the paper is the following.
  In Section  \ref{sec:Auxiliary} we develop Step 1 and Step 2 above.

  In Section \ref{sec:Th1}  we address Step 3 and Step 4 above and implement them to the proof of Theorem \ref{main1}.

  In the final Section \ref{sec:Final} we give examples of nonlinearities $f$ satisfying our assumptions.

\section{Auxiliary problem}\label{sec:Auxiliary}

\subsection{Statement of the auxiliary problem \eqref{AP}}

  In order to prove our results, the following auxiliary problem will play an important role: for each $k\in\{1, \ldots, K\}$ and any $\alpha\in (t_{k-1}, t_{k})$ fixed, consider

\begin{equation}\label{AP}\tag{$P_{k, \alpha}$}
  \left \{ \begin{array}{ll}
    -\Delta u = \widehat{f}_{\alpha}(u) & \mbox{in $\Omega$,}\\
    u>0 & \mbox{in $\Omega$,}\\
    u=0 & \mbox{on $\partial\Omega$,}
  \end{array}\right.
\end{equation}
  where
\begin{equation*}
  \widehat{f}_{\alpha}(s)=\left \{ \begin{array}{ll}
  f(0, \alpha) & \mbox{if $s\leq 0$,}\\
  f(s, \alpha) & \mbox{if $0<s\leq s_{\alpha}$,}\\
  0 & \mbox{if $s_{\alpha}<s$.}\\
\end{array}\right.
\end{equation*}

\begin{remark}\label{rem1}\rm
  Since $f$ is continuous, it is an immediate consequence of the definition of $\widehat{f}_{\alpha}$
  that if $\{\alpha_{n}\}\subset (t_{k-1}, t_{k})$ converges to $\alpha\in (t_{k-1}, t_{k})$
  and $s_{n}\in (0, s_{\alpha_{n}})$ is such that $s_{n}\to s$, for some $s\in (0, s_{\alpha})$,
  then $\widehat{f}_{\alpha_{n}}(s_{n})\to\widehat{f}_{\alpha}(s)$.
\end{remark}

\subsection{Existence of the solutions of \eqref{AP}}

  Since we have ``removed'' the nonlocal term of \eqref{P1}, we can treat problem \eqref{AP} using a variational approach.

\begin{proposition}\label{pro1}
  If conditions \eqref{f_{0}}, \eqref{f_{2}} and \eqref{f_{3}} hold, then for each $k\in\{1, \ldots, K\}$ and $\alpha\in (t_{k-1}, t_{k})$ fixed, problem \eqref{AP} has a
  unique classical solution $0<u_{\alpha}\leq s_{\alpha}$.
\end{proposition}

\begin{proof}
  Since, for each $k\in\{1, \ldots, K\}$ and $\alpha\in (t_{k-1}, t_{k})$ fixed, $\widehat{f}_{\alpha}$ is bounded and continuous, the energy functional
$$
  I_{k, \alpha}(u)=\frac{1}{2}\|u\|^{2}-\int_{\Omega}\widehat{F}_{\alpha}(u)dx
$$
  corresponding to problem \eqref{AP} is coercive and weakly lower semicontinuous.
  Here as usual,
$$\widehat{F}_{\alpha}(s)=\int_{0}^{s}\widehat{f}_{\alpha}(\sigma) d\sigma.$$
  Therefore $I_{k, \alpha}$ has a minimum point $u_{\alpha}$
  which is a weak solution of \eqref{AP}.
  Moreover, it follows from conditions \eqref{f_{2}} and \eqref{f_{3}} that
$$
  \frac{I_{k, \alpha}(s\varphi_{1})}{s^{2}}=\frac{1}{2}-\int_{\Omega}\frac{\widehat{F}_{\alpha}(s\varphi_{1})}{(s\varphi_{1})^{2}}\varphi_{1}^{2}dx\to
  \frac{1}{2}\left(1-\frac{\gamma_{\alpha}}{\lambda_{1}}\right)<0 \ \mbox{as $s\to 0^{+}$}.
$$
  The last inequality implies that the minimum point $u_{\alpha}$ of $I_{k, \alpha}$ is nontrivial, because for $s>0$ small enough
$$
  I_{k, \alpha}(u_{\alpha})\leq I_{k, \alpha}(s\varphi_{1})=(I_{k, \alpha}(s\varphi_{1})/s^{2})s^{2}<0.
$$
  Then $u_{\alpha}\in H^{1}_{0}(\Omega)$ satisfies
\begin{equation}\label{eq:weaksolution}
  \int_{\Omega} \nabla u_{\alpha} \nabla v dx= \int_{\Omega} \widehat f_{\alpha}(u_{\alpha}) v dx \quad
  \forall\ v\in H^{1}_{0}(\Omega).
\end{equation}
  Then  it is easy to see that any nontrivial weak solution $u_{\alpha}$ of problem \eqref{AP}
  satisfies $0\leq u_{\alpha}\leq s_{\alpha}$. Indeed by choosing $v=(u_{\alpha}-s_{\alpha})^{+}$ in \eqref{eq:weaksolution} we arrive
  (by the definition of $\widehat f_{\alpha}$) at
$$
  \int_{\Omega}\nabla u_{\alpha}\nabla (u_{\alpha}-s_{\alpha})^{+} dx=\int_{\Omega} \widehat{f}_{\alpha}(u_{\alpha}) (u_{\alpha}-s_{\alpha})^{+} dx =0
$$
  which redly  implies $u_{\alpha}\leq s_{\alpha}$. Analogously one deduces that $0\leq u_{\alpha}$ by taking $v=u_{\alpha}^{-}$ in  \eqref{eq:weaksolution}.
  Morover
  the weak solution of problem \eqref{AP} is unique by condition \eqref{f_{2}},
  see Brezis-Oswald \cite{BO}.
  Since $\widehat{f}_{\alpha}(u_{\alpha})=f(u_{\alpha},\alpha)$ is bounded and $f(\cdot,\alpha)\in C^{1}(\R)$, it follows
  from Agmon \cite{Ag} that $u_{\alpha}$ is a classical solution. Finally, the maximum principle completes the proof
  (see  Gilbarg-Trudinger \cite[Theorem 3.1]{GT}).
\end{proof}

\medskip

\subsection{Properties of the solutions of \eqref{AP}}

  Due to Proposition \ref{pro1}, we can set
\begin{equation}\label{eq:ca}
  c_{\alpha}:=I_{k,\alpha}(u_{\alpha})=\min_{u\in H_{0}^{1}(\Omega)}I_{k, \alpha}(u).
\end{equation}

  Since, by condition \eqref{f_{2}}, the map $(0, s_{\alpha})\ni s\mapsto\psi_{\alpha}(s)=\widehat{f}_{\alpha}(s)/s$ is decreasing
  (see \eqref{eq_psi} in Remark \ref{rem_hyp123}),
  there exists the inverse $\psi_{\alpha}^{-1}\colon (0, \gamma_{\alpha})\to (0, s_{\alpha})$.
  Thereby, by condition \eqref{f_{3}}, for
  each $\varepsilon\in (0, \inf\limits_{\alpha\in\mathcal{A}}\gamma_{\alpha}-\lambda_{1})$, it makes sense to consider the function
$$
  y_{\alpha}:=\psi_{\alpha}^{-1}(\lambda_{1}+\varepsilon)e_{1}.
$$

\begin{lemma}\label{odd}
  If conditions \eqref{f_{0}}, \eqref{f_{2}} and \eqref{f_{3}} hold, $\alpha\in (t_{k-1}, t_{k})$, then for each
  $\varepsilon\in (0, \inf\limits_{\alpha\in\mathcal{A}}\gamma_{\alpha}-\lambda_{1})$, we have
\begin{equation}\label{7}
  c_{\alpha}\leq -\frac{1}{2}\varepsilon\psi_{\alpha}^{-1}(\lambda_{1}+\varepsilon)^{2}\int_{\Omega}e_{1}^{2} dx.
\end{equation}
  where $c_{\alpha}$ is given in \eqref{eq:ca}.
\end{lemma}

\begin{proof}
  Note that, by condition \eqref{f_{2}}, we have
$$
  \widehat{F}_{\alpha}(t)\geq \frac12\widehat{f}_{\alpha}(s)s\quad \forall \ s\geq 0.
$$
  Hence,
$$
  \frac{I_{k,\alpha}(y_{\alpha})}{\psi_{\alpha}^{-1}(\lambda_{1}+\varepsilon)^{2}}
  \leq\frac{1}{2}\left[\|e_{1}\|^{2}-\int_{\Omega}\frac{\widehat{f}_{\alpha}(y_{\alpha})}{\psi_{\alpha}^{-1}(\lambda_{1}+\varepsilon)^{2}}y_{\alpha}dx\right],
$$
  or equivalently
$$
  \frac{I_{k, \alpha}(y_{\alpha})}{\psi_{\alpha}^{-1}(\lambda_{1}+\varepsilon)^{2}}
  \leq\frac{1}{2}\left[\|e_{1}\|^{2}-\int_{\Omega}\frac{\widehat{f}_{\alpha}(y_{\alpha})}{y_{\alpha}}e_{1}^{2}dx\right].
$$
  Using the definition of $e_{1}$ and condition \eqref{f_{2}}, we get
$$
  \frac{I_{k,\alpha}(y_{\alpha})}{\psi_{\alpha}^{-1}(\lambda_{1}+\varepsilon)^{2}}
  \leq\frac{1}{2}\left[\|e_{1}\|^{2}-\int_{\Omega}\frac{\widehat{f}_{\alpha}(\psi_{\alpha}^{-1}(\lambda_{1}+\varepsilon))}{\psi_{\alpha}^{-1}(\lambda_{1}+\varepsilon)}e_{1}^{2}dx\right].
$$
  Now, using the definition of $\psi_{\alpha}^{-1}$, we deduce
$$
  \frac{I_{k,\alpha}(y_{\alpha})}{\psi_{\alpha}^{-1}(\lambda_{1}+\varepsilon)^{2}}
  \leq\frac{1}{2}\left[\|e_{1}\|^{2}-(\lambda_{1}+\varepsilon)\int_{\Omega}e_{1}^{2}dx\right]=-\frac{1}{2}\varepsilon\int_{\Omega}e_{1}^{2}dx.
$$
  Therefore,
$$
  c_{\alpha}\leq I_{k, \alpha}(y_{\alpha})\leq -\frac{1}{2}\varepsilon\psi_{\alpha}^{-1}(\lambda_{1}+\varepsilon)^{2}\int_{\Omega}e_{1}^{2} dx,
$$
  which concludes the proof.
\end{proof}

  The next two technical results  will be helpful in what follows.

\begin{lemma}\label{odt}
  If conditions \eqref{f_{0}}, \eqref{f_{2}} and \eqref{f_{3}} hold, then
\begin{equation}\label{2}
  u_{\alpha}\geq z_{\alpha}:=\psi_{\alpha}^{-1}(\lambda_{1})e_{1}\quad \forall \ \alpha\in (t_{k-1}, t_{k}).
\end{equation}
\end{lemma}

\begin{proof}

  From condition \eqref{f_{2}} and definition of $\psi_{\alpha}^{-1}$ it follows that
$$
  \lambda_{1}= \frac{\widehat{f}_{\alpha}(\psi_{\alpha}^{-1}(\lambda_{1}))}{\psi_{\alpha}^{-1}(\lambda_{1})}\leq \frac{\widehat{f}_{\alpha}(z_{\alpha})}{z_{\alpha}}.
$$
  Thus
$$
  -\Delta(z_{\alpha})=\lambda_{1}z_{\alpha}\leq \widehat{f}_{\alpha}(z_{\alpha}) \ \mbox{in $\Omega$}.
$$

  Therefore $z_{\alpha}$ is a subsolution of \eqref{AP}. Inequality \eqref{2} follows now from condition
  \eqref{f_{2}} and
  Ambrosetti-Brezis-Cerami \cite[Lemma 3.3]{ABC}.
\end{proof}

\begin{lemma}\label{behhhh}
  If conditions \eqref{f_{0}}-\eqref{f_{2}} hold, then
\begin{equation}\label{232}
  \liminf\limits_{\alpha\to t_{k-1}^{+}}\psi_{\alpha}^{-1}(\lambda_{1})\geq \inf\limits_{\alpha\in (t_{k-1}, t_{k})}s_{\alpha}.
\end{equation}
  The same holds for $\alpha\to t_{k}^{-}$.
\end{lemma}

\begin{proof}
 Assuming the contrary, there would exist a sequence $\{\alpha_n\}\subseteq (t_{k-1},t_k)$ such that
 $\alpha_n\to t_{k-1}^+$ (or alternatively $\alpha_n\to t_{k}^-$) and
\[
  \psi_{\alpha_n}^{-1}(\lambda_{1})<T<\inf\limits_{\alpha\in (t_{k-1}, t_{k})}s_{\alpha},
\]
  for some $T\in\R$. Since $T\in [0,s_{\alpha_n}]$ for all $n\ge 1$ and
  $\psi_{\alpha_n}$ is decreasing on $[0,s_{\alpha_n}]$, we would have
\[
  \lambda_1>\psi_{\alpha_n}(T)=\frac{f(T,\alpha_n)}{T}.
\]
  But this contradicts hypothesis \eqref{f_{1}}.
  So the proof is done.
\end{proof}

\section{Proof of the main result}\label{sec:Th1}

\subsection{Auxiliary map $\mathcal P_{k}$}

  In virtue of Proposition \ref{pro1}, given $p\geq1$, for any $k\in \{1,\ldots, K\}$ we can define the map
\begin{equation}\label{eq:Pk1}
  \mathcal{P}_{k}: \alpha\in (t_{k-1}, t_{k}) \longmapsto \int_{\Omega}u_{\alpha}^{p}dx\in \R
\end{equation}
  where $u_{\alpha}$ is the unique solution of problem \eqref{AP}.
  The strategy of proving Theorem \ref{main1} will be to show that $\mathcal P_{k}$ is continuous and has
  two fixed points.
  Indeed any fixed point, let us say $\overline\alpha$, of $\mathcal P_{k}$ satisfies by definition
\begin{equation*}
\left \{ \begin{array}{ll}
  -\Delta u_{\overline\alpha} = \widehat{f}\left(u, \displaystyle\int_{\Omega} u^{p}_{ \overline\alpha} \,dx \right) & \mbox{in $\Omega$,}\\
  0<u_{\overline \alpha}\leq s_{\overline\alpha} & \mbox{in $\Omega$,}\\
  u_{\overline \alpha}=0 & \mbox{on $\partial\Omega$,}
\end{array}\right.
\end{equation*}
  and hence $u_{\overline\alpha}$ is a solution of \eqref{P1} with $L^{p}$-norm in $(t_{k-1}, t_{k})$.

\subsection{Continuity of $\mathcal P_{k}$}
\begin{proposition}\label{pp2}
  If conditions \eqref{f_{0}}, \eqref{f_{2}} and \eqref{f_{3}} hold, then for each $k\in \{1, 2, \ldots, K\}$,
  the map $\mathcal{P}_{k}$ defined in \eqref{eq:Pk1}

 is continuous.
\end{proposition}

\begin{proof}
  Let $\{\alpha_{n}\}\subset (t_{k-1}, t_{k})$ be a sequence such that $\alpha_{n}\to \alpha_{\ast}$,
  for some $\alpha_{\ast}\in (t_{k-1}, t_{k})$. Denote by $u_{n}$ the positive solution of $\eqref{AP}$ with $\alpha=\alpha_{n}$. Since,
\begin{equation}\label{1}
  \frac{1}{2}\|u_{n}\|^{2}-\int_{\Omega}F_{\alpha}(u_{n})dx=I_{k, \alpha}(u_{n})<0,
\end{equation}
  we get
$$
\|u_{n}\|\leq \left[2F_{\alpha}(s_{\alpha})|\Omega|\right]^{1/2},\quad \forall \ n\in\N.
$$
  Therefore, $\{u_{n}\}$ is bounded in $H_{0}^{1}(\Omega)$ and, up to a subsequence, there exists $u_{\ast}\in H_{0}^{1}(\Omega)$ such that
\begin{equation}\label{19}
  u_{n}\rightharpoonup u_{\ast} \ \mbox{in} \ H_{0}^{1}(\Omega).
\end{equation}
  Thus, from Remark \ref{rem1}, passing to the limit as $n\to\infty$ in
$$
  \int_{\Omega}\nabla u_{n}\nabla v dx=\int_{\Omega} \widehat{f}_{\alpha_{n}}(u_{n})v dx\quad \forall \ v\in H_{0}^{1}(\Omega),
$$
we get
$$
  \int_{\Omega}\nabla u_{\ast}\nabla v dx=\int_{\Omega} \widehat{f}_{\alpha_{\ast}}(u_{\ast})v dx\quad \forall \ v\in H_{0}^{1}(\Omega).
$$
  So, $u_{\ast}$ is a nonnegative weak solution of \eqref{AP} with $\alpha=\alpha_{\ast}$.
  We are going to show that $u_{\ast}\neq 0$. In fact, passing to the limit as
  $n\to\infty$ in
$$
  \int_{\Omega}\nabla u_{n}\nabla u_{\ast} dx=\int_{\Omega} \widehat{f}_{\alpha_{n}}(u_{n})u_{\ast} dx
$$
  and
$$
  \|u_{n}\|^{2}=\int_{\Omega} \widehat{f}_{\alpha_{n}}(u_{n})u_{n} dx,
$$
  we conclude that
\begin{equation}\label{20}
  \|u_{n}\|\to \|u_{\ast}\|.
\end{equation}
  From \eqref{19} and \eqref{20}, we have
\begin{equation}\label{21}
  u_{n}\to u_{\ast} \ \mbox{in $H_{0}^{1}(\Omega)$}.
\end{equation}

  By Lemma \ref{odd}, there exists $\varepsilon>0$, small enough, such that
$$
  I_{k, \alpha_{n}}(u_{n})\leq -\frac{1}{2}\varepsilon\psi_{\alpha}^{-1}(\lambda_{1}+\varepsilon)^{2}\int_{\Omega}e_{1}^{2} dx\quad \forall \ n\in\N.
$$
  So, passing to the limit as $n\to\infty$ and using \eqref{21}, we obtain
$$
  I_{k, \alpha_{\ast}}(u_{\ast})\leq -\frac{1}{2}\varepsilon\psi_{\alpha}^{-1}(\lambda_{1}+\varepsilon)^{2}\int_{\Omega}e_{1}^{2} dx<0.
$$
  Therefore $u_{\ast}\neq 0$. Arguing as in the proof of Proposition \ref{pro1} we can show that $u_{\ast}$ is a positive classical solution of \eqref{AP} with
  $\alpha=\alpha_{\ast}$. Since such a solution is unique, we conclude that
\begin{equation}\label{eq:continuity}
  u_{\ast}=u_{\alpha_{\ast}}.
\end{equation}
  Consequently,
\begin{equation}\label{21b}
  -\Delta(u_{n}-u_{\ast})=\widehat{f}_{\alpha_{n}}(u_{n})-\widehat{f}_{\alpha_{\ast}}(u_{\ast})=:g_{n}(x)\quad \forall \ n\in\N.
\end{equation}
  From \eqref{f_{0}}, the continuity of $f$ and inequalities $0\leq u_{n}\leq s_{\alpha_{n}}, 0\leq u_{\ast}\leq s_{\alpha_{\ast}}$, there exists a  constant $C>0$, such that
\begin{equation}\label{22}
  |g_{n}|_{\infty}\leq C_2\quad \forall \ n\in\N.
\end{equation}
  It follows from \eqref{21b}, \eqref{22} and
  Theorem 0.5 of Ambrosetti-Prodi \cite{AmPr} that there exists $\beta\in (0, 1)$ such that
$$
  \|u_{n}-u_{\ast}\|_{C^{1,\beta}(\overline{\Omega})}\leq C'\quad \forall \ n\in\N,
$$
  for some $C'>0$. By the compactness of embedding from $C^{1,\beta}(\overline{\Omega})$ into $C^{1}(\overline{\Omega})$ and \eqref{21}, up to a
  subsequence, we have
\begin{equation}\label{23}
  u_{n}\to u_{\ast} \ \mbox{in $C^{1}(\overline{\Omega})$}.
\end{equation}
  Convergence in \eqref{23} and inequality
$$
  ||u_{n}|_{p}-|u_{\ast}|_{p}|\leq |u_{n}-u_{\ast}|_{p}\leq |\Omega|^{1/p}|u_{n}-u_{\ast}|_{\infty}
$$
  lead us to
$$
  \mathcal{P}_{k}(\alpha_{n})\to\mathcal{P}_{k}(\alpha_{\ast}).
$$
  This proves the continuity of $\mathcal{P}_{k}$.
\end{proof}

\subsection{Existence of fixed points of $\mathcal{P}_{k}$}

\begin{proposition}\label{pp3}
  If conditions \eqref{f_{0}}-\eqref{f_{5}} hold, then the map $\mathcal{P}_{k}$ defined by \eqref{eq:Pk1}
  has at least two fixed points $t_{k-1}<\alpha_{1, k}<\alpha_{2, k}<t_{k}$.
\end{proposition}

\begin{proof}

  We start with two claims describing the behaviour of $\mathcal{P}_{k}$.

\medskip

\noindent {\bf Claim 1:}
  $\lim\limits_{\alpha\to t_{k-1}^{+}}\mathcal{P}_{k}(\alpha)>t_{k-1}$ and $\lim\limits_{\alpha\to t_{k}^{-}}\mathcal{P}_{k}(\alpha)>t_{k}$.

\medskip

  From Lemma \ref{odt}, we have
$$
  \mathcal{P}_{k}(\alpha)\geq (\psi_{\alpha}^{-1}(\lambda_{1}))^{p}\int_{\Omega}e_{1}^{p}dx\quad \forall \ \alpha\in (t_{k-1}, t_{k}).
$$
  Hence, by condition \eqref{f_{4}} and Lemma \ref{behhhh}, we get
\begin{eqnarray*}
  \liminf\limits_{\alpha\to t_{k-1}^{+}}\mathcal{P}_{k}(\alpha) & \geq & \left(\inf\limits_{\alpha\in (t_{k-1}, t_{k})}s_{\alpha}\right)^{p}\int_{\Omega}e_{1}^{p}dx>t_{K}>t_{k-1},\\
  \liminf\limits_{\alpha\to  t_{k}^{-}}\mathcal{P}_{k}(\alpha) & \geq & \left(\inf\limits_{\alpha\in (t_{k-1}, t_{k})}s_{\alpha}\right)^{p}\int_{\Omega}e_{1}^{p}dx>t_{K}>t_{k}.
\end{eqnarray*}

\medskip

\noindent {\bf Claim 2:}
  There exists $\alpha_0\in (t_{k-1}, t_{k})$ such that $\mathcal{P}_{k}(\alpha_0)<\alpha_0$.

\medskip

  For each $\alpha\in (t_{k-1}, t_{k})$, let $w_{\alpha}$ be the unique solution (which is positive) of the problem
\begin{equation*}
  \left \{ \begin{array}{ll}
  -\Delta u = u_{\alpha}^{p-1} & \mbox{in $\Omega$,}\\
  u=0 & \mbox{on $\partial\Omega$,}
  \end{array}\right.
\end{equation*}
  where $u_{\alpha}$ is the unique positive solution of \eqref{AP}. Hence, multiplying by $u_{\alpha}$ and integrating by parts, we have
$$
  \int_{\Omega}\nabla w_{\alpha}\nabla u_{\alpha} dx=\int_{\Omega}u_{\alpha}^{p}dx=\mathcal{P}_{k}(\alpha).
$$
  On the other hand, by the definition of $u_{\alpha}$, we get
\begin{equation}\label{12}
  \mathcal{P}_{k}(\alpha)=\int_{\Omega}\widehat{f}_{\alpha}(u_{\alpha})w_{\alpha} dx
\end{equation}
  thus,
\begin{equation}\label{16}
  \mathcal{P}_{k}(\alpha)\leq\left(\max\limits_{s\in [0, s_{\alpha}]}\widehat{f}_{\alpha}(s)\right)C_{1}\|w_{\alpha}\|,
\end{equation}
  (where $C_{1}>0$ is the best constant of the Sobolev embedding from $H_{0}^{1}(\Omega)$ into $L^{1}(\Omega)$).

  From the definition of $w_{\alpha}$, the fact that $0<u_{\alpha}\leq s_{\alpha}$ and H\"older's inequality, we obtain
\begin{equation}\label{15}
  \|w_{\alpha}\|\leq  \frac{1}{\sqrt{\lambda_{1}}}\left(\int_{\Omega}u_{\alpha}^{2(p-1)}dx\right)^{1/2}\leq  \frac{1}{\sqrt{\lambda_{1}}}s_{\alpha}^{p-1}|\Omega|^{1/2}.
\end{equation}
  Applying \eqref{15} in \eqref{16}, we obtain
$$
  \mathcal{P}_{k}(\alpha)\leq \left(\max\limits_{s\in [0, s_{\alpha}]}\widehat{f}_{\alpha}(s)\right)
  \frac{C_{1}}{\sqrt{\lambda_{1}}}s_{\alpha}^{p-1}|\Omega|^{1/2}\quad \forall \
  \alpha\in (t_{k-1}, t_{k}).
$$
  Using condition \eqref{f_{5}} we get the conclusion of Claim 2.

\medskip

  From the continuity of $\mathcal{P}_{k}$ (see Proposition \ref{pp2}),
  Claims 1 and 2 and the intermediate value theorem for continuous real functions,
  we conclude that $\mathcal{P}_{k}$ has at least two fixed points
  $\alpha_{1,k}$ and $\alpha_{2,k}$ in the interval $(t_{k-1},t_{k})$.
  The conclusions of Claims 1 and 2 are illustrated in Figure 1.

\begin{figure}[h]
\begin{center}
\begin{picture}(300,250)(0,0)
 \put(0,30){\vector(1,0){280}}
 \put(30,0){\vector(0,1){250}}
  \multiput(10,10)(2,2){115}{\circle*{0.01}}

 \multiput(80,30)(0,4){55}{\line(0,2){1}}
 \multiput(200,30)(0,4){55}{\line(0,2){1}}

 \multiput(102,30)(0,2){36}{\line(0,1){1}}
    \put(102,102){\circle*{3}}
 \multiput(126,30)(0,2){10}{\line(0,1){1}}
    \put(126,49){\circle*{3}}
 \multiput(175,30)(0,2){72}{\line(0,1){1}}
    \put(175,175){\circle*{3}}

 \bezier{3000}(84,250)(110,-70)(160,120)
 \bezier{3000}(160,120)(180,200)(200,240)
   \color{white}\put(200,240){\circle*{3}}
   \color{black}\put(200,240){\circle{3}}

  \put(80,26){\makebox(0,0)[ct]{{\large $t_{k-1}$}}}
  \put(102,24){\makebox(0,0)[ct]{{\large $\alpha_{1,k}$}}}
  \put(126,24){\makebox(0,0)[ct]{{\large $\alpha_0$}}}
  \put(175,24){\makebox(0,0)[ct]{{\large $\alpha_{2,k}$}}}
  \put(202,26){\makebox(0,0)[ct]{{\large $t_k$}}}

  \put(268,34){\makebox(0,0)[lb]{{\large $\alpha$}}}

  \put(25,238){\makebox(0,0)[rc]{{\large $y$}}}

  \put(140,238){\makebox(0,0)[rc]{{\large $y=P_k(\alpha)$}}}
  \put(265,230){\makebox(0,0)[rc]{\large $y=\alpha$}}

\end{picture}

\end{center}
\caption{Properties of $\mathcal{P}_{k}$ in $(t_{k-1},t_k)$ due to Claims 1 and 2.}
\end{figure}

\end{proof}

\subsection{Conclusion of the proof of Theorem \ref{main1}}\label{subsec:final}

\medskip

  From Propositions \ref{pro1} and \ref{pp3} it follows
  that problem \eqref{P1} has two classical positive solutions $u_{k, 1}$
  and $u_{k, 2}$ such that
$$
  t_{k-1}<\int_{\Omega}u_{k, 1}^{p}dx<\int_{\Omega}u_{k, 2}^{p}dx<t_{k},
$$
  for any fixed $k\in \{1, \ldots, K\}$.
  This finishes the proof.\hspace*{\fill} $\square$

\section{Examples of nonlinearities $f$}\label{se:exam}\label{sec:Final}

  Let us provide some examples of function $f$ satisfying hypotheses of Theorem \ref{main1}

\medskip

\noindent
 \textbf{(a)}
  Let us take an integer $K\geq 1$,
  $U\geq (K/\int_{\Omega}e_1^pdx)^{1/p}$

  and let
  $S\colon [0,K]\to (U,+\infty)$
  be a $C^1$-function.
  We put
\[
  \ol{S}:=\max\limits_{t\in [0,K]}S(t),\quad
  \ul{S}:=\min\limits_{t\in [0,K]}S(t)>U,
\]
  and
\[
  M:=\lambda_{1}^{1/2}/(2C_{1}|\Omega|^{1/2})

 .
\]
  Let $L\colon \R\to [0,\infty)$ be a continuous function such that
  $L(w)=0$ for $w\leq 0$, $L$ is increasing and $C^1$ on $(0,\infty)$, $\lim\limits_{w\to\infty}L(w)=+\infty$,
  $L(\ul{S})>\lambda_1$, and $\max\limits_{s\in [0,\ol{S}]}s\cdot L(\ol{S}-s)<M/\ol{S}^{p-1}$.

  Now, defining
\begin{equation}\label{ex_f}
  f(s,t):=s\cdot L\bigg(\frac{S(t)-s}{|\sin\pi t|}\bigg),
\end{equation}
  we will show that $f$ satisfies hypothesis \eqref{f_{0}}-\eqref{f_{5}}, with
  $t_i=i$ for $i\in\{0,1,\ldots,K\}$ and $s_{\alpha}=S(\alpha)$.

  Indeed, $f\colon\R\times\mathcal{A}\to\R$, where
  $\mathcal{A}=(0,\infty)\setminus \{1,\ldots,K\}$,
  $f(\cdot,\alpha)\in C^1(\R)$ for all $\alpha\in \mathcal{A}$,
  $f(s_{\alpha},\alpha)=0$,
  $f(s,\alpha)>0$ for $s\in (0,s_{\alpha})$. Thus hypothesis \eqref{f_{0}} holds.
  By the assumptions on $L$, it is also obvious that \eqref{f_{1}} is valid.

  Next, we have
\[
  \frac{f(s,\alpha)}{s}=L\bigg(\frac{s_{\alpha}-s}{|\sin\pi\alpha|}\bigg)
\]
  and so we see that the map $(0, s_{\alpha})\ni s\mapsto f(s, \alpha)/s$ is decreasing (as $L$ is increasing), hence \eqref{f_{2}} holds.

  For any $k\in \{1,\ldots, K\}$ and $\alpha\in (t_{k-1},t_k)$, we have
\[
  \gamma_{\alpha}
  =\lim\limits_{s\to 0^{+}}\frac{f(s, \alpha)}{s}
  =\lim\limits_{s\to 0^{+}}L\bigg(\frac{s_{\alpha}-s}{|\sin\pi\alpha|}\bigg)
  =L\bigg(\frac{s_{\alpha}}{|\sin\pi\alpha|}\bigg)
  \geq L(s_{\alpha})
  \geq L(\ul{S})
  >\lambda_1,
\]
  thus hypothesis \eqref{f_{3}} holds.

  Hypothesis \eqref{f_{4}}
  is satisfied, by the assumptions on the map $S$ (see the definition of $U$).

  Finally, for a fixed $k\in\{1, \ldots, K\}$, we have
\begin{eqnarray*}
  \inf\limits_{\alpha\in (t_{k-1}, t_{k})}\max\limits_{s\in[0, s_{\alpha}]} \frac{f(s,\alpha)s_{\alpha}^{p-1}}{\alpha}
  &=&
  \inf\limits_{\alpha\in (k-1, k)}\max\limits_{s\in[0, s_{\alpha}]}\frac{ss_{\alpha}^{p-1}}{\alpha}L\bigg(\frac{S(\alpha)-s}{|\sin\pi\alpha|}\bigg)\\
  & \leq &
    \max\limits_{s\in[0, \ol{S}]}\frac{s\ol{S}^{p-1}}{k-\frac{1}{2}}L\bigg(\frac{S(k-\frac{1}{2})-s}{|\sin\pi/2|}\bigg) \\
  &\leq&
    2\ol{S}^{p-1}\max\limits_{s\in[0, \ol{S}]}s L(\ol{S}-s)\\
  & < &
    2\ol{S}^{p-1}M/\ol{S}^{p-1}
  =
   2M
  =
   \frac{\lambda_{1}^{1/2}}{C_{1}|\Omega|^{1/2}}
\end{eqnarray*}

  thus hypothesis \eqref{f_{5}}

  holds.

\medskip

\noindent
 \textbf{(b)}
  Let us now provide some particular examples of functions $S$ and $L$ satisfying the above assumptions.

  Take $K$, $U$, and $M$ as defined in \textbf{(a)} and put
 \[
  A:=U+1,\quad
  n>\frac{(\lambda_1+1)(A+1)^p}{M}-1,\quad\textrm{and}\quad
  \beta:=\frac{\lambda_1+1}{A^n}.
 \]
   Next choose $B>A$ such that
 \[
   B<\min\left\{A\sqrt[n]{\frac{(n+1)M}{(\lambda_1+1)(A+1)^p}},\ A+1\right\}.
 \]
  Now we define function $S\colon [0,K]\to (0,+\infty)$, by
\[
  S(t):=\frac{B-A}{K}t+A\quad\forall t\in [0,K],
\]
  and function $L\colon \R\to [0,\infty)$, by
\[
  L(w):=\left\{
  \begin{array}{lll}
    \beta w^n & \textrm{for} & w\geq 0,\\
    0         & \textrm{for} & w<0.
  \end{array}
  \right.
\]
  It is easy to check, that $\ul{S}=A>U$ and $\ol{S}=B$.
  Moreover  the function $L$ is continuous,
  increasing and $C^1$ on $(0,\infty)$, $\lim\limits_{w\to\infty}L(w)=+\infty$,
\[
  L(\ul{S})
  =
  L(A)
  =
  \beta A^n
  =
  \frac{\lambda_1+1}{A^n}A^n
  =
  \lambda_1+1
  >
  \lambda_1,
\]
  and finally the maximum of the function $[0,\ol{S}]\ni s\longmapsto s\cdot L(\ol{S}-s)=\beta s(B-s)^n$
  is attained at $s_{max}=\frac{B}{1+n}$ and so
\begin{eqnarray*}
  \max\limits_{s\in [0,\ol{S}]}s\cdot L(\ol{S}-s)
 & =&
  \beta\frac{B}{1+n}\bigg(\frac{nB}{1+n}\bigg)^n \\
&  =&\frac{\lambda_1+1}{1+n}\bigg(\frac{B}{A}\bigg)^n B \bigg(\frac{n}{1+n}\bigg)^n\\
  & < &
    \frac{\lambda_1+1}{1+n}\frac{(n+1)M}{(\lambda_1+1)(A+1)^p}(A+1)\\
&  =& M/(A+1)^{p-1}
  <
  M/B^{p-1}=M/\ol{S}^{p-1},
\end{eqnarray*}
  by the definition of $B$. Thus both functions $S$ and $L$ fit into the framework of example \textbf{(a)},
  and so the function $f$ defined by \eqref{ex_f} satisfies hypotheses \eqref{f_{0}}-\eqref{f_{5}}.

\medskip

\noindent
 \textbf{(c)}
  For another choice of functions $S$ and $L$ satisfying the assumptions of \textbf{(a)}, take again $K$, $U$, and $M$ as in \textbf{(a)} and put
  $A:=U+1$. Next fix two numbers $B$ and $D$ such that
\[
   A<B<D<\min\left\{A+\frac{MA}{(\lambda_1+1)(A+1)^p},\ A+1\right\}
\]
  and define
\[
  C:=\frac{(\lambda_1+1)(D-A)}{A}.
\]
  Let $S\colon [0,K]\to [A,B]$ be any $C^1$-function.
  Then $\ul{S}\geq A$ and $\ol{S}\leq B$.

  On the interval $[0,B]$ we defined an increasing $C^1$-function
\[
  L(w)=-C\frac{w}{w-D}
\]
  and extend it outside $[0,B]$ in such a way that it is still $C^1$ and increasing.
  First, we have
\[
  L(\ul{S})\ge L(A)=-\frac{(\lambda_1+1)(D-A)}{A}\cdot\frac{A}{A-D}=\lambda_1+1>\lambda_1.
\]
  Next
\[
  \max_{s\in [0,\ol{S}]}s\cdot L(\ol{S}-s)
  \leq
    \max_{s\in [0,B]}s\cdot L(B-s)
  =
    \max_{s\in [0,B]} -Cs\frac{B-s}{B-s-D}
  =
    \max_{s\in [0,B]} \frac{Cs(B-s)}{s+(D-B)}.
\]
  The last maximum is attained at
  $s_{max}=\sqrt{D(D-B)}-(D-B)\in (0,B)$,  with the value
\begin{eqnarray*}
  \max_{s\in [0,B]} \frac{Cs(B-s)}{s+(D-B)}
    &=&\frac{C(\sqrt{D(D-B)}-(D-B))(D-\sqrt{D(D-B)})}{\sqrt{D(D-B)}}\\
  & = &
    \frac{(\lambda_1+1)(D-A)}{A}\bigg(1-\sqrt{\frac{D-B}{D}}\bigg)(D-\sqrt{D(D-B)})\\
  &\le&
    \frac{(\lambda_1+1)}{A}(D-A)D\\
  & < &
    \frac{(\lambda_1+1)}{A}\frac{MA}{(\lambda_1+1)(A+1)^p}(A+1)\\
  &=&\frac{M}{(A+1)^{p-1}}
  \leq\frac{M}{B^{p-1}}
  \leq\frac{M}{\ol{S}^{p-1}}.
\end{eqnarray*}
  Thus both functions $S$ and $L$ fit into the framework of example \textbf{(a)}.

config: arxiv


TITLE: Questions about linear independent and spanning set and basis
QUESTION [0 upvotes]: True or false? Given reason

A set of 5 Vectors in R5 must be a basis for R5  (R5 means 5th dimensional space)
A set of 6 Vectors in R5 cannot be a basis for R5
A set of 7 vectors in R5 must be a spanning set for R5
A set of 6 polynomials in R5 must be a basis for R5
A set of 6 polynomials in R5 may be a basis for R5

Anyone help me to explain these question.. I want to know the reason cz it's confused me a lot and i always get mess about these kind of question
appericated it!

REPLY [1 votes]: 1) False: Use the zero vector and any other 4 vectors.
2) True: For a set of vectors to be a basis, all vectors must be linearly independent. It's not possible to have 6 linearly independent vectors in R5 (max is 5 linearly independent vectors).
3) False: Pick any vector $v$, then $2v, 3v, 4v, 5v, 6v, 7v$. These do not span R5.
4) Not sure what this means, R5 is a vector space, not a function space.
5) Same as 4.


TITLE: Can a series be empty?
QUESTION [8 upvotes]: It sounds contra-intuitive, since it won't exist without elements. However, I am thinking in empty sets, null sequences and empty lists, which indeed exists.

REPLY [3 votes]: A basic rule about sums is that $\sum\limits_{\alpha\in A}x_\alpha+\sum\limits_{\alpha\in B}x_\alpha=\sum\limits_{\alpha\in A\cup B}x_\alpha$ whenever these sums exist and $A\cap B=\varnothing$. Using this for $B=\varnothing$ and any $A$ imposes that $\sum\limits_{\alpha\in \varnothing}x_\alpha=0$.
(Note that, by the same logic, $\prod\limits_{\alpha\in \varnothing}x_\alpha=1$.)


TITLE: How does the constant proportionality balance an equation?
QUESTION [0 upvotes]: The Newtonian universal law of gravitation,

Every object in the universe attracts every other object with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them

Therefore, $F=GMm/r^2$ where $G$ is the universal gravitational constant that has been identified to have a value of $6.67 \ 10^{-11}\mathrm{\ N\ kg^{-2}\ m^2}$ by Henry Cavendish.
To convert a proportion to an equation, there is an inclusion of a constant

Reason: To balance the segment at the left to the segment at the right thereby, comprehension of a constant in an alleged equation.

In a general perspective of equations, How is a proportionality constant able to balance an equation?
To the public, I request to answer as per the following divisions of uncertainty over this concept.

How do we define a certain value to the proportionality constant?

How does it balance the equation?

What difficulties did the discoverer face when he tried to define or understand the nature of the proportionality constant?

REPLY [1 votes]: 1)How do we define a certain value to the proportionality constant?

This is done experimentally. We take many different objects at different sizes and distances and we measure the gravitational force between them. Then, for all of these objects, we plot $F$ on the vertical axis and $Mm/r^2$ on the horizontal axis. Then the slope of the resulting line is $G$.

How does it balance the equation?

The phrase “$a$ is proportional to $b$” means that there is some constant $k$ such that $a=kb$ for all $a$ and $b$. Furthermore, if $a$ has units of $[a]$ and $b$ has units of $[b]$ then in order for the units to balance we see that $[k]=[a]/[b]$ where $k$ has units of $[k]$.

What difficulties does a person face when he tries to define or understand the nature of the proportionality constant?

That is a case by case question. Not all students will have the same difficulties. For the discoverer, the only difficulty is experimental: how to measure the various quantities with sufficient accuracy and precision. Once that is done then the analysis is straightforward


text:
\begin{document}
\maketitle
\bigskip
\begin{abstract}

We consider the regularity question of solutions for the dynamic initial-boundary value problem for the linear relaxed micromorphic model. This generalized continuum model couples a wave-type equation for the displacement with a generalized Maxwell-type wave equation for the micro-distortion. Naturally solutions  are found in ${\rm H}^1$ for the displacement $u$ and ${\rm H}(\Curl)$ for the microdistortion $P$.  Using energy estimates for difference quotients, we improve this regularity. We show
 ${\rm H}^1_{\rm loc}$--regularity for the displacement field, ${\rm H}^1_{\rm loc}$--regularity for the micro-distortion tensor $P$ and  that   ${\rm Curl}\,P$  is ${\rm H}^1$--regular if the data is sufficiently smooth.

\bigskip

\noindent\textit{Mathematics Subject Classification}:
35M33, 	35Q74, 	74H20, 	74M25, 	74B99
\smallskip

\noindent\textit{Keywords}:
tangential trace, extension operator, generalized continua, inhomogeneous boundary conditions
\end{abstract}

\section{Introduction}

Generalized continuum theories like the micromorphic or Cosserat model have a long history \cite{Mindlin64,Eringen99}. These models are endowed with additional degrees of freedom (as compared to standard linear elasticity) which are meant to capture effects from a microscale on a continuum level. In the micromorphic family, each macroscopic material point is displaced with the classical displacement $u:\Omega\times[0,T]\rightarrow\mathbb{R}^3$ and attached to the macroscopic point there is an affine micro-distortion field $P:\Omega\times[0,T]\rightarrow\mathbb{R}^{3\times3}$  describing the interaction with the microstructure.

The equations of motion are obtained from Hamilton's principle and the introduction of suitable kinetic and elastic energy expressios. Typical for the classical micromorphic model is a quadratic term  $\frac{1}{2}\lVert \nabla P\rVert^2$ leading to an equilibrium problem of the type
\begin{eqnarray}
\label{1}
u_{,tt}&=&\Dyw\big(\nabla u-P\big)+f=\Delta u-\Dyw P+f,\\[1ex]
P_{,tt}&=&(\nabla u-P)- \sym P+\Delta P+M.\notag
\end{eqnarray}
Unique  solutions are found in classical Hilbert spaces, $u\in {\rm C}([0,T];{\rm H}
^1(\Omega, \mathbb{R}^3))$ and $P\in {\rm C}([0,T];{\rm H}
^1(\Omega, \mathbb{R}^{3\times 3}))$ for all times $T>0$, under suitable initial and boundary conditions and for these models \cite{picard,IesanNappa2001}, higher regularity follows from the standard approach for the wave equation.

Departing from this framework, Neff and his co-workers have introduced the so called relaxed micromorphic model \cite{NeffGhibaMicroModel}. Here, the ``curvature energy'' $\frac{1}{2}\lVert \nabla P\rVert^2$ is replaced by $\frac{1}{2}\lVert \Curl P\rVert^2$, which represents a sort of ``relaxation'' since the interaction-strength of the microstructure with itself is getting much weaker. The typical set of equations turns into
\begin{align}
\label{2}
u_{,tt}&=\Dyw\big(\nabla u-P\big)+f,\\[1ex]
P_{,tt}&=(\nabla u-P)- \sym P+\Curl \Curl P+M,\notag
\end{align}
under suitable initial and boundary conditions, which for $P$ has to be a tangential boundary condition. The second equation can be seen as a generalized Maxwell-problem due to the $\Curl \Curl$-operator. Unique solutions are now found in the natural setting $u\in {\rm C}([0,T];{\rm H})
^1(\Omega, \mathbb{R}^3)$ and $P\in {\rm C}([0,T];{\rm H}
(\Curl; \Omega))$ for all times $T>0$, see \cite{GhibaNeffExistence,Sebastian1}.

The current interest for the relaxed micromorphic model mainly stems from the fact that it is able to describe frequency-band gaps as observed in real meta-materials  (see e.g. \cite{blanco2000large,liu2000locally} and \cite{MadeoNeffGhibaW,madeo2016reflection,d2017panorama,aivaliotis2019microstructure,barbagallo2019relaxed,aivaliotisfrequency,d2019effective}).
When it comes to the numerical implementation, it would be natural to use the pair of function spaces  ${\rm H}
^1(\Omega, \mathbb{R}^3)\times {\rm H}
(\Curl; \Omega)$. If, on the other hand, a standard ${\rm H}
^1(\Omega, \mathbb{R}^3)\times {\rm H}
^1(\Omega, \mathbb{R}^{3\times 3})$ framework is used (which is more easily available from relevant software packages) the occurring approximation error hinges on the local regularity of solutions.
Indeed, in this paper we show (Theorem \ref{mr}) that $u\in {\rm H}^2_{\rm loc}(\Omega)$ and $P\in {\rm H}^1_{\rm loc}(\Omega)$ can be achieved under suitable regularity assumptions on the data. This establishes convergence of standard numerical FEM-implementations in the interior and constitutes a major motivation for our work. While in the static case, this kind of improved regularity (from $H(\Curl;\Omega)$ to ${\rm H}^1_{\rm loc}(\Omega)$) for the microdistortion field $P$ is illusory, the dynamic formulation provides much more control of the appearing fields through the kinematic terms.

In a previous paper \cite{Sebastian1}, starting from some results established by Alonsi and Valli \cite{tantrace}, we have seen that for all $G\in \widetilde{\chi}_{\partial\Omega}:= \{G\in {\rm H}^{\frac{1}{2}}(\partial\Omega)\,\,;\,\, \bigl\langle G_i\big|_{\partial\Omega}, n\bigr\rangle=0\}$ there exists an  extension $\widetilde{G}\in {\rm H}(\Curl;\Omega)$ such that $\Curl \Curl \widetilde{G}=0$ and which belongs actually  to ${\rm H}^1(\Omega)$. This result is useful in order to prove that the initial-boundary value problem with non-homogeneous boundary condition admits a unique solution $(u, P)$ with the regularity:
$
u\in {\rm C}^1([0,T];{\rm H}^1(\Omega ))\,, u_{,tt}\in {\rm C}([0,T];{\rm L}^2(\Omega ))\,,$ $
P\in {\rm C}^1([0,T]; {\rm H}(\Curl;\Omega )),\  P_{,tt}\in {\rm C}([0,T];{\rm L}^2(\Omega)), $ $  \Curl\Curl P\in {\rm C}([0,T];{\rm L}^2(\Omega  )),$ for all times $T>0$.
Moreover, we have shown that any extension
 $\widetilde{v}\in{\rm H}(\curl;\Omega)$ of $v\in \tilde{\bigchi}_{\partial\Omega}$ is such that $\nabla\curl \tilde {v}\in {\rm L}^2(\Omega)$, see \cite{Sebastian1}. In the present paper, starting from this remark and by using some standard techniques, we prove  that under suitable assumptions on the data, the solution $(u, P)$ is in fact smoother, i.e.
$
 u\in {\rm L}^{\infty}(0,T;{\rm H}^2_{\rm loc}(\Omega))\,,  P\in {\rm L}^{\infty}(0,T;{\rm H}^1_{\rm loc}(\Omega))   \mathrm{\,\,and} \
 \Curl P\in {\rm L}^{\infty}(0,T;{\rm H}^1(\Omega))\,,
 $
 for all times $T>0$. We point out that this result may seem surprisingly  unusual, since we have no information on ${\rm Div} \,P$ in $\Omega$ and not even on $P_i\cdot n$ ($i=1,2,3$) on $\partial \Omega$.

\section{The relaxed micromorphic model}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\setcounter{equation}{0}

We consider $\Omega$ to be a connected, bounded, open subset of $\mathbb{R}^3$ with a  ${\rm C}^{1,1}$  boundary $\partial\Omega$ and $T > 0$ is a fixed length of the time interval. The domain $\Omega$ is occupied by a micromorphic continuum whose motion  is referred to a fixed system of rectangular Cartesian axes $Ox_i$ $(i=1,2,3)$.
\subsection{Notations}
 Throughout this paper {(if we do not specify otherwise)} Latin subscripts take the values $1,2,3$.
We denote by $\mathbb{R}^{3\times 3}$ the set of real $3\times 3$ matrices.   For all $X\in\mathbb{R}^{3\times3}$ we set ${\rm sym}\, X=\frac{1}{2}(X^T+X)$ and ${\rm skew} X=\frac{1}{2}(X-X^T)$.
The standard Euclidean scalar product on $\mathbb{R}^{3\times 3}$ is given by
$\langle {X},{Y}\rangle_{\mathbb{R}^{3\times3}}=\tr({X Y^T})$, and thus the Frobenius tensor norm is
$ \lVert {X}\rVert^2=\langle{X},{X}\rangle_{\mathbb{R}^{3\times3}}$. In the following we omit the index
$\mathbb{R}^{3\times3}$. The identity tensor on $\mathbb{R}^{3\times3}$ will be denoted by $\id$, so that
$\tr({X})=\langle{X},{\id}\rangle$. Typical conventions for differential
operations are implied such as comma followed
by a subscript to denote the partial derivative with respect to
the corresponding Cartesian coordinate, while $t$ after a comma denotes the partial derivative with respect to the time. A matrix having the  three  column vectors $A_1,A_2, A_3$ will be written as
$
(A_1\,|\, A_2\,|\,A_3).
$

We denote by $u:\Omega\times [0,T]\rightarrow \R^3$  the displacement vector of the material point, while $P:\Omega\times [0,T]\rightarrow \R^{3\times 3}$ describes the substructure of the material which can rotate, stretch, shear and shrink (the micro-distortion). Here, $T > 0$ is a fixed length of the time interval.
For vector fields $u=\left(    u_1, u_2,u_3\right)$ with  $u_i\in {\rm H}^{1}(\Omega)$, $i=1,2,3$,
we define
$
\nabla \,u:=\left(
\nabla\,  u_1\,|\,
\nabla\, u_2\,|\,
\nabla\, u_3
\right)^T,
$
while for tensor fields $P$ with rows in ${\rm H}({\rm curl}\,; \Omega)$, i.e.
$
P=\begin{pmatrix}
(P^T.e_1)^T\,|\,
(P^T.e_2)^T\,|\,
(P^T.e_3)^T
\end{pmatrix}^T$, $P^T.e_i\in {\rm H}({\rm curl}\,; \Omega)$, $i=1,2,3$,
we define
$$
{\rm Curl}\,P:=\begin{pmatrix}
{\rm curl}\, (P^T.e_1)^T\,|\,
{\rm curl}\, (P^T.e_2)^T\,|\,
{\rm curl}\, (P^T.e_3)^T
\end{pmatrix}^T
.
$$
 The corresponding Sobolev spaces for the second order tensor fields $P$, ${\rm Curl}\, P$ and $\nabla\, u$  will be denoted by
$
{\rm H}^1(\Omega) \ \ \text{and}\  \ {\rm H}({\rm Curl}\,; \Omega)\, ,
$ and $
{\rm H}^1_0(\Omega) \ \ \text{and}\  \ {\rm H}_0({\rm Curl}\,; \Omega)\, ,
$ respectively.

\subsection{The initial-boundary value problem in the linear relaxed micromorphic theory}\label{sectmodel}

The partial differential equations associated to the dynamical relaxed  micromorphic model \cite{NeffGhibaMicroModel} are
\begin{eqnarray}
\label{1.2}
u_{,tt}&=&\Dyw\big(2\,\mu_{\rm e}\sym(\nabla u-P)+2\,\mu_{\rm c}\skyw(\nabla u-P)+\lambda_{\rm e}\tr(\nabla u-P)\id\big)+f\nn\\[1ex]
P_{,tt}&=&2\,\mu_{\rm e}\sym(\nabla u-P)+2\,\mu_{\rm c}\skyw(\nabla u-P)+\lambda_{\rm e}\tr(\nabla u-P)\id\nn\\[1ex]
&&-(2\,\mu_{\mathrm{micro}} \sym P+\lambda_{\mathrm{micro}}(\tr P)\id)-\mu_{\rm micro}\, L_{\rm c}^2\,\Curl\Curl P+M\,,
\end{eqnarray}
in $\Omega\times (0,T)$, where $f:\Omega\times (0,T)\rightarrow \R^3$ is a given body force and $M:\Omega\times (0,T)\rightarrow \R^{3\times 3}$ is a given body moment tensor.

Here, the constants  $\mu_{\rm e},\lambda_{\rm e},\mu_{\rm c}, \mu_{\rm micro}, \lambda_{\rm micro}$  are constitutive parameters describing the isotropic elastic response of the material, while $L_{\rm c}>0$ is the characteristic length of the relaxed micromorphic
model. We assume that the constitutive parameters are such that
\begin{align}\label{condpara}
\mu_{\rm e}>0,\quad\quad  2\,\mu_{\rm e}+3\,\lambda _{\rm e}>0,\quad\quad  \mu_{\rm c}\geq 0,\quad\quad  \mu_{\rm micro}>0, \quad\quad  2\,\mu_{\rm micro}+3\,\lambda _{\rm micro}>0.
\end{align}

The system \eqref{1.2} is considered with the boundary conditions
\begin{equation}
\begin{split}
u(x,t)=g(x,t),\quad \quad P_i(x,t)\times n(x)=G_i(x,t)
\end{split}
\label{1.3}
\end{equation}
for $(x,t)\in \partial\Omega\times [0,T]$, where $n$ is the unit normal vector at the surface $\partial\Omega$, $\times$ denotes the vector product and $P_i$ ($i=1,2,3$) are the rows of $P$. The model is also driven by the following initial conditions
\begin{equation}
\begin{split}
u(x,0)=u^{(0)}(x)\,,\quad u_{,t}(x,0)=u^{(1)}(x)\,,\quad P(x,0)=P^{(0)}(x)\,,\quad P_{,t}(x,0)=P^{(1)}(x)
\end{split}
\label{1.4}
\end{equation}
for $x\in\Omega$.

\begin{de}
We say that the initial data $(u^{(0)},u^{(1)},P^{(0)},P^{(1)})$ satisfy the compatibility condition if
\begin{equation}
u^{(0)}(x)=g(x,0)\,,\quad \ \,
u^{(1)}(x)=g_{,t}(x,0)\,, \quad
P^{(0)}_{i}(x)\times n(x)=G_i(x,0)\,,\quad
P^{(1)}_{i}(x)\times n(x)=G_{i,t}(x,0)
\label{comcon}
\end{equation}
for $x\in\partial\Omega$ and $i=1,2,3$, where $G_{i,t}$ denotes the time derivative of the function $G_i$.
\end{de}

\subsection{Preliminary results}

In a previous paper \cite{Sebastian1} we have considered the space
\begin{equation}
\tilde{\bigchi}_{\partial\Omega}:=\{v\in {\rm H}^{\frac{1}{2}}(\partial\Omega)\mid\bigl\langle v\big|_{\partial\Omega}, n\bigr\rangle=0\}
.
\end{equation}
This space is related to the fact that, according to \cite{tantrace} and  \cite[p. 34]{Giraultbook}, for all $v\in {\rm H}(\curl;\Omega)$  the tangential trace $n\times v\big|_{\partial \Omega}$  belongs to  a proper subspace of
${\rm H}^{-\frac{1}{2}}(\partial\Omega)$ defined  by
\begin{equation}
\bigchi_{\partial\Omega}:=\{v\in {\rm H}^{-\frac{1}{2}}(\partial\Omega)\mid\bigl\langle v\big|_{\partial\Omega}, n\bigr\rangle=0\,\,\mathrm{and}\,\,\dyw_{\tau} v\in {\rm H}^{-\frac{1}{2}}(\partial\Omega)\}
\end{equation}
and equipped with the norm
\begin{equation}
 \lVert v \rVert_ {\bigchi_{\partial\Omega}}= \lVert v \rVert_ {{\rm H}^{-\frac{1}{2}}(\partial\Omega)}+ \lVert\dyw_{\tau} v \rVert_ {{\rm H}^{-\frac{1}{2}}(\partial\Omega)}\,,
\end{equation}
where $\dyw_{\tau} v$ is the tangential divergence of the vector $v$. Let us recall that the tangential divergence $\dyw_{\tau} v$ of the vector $v$ is the distribution in ${\rm H}^{-\frac{3}{2}}(\partial\Omega)$ which satisfies

	\begin{equation*}
	[[\dyw_{\tau} v,w]]_{\partial\Omega}=-\big[ v,(\nabla w^{\ast})\big|_{\partial\Omega}\big]_{\partial\Omega}\quad \forall\quad w\in {\rm H}^{\frac{3}{2}}(\partial\Omega)\,,
	\label{2.3}
	\end{equation*}
	where $w^{\ast}\in {\rm H}^{2}(\Omega)$ is any extension of $w$ in $\Omega$. Here, $\big[\cdot,\cdot\big]$ denotes the duality pair between the space ${\rm H}^{-\frac{1}{2}}(\partial\Omega)$ and ${\rm H}^{\frac{1}{2}}(\partial\Omega)$.  $[[\cdot,\cdot]]$ denotes the duality pair between the space ${\rm H}^{-\frac{3}{2}}(\partial\Omega)$ and ${\rm H}^{\frac{3}{2}}(\partial\Omega)$ (more information can be found in \cite{tandyw}).

We observe that $\tilde{\bigchi}_{\partial\Omega}\subset \bigchi_{\partial\Omega}$. In \cite{Sebastian1}, summarising some results presented in  \cite{tantrace} and \cite[Theorem 6 of Section 2]{electrobook}, we have concluded that the following results hold true.
\begin{tw}
	\label{lem:2.2}
	Assume that the boundary $\partial\Omega$ is of class ${\rm C}^{1,1}$  or that $\Omega$ is a convex polyhedron. Moreover, let us assume that $v\in {\bigchi}_{\partial\Omega}$. Then there exists an extension $\widetilde {v}\in {\rm H}(\curl;\Omega)$ of $v$ in $\Omega$ such that
		\begin{enumerate}
			\item[1.] $\curl\curl\widetilde {v}=0\,;$ \qquad \qquad
			2. $\dyw\,\widetilde {v}=0$\,;
			\qquad \qquad  3.  $\widetilde {v}\in {\rm H}^1(\Omega)$ \ \ for all\ \  $v\in \widetilde{\chi}_{\partial\Omega}$.
		\end{enumerate}

\end{tw}

\begin{col} 	\label{collem}
The construction of the extension operator defined in the proof of Theorem \ref{lem:2.2} yields that $\nabla\curl \widetilde {v}\in {\rm L}^2(\Omega)$ (we refer to \cite{Sebastian1} for more details).
\end{col}

Using these results, we have proven the existence and uniqueness of the solution of the initial-boundary value problem arising in the linear relaxed theory for non-homogeneous boundary conditions, see \cite{Sebastian1}.
\begin{tw} {\rm (Existence of solution with non-homogeneous boundary conditions)}\label{existenceresult}
	Let us assume that the constitutive parameters satisfy  \eqref{condpara} and the initial data are such that
\begin{equation}
(u^{(0)}, u^{(1)}, P^{(0)}, P^{(1)})\in {\rm H}^1(\Omega; \R^3 )\times {\rm H}^1(\Omega; \R^3 )\times {\rm H}(\Curl;\Omega )\times {\rm H}(\Curl;\Omega )\,
\label{2.27}
\end{equation}
and that the compatibility condition \eqref{comcon} holds.
Additionally,
\begin{equation}
\begin{split}
\Dyw\big(2\,\mu_{\rm e}\sym(\nabla u^{(0)}-P^{(0)})+2\,\mu_{\rm c}\skyw(\nabla u^{(0)}-P^{(0)})+\lambda_{\rm e}\tr(\nabla u^{(0)}-P^{(0)})\id\big)&\in {\rm L}^2(\Omega)\,,
\\
\Curl\Curl P^{(0)}&\in {\rm L}^2(\Omega)
\end{split}
\end{equation}
and
$
f\in {\rm C}^1([0,T];{\rm L}^2(\Omega))\,,\  M\in {\rm C}^1([0,T];{\rm L}^2(\Omega))$, $
g\in {\rm C}^3([0,T];{\rm H}^{\frac{3}{2}}(\partial\Omega))\,,\  G_i\in {\rm C}^3([0,T];\tilde{\chi}_{\partial\Omega})\,,\,\, i=1,2,3\,.
$
Then, the system \eqref{1.2} with boundary conditions \eqref{1.3} and initial conditions \eqref{1.4} possesses a global in time, unique solution $(u, P)$ with the regularity: for all times $T>0$
\begin{equation}
\begin{split}
u\in {\rm C}^1([0,T];{\rm H}^1(\Omega ))\,,&\  u_{,tt}\in {\rm C}([0,T];{\rm L}^2(\Omega ))\,,\
P\in {\rm C}^1([0,T]; {\rm H}(\Curl;\Omega )),\  P_{,tt}\in {\rm C}([0,T];{\rm L}^2(\Omega))
\end{split}
\label{2.32}
\end{equation}
Moreover,
\begin{equation}
\Dyw\big(2\,\mu_{\rm e}\sym(\nabla u-P)+2\,\mu_{\rm c}\skyw(\nabla u-P)+\lambda_{\rm e}\tr(\nabla u-P)\id\big)\in {\rm C}([0,T];{\rm L}^2(\Omega  ))\\[1ex]
\label{2.33}
\end{equation}
and
\begin{equation}
\Curl\Curl P\in {\rm C}([0,T];{\rm L}^2(\Omega  ))\,.
\label{2.34}
\end{equation}
\label{tw:nonhomo}
\end{tw}
\begin{col} 	\label{collem1}
Theorem \ref{tw:nonhomo} assumes that $G_i\in {\rm C}^3([0,T];\tilde{\chi}_{\partial\Omega})$ for $i=1,2,3$, therefore the Theorem \ref{lem:2.2} and Corollary \ref{collem} yield that there exists an extension $\tilde{G}_i\in {\rm C}^3([0,T]; {\rm H}(\curl;\Omega ))$ such that $n\times \tilde{G}_i=G_i$ on $\partial\Omega$ in the sense of ${\rm H}^{-\frac{1}{2}}(\partial\Omega)$, $\curl\curl  \tilde{G}_i=0$ and $\nabla \tilde{G}_i\in {\rm C}^3([0,T]; {\rm H}^1(\Omega ))$.
\end{col}
The assumption on the constitutive parameters was used in the proof since we need to know that there exists a constant $C>0$ such that
\begin{align}
{\rm C}( \lVert\nabla u\rVert^2_{{\rm L}^2(\Omega)}+ \lVert P\rVert^2_{{\rm H}(\Curl;\Omega)})\leq \int_{\Omega}\Big(&\mu_{\rm e} \lVert\sym(\nabla u-P)\rVert^2+\mu_{\rm c} \lVert {\rm skew}(\nabla u-P)\rVert^2+\frac{\lambda_{\rm e}}{2}[\tr(\nabla u-P)]^2\\
&+ \mu_{\mathrm{micro}}  \lVert\sym P\rVert^2+\frac{\lambda_{\mathrm{micro}}}{2}\,[\tr (P)]^2+\frac{\mu_{\rm micro} \,L_{\rm c}^2}{2} \,\lVert\Curl P\rVert^2\Big)\,\di x\,\notag
\end{align}
for all $u\in {\rm H}^1_0(\Omega)$ and $P\in{\rm H}_0({\rm Curl}\, ; \Omega)$.
This coercivity follows even when $\mu_{\rm c}=0$, due to the fact that
\cite{NeffPaulyWitsch,BNPS2,LMN}
there exists a positive constant $C$, only depending on $\Omega$, such that for all $P\in{\rm H}_0({\rm Curl}\, ; \Omega)$ the following estimate holds
	\begin{align}
	 	{ \lVert  P \rVert_ {{\rm H}(\mathrm{Curl})}^2}:= \lVert  P \rVert_ {{\rm L}^2(\Omega)}^{ {2}}+ \lVert  \Curl P \rVert_ {{\rm L}^2(\Omega)}^{ {2}}&\leq C\,( \lVert  {\rm sym} P\rVert^2_{{\rm L}^2(\Omega)}+ \lVert  \Curl P\rVert^2_{{\rm L}^2(\Omega)}).
	\end{align}

Let us adjoin the total energy to a solution of the initial-value problem
\begin{align}\label{1.5}
\E(u, P)(t)=&\frac{1}{2}\int_{\Omega}( \lVert u_{,t}\rVert^2+ \lVert P_{,t}\rVert^2)\,\di x + \int_{\Omega}\Big(\mu_{\rm e} \lVert \sym(\nabla u-P)\rVert^2+\frac{\lambda_{\rm e}}{2}\,[\tr(\nabla u-P)]^2
+\mu_{\rm c}\, \lVert \skyw(\nabla u-P)\rVert^2
\notag\\[1ex]
&\qquad \quad \qquad \quad \qquad \quad\qquad  \  +\mu_{\rm micro} \lVert \sym P\rVert^2+\frac{\lambda_{\mathrm{micro}}}{2}\,[\tr (P)]^2
+\frac{\mu_{\rm micro}\, L_{\rm c}^2}{2}\, \lVert \Curl P\rVert^2\Big)\,\di x\,.
\end{align}

\section{Higher local regularity }
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\setcounter{equation}{0}
The aim of this section is to show the higher local regularity of the solution of problem \eqref{1.2}. We will use the difference quotient method. Let us consider bounded open subsets of $\Omega$ with smooth boundaries (so we can use Korn's inequality) such that $V\Subset U\Subset\Omega$. We consider  a cutoff function $\eta\,:\R^3\rightarrow [0,1]$ with the following properties
\begin{equation}
\eta\,:\Omega\rightarrow [0,1]\,,\qquad  \eta\,\in C_0^{\infty}(\R^3)\,,\qquad
\eta\,=1\quad \mathrm{on}\,\, V  \quad \mathrm{and}\quad \eta\,=0\quad \mathrm{on}\,\, \Omega\setminus U\,.
\label{4.1}
\end{equation}
We will denote by $D^h_k$ the difference quotient in the direction $\vec{e}_k$ with the step $h$ i.e. for any function $\phi$ defined on $\Omega$, any $h\in\R$ sufficiently small ($k=1,2,3$), set
\begin{equation}
\begin{split}
D^h_k\phi(x):=\frac{\phi(x+h\vec{e}_k)-\phi(x)}{h}\,.
\end{split}
\label{4.2}
\end{equation}
Observe that for $0<|h|<\frac{1}{2}\mathrm{dist}(U,\partial\Omega)$ and $x\in U$ the difference quotient is well-defined. Moreover, the products $\eta\, D^h_k(\cdot)$ are equal to zero for $x\notin U$. Let us recall  Theorem 3 from  Section 5.8.2 of \cite{evansbook} on the relation between the difference quotient and weak derivatives.
\begin{tw}
		\begin{description}
			\item[]
			\item[(i)] Assume that $1\leq p < \infty$ and $\phi\in W^{1,p}(\Omega)$. Then for all
			$k=1,2,3$ and all $V\Subset U\Subset\Omega$ it holds
			\begin{equation}
			\begin{split}
			 \lVert D^h_k\phi \rVert_ {L^{p}(V)}\leq C\, \lVert \nabla\phi \rVert_ {L^{p}(U)}
			\end{split}
			\label{difguo1}
			\end{equation}
			for some constant $C={\rm C}(p,U)$ and all $0<|h|<\frac{1}{2}\mathrm{dist}( V,\partial U)$.
			\item[(ii)] In turn, if $1<p<\infty$, $\phi\in L^p(\Omega)$ and there exists constant $C>0$ such that for all $k=1,2,3$ and all $0<|h|<\frac{1}{2}\mathrm{dist}( V,\partial U)$ the following inequality
			\begin{equation}
			\begin{split}
			 \lVert D^h_k\phi \rVert_ {L^{p}(V)}\leq C
			\end{split}
			\label{difguo2}
			\end{equation}
			is fulfilled. Then,
			\begin{equation}
			\phi\in W^{1,p}(V)\quad\mathrm{and}\quad  \lVert \nabla\phi \rVert_ {L^{p}(V)}\leq C\,.
			\label{difguo3}
			\end{equation}
		\end{description}
	\label{tw:difquo}
\end{tw}
\noindent
In the proof of the main estimate we will need the following information about tensor $P$
\begin{equation}
\begin{split}
 \lVert \nabla\curl P_i \rVert_ {{\rm C}([0,T];{\rm L}^2(\Omega))}\leq C\,,
\end{split}
\label{ineq}
\end{equation}
where $i=1,2,3$, $
P=\left(\begin{array}{c}
P_1
\,|\,
P_2
\,|\,
P_3
\end{array}\right)^T
$ is the unique solution of the problem \eqref{1.2}-\eqref{1.4} and the constant $C>0$ depends on $\Omega$ ($\Curl P$ is calculated with respect to rows of the matrix $P$). Observe that it is sufficient to prove the inequality \eqref{ineq} with homogeneous tangential boundary condition $P_i\times n =0$ on $\partial\Omega$ since the estimates in the case of inhomogeneous tangential boundary condition  follows as a consequence of Corollary \ref{collem1}.

\noindent
Assuming that $P_i\times n=0$ on $\partial\Omega$ we obtain from Theorem \ref{existenceresult} that $P_i\in {\rm C}([0,T];{\rm H}_0(\curl;\Omega))$, which means, that $P_i\in {\rm C}([0,T];{\rm H}(\curl;\Omega))$ and it has vanishing tangential component on the boundary $\partial\Omega$. Using ${\rm C}_0^{\infty}(\Omega)$- vector fields, one can show by a standard closure procedure that $\curl P_i\in {\rm C}([0,T];{\rm H}_0(\mathrm{div};\Omega))$ and $\mathrm{div}\curl P_i=0$ ($\curl P_i$ has vanishing normal component on the boundary $\partial\Omega$)-see for example \cite[Proposition 6.1.5]{Seifert}. Now the inequality \eqref{ineq} is a consequence of the inequality know in the literature as  Gaffney's inequality: for $v\in {\rm H}(\curl;\Omega)\cap {\rm H}_0(\dyw;\Omega)$ the following inequality
\begin{equation}
\begin{split}
 \lVert \nabla v \rVert_ {{\rm L}^2(\Omega)}\leq {\rm C}\,( \lVert \curl v \rVert_ {{\rm L}^2(\Omega)}+ \lVert \dyw v \rVert_ {{\rm L}^2(\Omega)}+ \lVert v \rVert_ {{\rm L}^2(\Omega)})\,,
\end{split}
\label{ineq2}
\end{equation}
is satisfied, where the constant $C>0$ does not depend on $v$. Some standard references on \eqref{ineq2} are Amrouche-Bernardi-Dauge-Girault \cite{Dauge}, Costabel \cite{Costabel}, Dautray-Lions \cite{Lions}, and Grisvard \cite[p.~318]{electrobook} (see also \cite{Costabel90} and \cite{Costabel99}). The regularity \eqref{2.34} entails that $\curl P_i\in {\rm H}(\curl;\Omega)\cap {\rm H}_0(\dyw;\Omega)$, hence applying inequality \eqref{ineq2} with $v=\curl P_i$ we obtain the inequality \eqref{ineq}. Now we are ready to prove the main estimate of this article.
\begin{tw} {\rm (Main estimate) }
	Suppose that  $(u,P)$ is the solution of the problem \eqref{1.2} with $P_i(x,t)\times n(x)=0$ on $\partial \Omega$ and under the hypotheses  of Theorem \ref{existenceresult}.
	Moreover, assume that the given forces have the regularity
	$
	f\in {\rm L}^2(0,T;{\rm H}^1_{\rm loc}(\Omega))$, $ M\in {\rm L}^2(0,T;{\rm H}^1_{\rm loc}(\Omega))
	$ for all $T>0$
	and the initial data admit the regularity
	$
	\nabla u^{(0)}-P^{(0)}\in {\rm H}^1_{\rm loc}(\Omega)\,,  \sym P^{(0)}\in {\rm H}^1_{\rm loc}(\Omega)\,,$ $
	\tr P^{(0)}\in {\rm H}^1_{\rm loc}(\Omega)\,, $ $ \Curl P^{(0)}\in {\rm H}^1_{\rm loc}(\Omega)\,, $ $
	u^{(1)}\in {\rm H}^1_{\rm loc}(\Omega;\R^{3})\,$ $  \mathrm{and}\quad P^{(1)}\in {\rm H}^1_{\rm loc}(\Omega)\,.
	$
	Then, for all $k\in\{1,2,3\}$, $t\in [0,T]$ and sufficiently small $h\in\R$ the following inequality
	\begin{equation}
	\E(\eta\, D^h_k u, \eta\, D^h_k P)(t)\leq C
	\label{estimate}
	\end{equation}
	holds, where $(u,P)$ is the solution of the system \eqref{1.2}, $\eta$ is the function defined by \eqref{4.1} and the constant $C>0$ does not depend on $h$ or on $\eta$.
	\label{tw:regu}
\end{tw}
\begin{proof} From Theorem \ref{existenceresult} we got that a solution $(u,P)$  of the problem \eqref{1.2}  with $P_i(x,t)\times n(x)=0$ on $\partial \Omega$ has the following regularity
	\begin{equation}
	\begin{split}
	u\in {\rm C}^1([0,T];{\rm H}^1(\Omega))\,,&\quad u_{,tt}\in {\rm C}([0,T];{\rm L}^2(\Omega ))\,,\\[1ex]
	P\in {\rm C}^1([0,T]; {\rm H}_0(\Curl;\Omega ))\,, &\quad P_{,tt}\in {\rm C}([0,T];{\rm L}^2(\Omega)),\\[1ex]
	\Curl\Curl P\in {\rm C}([0,T];{\rm L}^2(\Omega  ))\,\quad &\mathrm{and}\quad \nabla\Curl P\in {\rm C}([0,T];{\rm L}^2(\Omega))\,,
	\end{split}
	\label{assu}
	\end{equation}
where the last regularity statement in \eqref{assu} follows from inequality \eqref{ineq}. Fix $k\in\{1,2,3\}$ and assume that $h$ is sufficiently small. Calculating the time derivative of the energy \eqref{1.5} evaluated on localised differences provides
\begin{align}\label{4.3}
\frac{d}{dt}\big(\E(\eta\, D^h_k u, \eta\, D^h_k P)(t)\big)=&\int_{\Omega}\Big[\langle\eta\, D^h_k u_{,t},\eta\, D^h_k u_{,tt}\rangle + \langle\eta\, D^h_k P_{,t},\eta\, D^h_k P_{,tt}\rangle \Big]\,\di x\notag\\[1ex]
&+ \int_{\Omega}\Big[2\,\mu_{\rm e}\langle\eta\,\sym(\nabla D^h_k u-D^h_k P),\eta\,\sym(\nabla \dhk u_{,t}-D^h_k P_{,t})\rangle\notag\\[1ex]
&\qquad \quad +\lambda_{\rm e}\,\eta\,\tr(\nabla\dhk u-D^h_k P)\,\eta\,\tr(\nabla\dhk u_{,t}-D^h_k P_{,t})\\[1ex]
&\qquad \quad +2\,\mu_{\rm c}\langle\eta\,\skyw(\nabla\dhk u-D^h_k P),\eta\,\skyw(\nabla\dhk u_{,t}-D^h_k P_{,t})\rangle\notag\\[1ex]
&\qquad \quad +2\,\mu_{\mathrm{micro}} \,\eta^2\langle\sym \dhk P,\sym\dhk P_{,t}\rangle+\lambda_{\mathrm{micro}}\tr(\dhk P)\,\tr(\dhk P_{,t})\notag\\[1ex]
&\qquad \quad +\mu_{\rm micro} L_{\rm c}^2\langle\eta\,\curl \dhk P, \eta\,\curl \dhk P_{,t}\rangle\Big]\,\di x\notag
\end{align}
\begin{align}
\qquad \qquad\qquad\qquad\quad=&\int_{\Omega}\Big[\langle\eta\, D^h_k u_{,t},\eta\, D^h_k u_{,tt}\rangle+\langle 2\,\mu_{\rm e}\,\eta\,\sym(\nabla\dhk u-D^h_k P)+ \lambda_{\rm e}\,\eta\,\tr(\nabla\dhk u-D^h_k P)\id\notag\\[1ex]
&\qquad \qquad \qquad \qquad\qquad\qquad+ 2\,\mu_{\rm c}\,\eta\,\skyw(\nabla\dhk u-D^h_k P),\eta\,\nabla\dhk u_{,t}\rangle\Big]\,\di x\notag\\[1ex]
&+\int_{\Omega}\Big[\langle\eta\, D^h_k P_{,t},\eta\, D^h_k P_{,tt}\rangle-\langle 2\,\mu_{\rm e}\,\eta\,\sym(\nabla\dhk u-D^h_k P)+ \lambda_{\rm e}\,\eta\,\tr(\nabla\dhk u-D^h_k P)\id\notag\\[1ex]
&\quad \qquad  + 2\,\mu_{\rm c}\,\eta\,\skyw(\nabla\dhk u-D^h_k P)-2\,\mu_{\mathrm{micro}} \,\eta\,\sym \dhk P-\lambda_{\mathrm{micro}}\tr(\dhk P)\id,\eta\,\dhk P_{,t}\rangle\notag\\[1ex]
&\quad \quad \quad +\mu_{\rm micro} L_{\rm c}^2\langle \eta\,\Curl \dhk P, \eta\,\Curl \dhk P_{,t}\rangle \Big]\,\di x\,.\notag
\end{align}

It is worth to underline that the regularity \eqref{assu} of the solution $(u,P)$ implies that all integrals in \eqref{4.3} are well-defined. Let
\begin{equation}
\eta\,\dhk\sigma=2\,\mu_{\rm e}\,\eta\,\sym(\nabla\dhk u-D^h_k P)+ \lambda_{\rm e}\,\eta\,\tr(\nabla\dhk u-D^h_k P)\,\id+2\,\mu_{\rm c}\,\eta\,\skyw(\nabla\dhk u-D^h_k P)\,,
\label{4.4}
\end{equation}
where
\begin{align}
\sigma=&\,2\,\mu_{\rm e}\sym(\nabla u- P)+ \lambda_{\rm e}\tr(\nabla u- P)\,\id+2\,\mu_{\rm c}\skyw(\nabla u- P)
\end{align}
is the  (non-symmetric) Cauchy-stress  tensor.
Then,
\begin{align}\label{4.5}
\frac{d}{dt}\big(\E(\eta\, D^h_k u, \eta\, D^h_k P)(t)\big)=&\int_{\Omega}\Big[\langle \eta\, D^h_k u_{,t},\eta\, D^h_k u_{,tt}\rangle+\langle\eta\,\dhk\sigma,\eta\,\nabla\dhk u_{,t}\rangle\Big]\,\di x \\[1ex]
&+\int_{\Omega}\Big[\langle \eta\, D^h_k P_{,t},\eta\, D^h_k P_{,tt}\rangle-\langle\eta\,\dhk\sigma-2\,\mu_{\mathrm{micro}} \,\eta\,\sym \dhk P-\lambda_{\mathrm{micro}}\tr(\dhk P)\id,\eta\,\dhk P_{,t}\rangle\notag\\[1ex]
&\qquad \quad +\mu_{\rm micro} L_{\rm c}^2\langle\eta\,\Curl \dhk P, \eta\,\Curl \dhk P_{,t}\rangle \Big]\,\di x\,.\notag
\end{align}
Denote by $(\cdot)_i$ ($i=1,2,3$) the rows of a $3\times 3$ matrix and set $u=(u_1,u_2,u_3)$. Therefore, for $i=1,2,3$ we conclude that
\begin{equation}
\begin{split}
\dyw\big(\eta^2\dhk u_{i,t}\,\dhk\sigma_i\big)&= \langle\dhk\sigma_i,\eta^2\nabla\dhk u_{i,t}+2\,\eta\,\nabla\eta\,\dhk u_{i,t}\rangle+\eta^2\dhk u_{i,t}\dyw(\dhk\sigma_i)\,
\end{split}
\label{4.6}
\end{equation}
and
\begin{equation}
\begin{split}
\sum_{i=1}^{3} \langle\eta\, \dhk\sigma_i,\eta\,\nabla\dhk u_{i,t}\rangle=&\sum_{i=1}^{3} \dyw\big(\eta^2\dhk u_{i,t}\,\dhk\sigma_i\big)-\sum_{i=1}^{3} \langle\eta\,\dhk\sigma_i,2\nabla\eta\,\dhk u_{i,t}\rangle-\sum_{i=1}^{3} \eta^2\dhk u_{i,t}\dyw(\dhk\sigma_i)\,.
\end{split}
\label{4.7}
\end{equation}
Moreover, we have
\begin{align}\label{4.8}
\dyw\big(\eta^2\,\mu_{\rm micro} L_{\rm c}^2\curl\dhk P_{i}\times\dhk P_{i,t}&\big)\!\!= \langle\dhk P_{i,t},\curl\big(\eta^2\,\mu_{\rm micro} L_{\rm c}^2\curl\dhk P_{i}\big)\rangle\notag\\[1ex]
&\qquad \qquad -\langle\eta^2\,\mu_{\rm micro} L_{\rm c}^2\curl\dhk P_{i},\curl\dhk P_{i,t}\rangle\\[1ex]
&=
\langle\dhk P_{i,t},2\eta\,\nabla\eta\,\times\mu_{\rm micro} L_{\rm c}^2\curl\dhk P_{i}+\eta^2\curl\big(\mu_{\rm micro} L_{\rm c}^2\curl\dhk P_{i}\big)\rangle\notag\\[1ex]
&\qquad \qquad-\langle \eta^2\,\mu_{\rm micro} L_{\rm c}^2\curl\dhk P_{i},\curl\dhk P_{i,t}\rangle\,.\notag
\end{align}
This leads to
\begin{equation}
\begin{split}
\sum_{i=1}^{3}& \mu_{\rm micro} L_{\rm c}^2\langle\eta\,\curl\dhk P_{i},\eta\,\curl\dhk P_{i,t}\rangle=-\sum_{i=1}^{3} \dyw\big(\eta^2\,\mu_{\rm micro} L_{\rm c}^2\curl\dhk P_{i}\times \dhk P_{i,t}\big)\\[1ex]
&+\sum_{i=1}^{3}\langle\eta\, \dhk P_{i,t},\eta\,\curl\big(\mu_{\rm micro} L_{\rm c}^2\curl\dhk P_{i}\big)\rangle+\sum_{i=1}^{3} \langle\eta\, \dhk P_{i,t},2\,\nabla\eta\,\times\mu_{\rm micro} L_{\rm c}^2\curl\dhk P_{i}\rangle\,.
\end{split}
\label{4.9}
\end{equation}
Inserting \eqref{4.7} and \eqref{4.9} into \eqref{4.5} and using equations \eqref{1.2} we arrive at
\begin{align}\label{4.10}
\frac{d}{dt}\big(\E(\eta\, D^h_k u, \eta\, D^h_k P)(t)\big)=&\int_{\Omega}\langle\eta\, D^h_k u_{,t},\eta\, \dhk f)\rangle\,\di x -2\int_{\Omega}\sum_{i=1}^{3} \langle\eta\,\dhk\sigma_i,\nabla\eta\,\dhk u_{i,t}\rangle,\di x\\[1ex]
&+\int_{\Omega}\langle\eta\, D^h_k P_{,t},\eta\, D^h_k M\rangle\,\di x+2\int_{\Omega}\sum_{i=1}^{3} \langle\eta\, \dhk P_{i,t},\nabla\eta\,\times\mu_{\rm micro} L_{\rm c}^2\curl\dhk P_{i}\rangle\,\di x\,.\notag
\end{align}
Notice that the divergence theorem and the properties \eqref{4.1} of the function $\eta$ show that the integrals over $\Omega$ of the first terms on the right-hand side of \eqref{4.7} and \eqref{4.9} are equal to zero. Now integrating with respect to time we obtain
\begin{equation}
\begin{split}
\E(\eta\, D^h_k u,& \eta\, D^h_k P)(t)=\E(\eta\, D^h_k u, \eta\, D^h_k P)(0)+\int_0^t\int_{\Omega}\langle\eta\, D^h_k u_{,t},\eta\, \dhk f)\rangle\,\di x\di \tau\\[1ex]
&-2\int_0^t\int_{\Omega}\sum_{i=1}^{3} \langle\eta\,\dhk\sigma_i,\nabla\eta\,\dhk u_{i,t}\rangle\,\di x\di\tau+\int_0^t\int_{\Omega}\langle\eta\, D^h_k P_{,t},\eta\, D^h_k M\rangle\,\di x\di\tau\\[1ex]
&-2\int_0^t\int_{\Omega}\sum_{i=1}^{3} \langle\eta\, \dhk P_{i,t},\nabla\eta\,\times\mu_{\rm micro} L_{\rm c}^2\curl\dhk P_{i}\rangle\,\di x\di\tau\,.
\end{split}
\label{4.11}
\end{equation}
The first integral on the right-hand side of \eqref{4.11} is estimated as follows
\begin{equation}
\begin{split}
\int_0^t\int_{\Omega}(\eta\, D^h_k u_{,t},\eta\, \dhk f)\,\di x\di \tau\leq \int_0^t  \lVert \nabla u_{,t} \rVert_ {{\rm L}^2(\Omega)} \lVert \nabla f \rVert_ {{\rm L}^2(\mathrm{supp\,\eta})}\,\di\tau
\end{split}
\label{4.12}
\end{equation}
and the assumption on $f$ and the regularity of $u_{,t}$ yield that it is finite. Young's inequality implies that the second integral on the right-hand side of \eqref{4.11} can be estimated as follows
\begin{equation}
\begin{split}
\int_0^t\int_{\Omega}\sum_{i=1}^{3} \langle\eta\,\dhk\sigma_i,\nabla\eta\,\dhk u_{i,t}\rangle\,\di x\di\tau&\leq \int_0^t\int_{\Omega} \eta^2  \lVert \dhk\sigma\rVert^2\,\di x\di \tau +C \lVert \nabla\eta\,\rVert^2_{L^{\infty}(\Omega)}\int_0^t  \lVert \nabla u_{,t} \rVert_ {{\rm L}^2(\Omega)}^2\,\di\tau\\[1ex]
&\leq \int_0^t\E(\eta\, D^h_k u, \eta\, D^h_k P)(\tau)\,\di\tau+C \lVert \nabla\eta\,\rVert^2_{L^{\infty}(\Omega)}\int_0^t  \lVert \nabla u_{,t} \rVert_ {{\rm L}^2(\Omega)}^2\,\di\tau\,.
\end{split}
\label{4.13}
\end{equation}
Again, using the regularity of $u_{,t}$ we have that the second integral on the right-hand side of \eqref{4.13} is finite. In turn, the third integral of the right-hand side of \eqref{4.11} is evaluated as
\begin{equation}
\begin{split}
\int_0^t\int_{\Omega}\langle\eta\, D^h_k P_{,t},\eta\, D^h_k M)\rangle\,\di x\di\tau&\leq \int_0^t\int_{\Omega} \eta^2  \lVert \dhk P_{,t}\rVert^2\,\di x\di\tau +C\int_0^t\int_{\Omega} \lVert \nabla M\rVert^2\,\di x\di \tau\\[1ex]
&\leq \int_0^t\E(\eta\, D^h_k u, \eta\, D^h_k P)(\tau)\,\di\tau+C\,.
\end{split}
\label{4.14}
\end{equation}
Additionally, since
$
\curl\dhk P_{i}=\dhk(\curl P_i)\,,
$
we obtain
\begin{equation}
\begin{split}
\int_0^t\int_{\Omega}&\sum_{i=1}^{3} \langle\eta\, \dhk P_{i,t},\nabla\eta\,\times\mu_{\rm micro} L_{\rm c}^2\curl\dhk P_{i}\rangle\,\di x\di\tau\\[1ex]
&\leq \int_0^t\int_{\Omega} \eta^2  \lVert \dhk P_{,t}\rVert^2\,\di x\di \tau+ C \lVert \nabla\eta\,\rVert^2_{L^{\infty}(\Omega)}\int_0^t  \lVert \nabla\Curl P \rVert_ {{\rm L}^2(\Omega)}^2\,\di\tau\\[1ex]
&\leq \int_0^t\E(\eta\, D^h_k u, \eta\, D^h_k P)(\tau)\,\di\tau+C \lVert \nabla\eta\,\rVert^2_{L^{\infty}(\Omega)}\int_0^t  \lVert \nabla\Curl P \rVert_ {{\rm L}^2(\Omega)}^2\,\di\tau\,.
\end{split}
\label{4.16}
\end{equation}
Inequality \eqref{ineq} yields that the second term on the right-hand side of \eqref{4.16} is bounded. Substituting \eqref{4.12}-\eqref{4.16} into \eqref{4.11} we get
\begin{equation}
\begin{split}
\E&(\eta\, D^h_k u, \eta\, D^h_k P)(t)\leq\E(\eta\, D^h_k u, \eta\, D^h_k P)(0)+C\big(\int_0^t\E(\eta\, D^h_k u, \eta\, D^h_k P)(\tau)\,\di\tau+1\big)\,.\\[1ex]
\end{split}
\label{4.17}
\end{equation}
Thus, from Gronwall's inequality
\begin{equation}
\begin{split}
\E&(\eta\, D^h_k u, \eta\, D^h_k P)(t)\leq C\big(\E(\eta\, D^h_k u, \eta\, D^h_k P)(0)+1\big)\,.\\[1ex]
\end{split}
\label{4.18}
\end{equation}
for all $t\in (0,T)$. Applying the regularity of the initial data we obtain
$
\E(\eta\, D^h_k u, \eta\, D^h_k P)(t)\leq C\,,
$
where the constant $C>0$ depends on the length of time interval $(0,T)$ ($C$ does not depend on $h$).
\end{proof}
\subsection{The main result}
\begin{tw}	\label{regularity} \label{mr} {\rm (Regularity of the solution)} Suppose that all hypotheses of Theorem \ref{tw:regu} hold. Moreover, let $P^{(0)}\in {\rm H}^1_{\rm loc}(\Omega)$. Then,
	\begin{equation}
	u\in {\rm L}^{\infty}(0,T;{\rm H}^2_{\rm loc}(\Omega))\,,\quad P\in {\rm L}^{\infty}(0,T;{\rm H}^1_{\rm loc}(\Omega))\quad \mathrm{and}\quad
	\Curl P\in {\rm L}^{\infty}(0,T;{\rm H}^1(\Omega))\,.
	\label{regu1}
	\end{equation}
\end{tw}
\begin{proof} The proof is divided into two parts. First we are going to prove that $P\in {\rm L}^{\infty}(0,T;{\rm H}^1_{\rm loc}(\Omega))$ and $\Curl P\in {\rm L}^{\infty}(0,T;{\rm H}^1(\Omega))$.
\begin{itemize}
	\item Theorem \ref{tw:regu} implies that $ \lVert \dhk P_{,t}\rVert^2_{{\rm L}^{\infty}(0,T;{\rm L}^2(V))}\leq C$, thus from Theorem \ref{tw:difquo} we deduce
	\begin{equation}
	\begin{split}
	 \lVert \nabla P_{,t}\rVert^2_{{\rm L}^{\infty}(0,T;{\rm L}^2(V))}\leq C\,.
	\end{split}
	\label{4.20}
	\end{equation}
	Moreover, using the formula
	$
	\nabla P(t)=\nabla P^{(0)}+\int_0^t \nabla P_{,\tau}(\tau) \,\di\tau
	$
	we get
	\begin{equation}
	\begin{split}
	 \lVert \nabla P(t) \rVert_ {{\rm L}^2(V)}= \lVert \nabla P^{(0)} \rVert_ {{\rm L}^2(V)}+\int_0^T \lVert \nabla P_{,\tau}(\tau) \rVert_ {{\rm L}^2(V)} \,\di\tau\,.
	\end{split}
	\label{4.22}
	\end{equation}
	The regularity of $P^{(0)}$ and inequality \eqref{4.20} yield that
	$
	 \lVert \nabla P\rVert^2_{{\rm L}^{\infty}(0,T;{\rm L}^2(V))}\leq C
	$
	and $P\in {\rm L}^{\infty}(0,T;{\rm H}^1_{\rm loc}(V))$. Notice that inequality \eqref{ineq} gives us also that $\Curl P\in {\rm L}^{\infty}(0,T;{\rm H}^1(\Omega))$.
	\item Observe that
	\begin{equation}
	\begin{split}
	 \lVert \eta\,\sym(\nabla\dhk(u))\rVert^2_{{\rm L}^2(\Omega)}&\leq  2\Big(\lVert \eta\,\sym(\nabla\dhk(u)-\dhk P)\rVert^2_{{\rm L}^2(\Omega)}+ \lVert \eta\,\sym(\dhk P)\rVert^2_{{\rm L}^2(\Omega)}\Big)
	\end{split}
	\label{4.24}
	\end{equation}
	and Theorem \ref{tw:regu} implies that $ \lVert \eta\,\sym(\nabla\dhk(u))\rVert^2_{{\rm L}^2(\Omega)}$ is bounded independently on $h$. Now, the regularity of $u$ follows from the identities
	\begin{equation}
	\begin{split}
	\eta\,(\partial_k\nabla u)&=\nabla(\eta\,\partial_k u)-\nabla\eta\,\otimes\partial_k u\,,\\[1ex]
	\sym(\nabla(\eta\,\partial_k u))&=\eta\,(\partial_k\sym(\nabla u))+\sym(\nabla\eta\,\otimes\partial_k u)
	\end{split}
	\label{4.25}
	\end{equation}
	and from Korn's inequality \cite{Neff00b},  in the following form
	\begin{equation}
	\begin{split}
	 \lVert \eta\,(\partial_k\nabla u)\rVert^2_{{\rm L}^2(\Omega)}&\leq  2\Big(\lVert \nabla(\eta\,\partial_k u)\rVert^2_{{\rm L}^2(\Omega)}
	+ \lVert \nabla\eta\,\otimes\partial_k u\rVert^2_{{\rm L}^2(\Omega)}\Big)\\[1ex]
	&\leq C \Big(\lVert \sym(\nabla(\eta\,\partial_k u))\rVert^2_{{\rm L}^2(\Omega)} +  \lVert \nabla\eta\, \rVert_ {L^{\infty}(\Omega)} \lVert \nabla u\rVert^2_{{\rm L}^2(\Omega)}\Big)\\[1ex]
	&\leq C\Big( \lVert \eta\,(\partial_k\sym(\nabla u))\rVert^2_{{\rm L}^2(\Omega)}+ \lVert \nabla u\rVert^2_{{\rm L}^2(\Omega)}\Big)\,.
	\end{split}
	\label{4.26}
	\end{equation}
	This proves that $u\in {\rm L}^{\infty}(0,T;{\rm H}^2_{\rm loc}(\Omega))$. \qedhere
\end{itemize}
\end{proof}
\begin{remark} Note that the regularity for the displacement vector $u$ is obtained from the isotropic elastic energy and microstrain self energy. Comparing this with the general regularity theory for hyperbolic equations, it is a standard approach \cite{neff2008regularity}. It is clear that locally the dislocation energy $\lVert \Curl P\rVert^2$ does not control all weak derivatives of the tensor $P$ in ${\rm L}^2$. But, in the dynamic case the total energy contains also the kinematic energy. From the difference quotient method and the energy estimate \eqref{estimate} we are able to control all weak partial  derivatives of the micro-distortion tensor $P_{,t}$ locally in ${\rm L}^2$. Assuming that the initial tensor $P^{(0)}$ has a better regularity, we also control all weak partial derivatives of the tensor $P$ locally in ${\rm L}^2$.
\end{remark}
\bibliographystyle{plain}
\begin{footnotesize}
	\noindent{\bf Acknowledgements:}  	The  work of I.D. Ghiba  was supported by a grant of the Romanian Ministry of Research
	and Innovation, CNCS--UEFISCDI, Project no.
	PN-III-P1-1.1-TE-2019-0348, Contract No. TE 8/2020, within PNCDI III.

	\medskip

	\noindent \textbf{Conflicts of interest:}\ This work does not have
	any conflicts of interest.

\addcontentsline{toc}{section}{References}


TITLE: Asymptotics of Gaussian integral along vertical line in complex plane
QUESTION [1 upvotes]: In Marsden / Hoffman, an exercise (14 on p446) asks for an asymptotic analysis of the Gaussian integral
$$\int_{\gamma} \textrm{e}^{-\zeta^{2}}d\zeta$$
along the contour $\gamma$ defined by the vertical line from $x$ to $x + ix$ with $x \in \mathbb{R}$.
By parameterising the contour $\gamma : [0, x] \rightarrow \mathbb{C}$ by $\gamma(t) = x + it$ I came to
$$I(x) := \textrm{e}^{-x^{2}}\int_{0}^{x}\textrm{e}^{-2ixt + t^{2}}dt$$
and I would like to show this vanishes as $x \rightarrow \infty$ along the real axis.
I found a way tha t works by considering the absolute value of the integral and using the trick (the same trick works even without the absolute value)
$$ \textrm{e}^{t^{2}} = \frac{1}{2t}\frac{d}{dt} \textrm{e}^{t^{2}}$$
which allows an integration by parts that gives
$$|I(x)| \leqslant \frac{1}{2x} + \textrm{e}^{-x^{2}}\int_{1}^{x} dt \frac{\textrm{e}^{t^{2}}}{2t^{2}} + \mathcal{O}(\textrm{e}^{-x^{2}})$$
where the trailing factor comes from the region of integration $t \in [0, 1]$.
To write $|I(x)| \sim \frac{1}{2x}$ but for this to count as an asymptotic relation the remaining integral must be $\textit{o}(\frac{1}{x^{2}})$. This seems as difficult to show from the integral as it was to show the leading behaviour of $|I(x)|$ in the first place. I found two methods:

Integrate by parts a second time to get $\textrm{e}^{-x^{2}}\int_{1}^{x} dt \frac{\textrm{e}^{t^{2}}}{2t^{2}} = \frac{1}{4x^{3}} + \frac{3}{4}\textrm{e}^{-x^{2}} \int_{0}^{x}\frac{\textrm{e}^{t^{2}}} {t^{4}}dt$ and I can show the final integral is less than $\frac{3}{4x^{3}}$.
Use L'Hopital on $\lim_{x\rightarrow \infty} \frac{\int_{0}^{x}\frac{\textrm{e}^{t^{2}}} {t^{2}}dt}{\textrm{e}^{x^{2}} / x}$ to get limit zero.

Question: Is there a better or easier way to bound this integral from its definition?
I would prefer, however, to apply some other methods from the relevant chapter such as steepest descent or stationary phase. Yet there are various problems I'd like some help understanding:

Steepest descent using change of variables $t = yx$ for
$$ x\textrm{e}^{-x^{2}}\int_{0}^{1} \textrm{e}^{x^{2}(t^{2} -2it)}$$
whose exponent has a stationary point at $t = i$.

Question: Am I free to deform the contour in anyway required to pass through the stationary point in such a way that $\mathcal{Im}(x^{2} (\zeta^{2} - 2i\zeta))$ is constant around $\zeta = i$?
Problem: If I do this, I get that the asymptotic behaviour of $I(x) \sim 1$. What has gone wrong here?

Stationary phase with $\textrm{e}^{-x^{2}}\int_{0}^{x} \textrm{e}^{-2itx}\textrm{e}^{t^{2}}$. Now the problem is that the stationary point is at $t = 0$. In any case, by my calculation stationary phase gives the (incorrect) asymptotic behaviour $I(x) \sim \textrm{e}^{x^{2}}/\sqrt{x}$ (actually since the phase exponent is linear in $t$ the second derivative is vanishing so perhaps this result is incorrectly obtained).

Can anyone confirm these approaches and point out where things have gone wrong?

REPLY [1 votes]: The problem with OP's approach is essentially that there isn't a stationary point nearby. Instead let us employ the same strategy as in my Math.SE answer here. Write OP's integral as
$$\begin{align}I(x)
~=~&\int_{x}^{x+ix} \! \mathrm{d}z~e^{-z^2}~=~[e^{-a^2}J(a)]^{a=x}_{a=x+ix}\cr
~\stackrel{(D)}{=}~&\left(\frac{e^{-x^2}}{2x} -\frac{e^{-2ix^2}}{2x(1+i)}\right)\left( 1+O(x^{-2}) \right),\end{align}\tag{A}$$
where
$$J(a)~:=~\int_{\mathbb{R_+}} \!\mathrm{d}z~e^{-z(z+2a)}.\tag{B}$$
Next change integration variable
$$u~=~z(z+2a)\qquad \Leftrightarrow\qquad z~=~\sqrt{u+a^2}-a.\tag{C}$$
Then
$$ \begin{align}J(a)
~=~&\int_{\mathbb{R_+}} \!\mathrm{d}u~ \frac{e^{-u}}{2\sqrt{u+a^2}}
~=~\frac{1}{2a}\int_{\mathbb{R_+}} \!\mathrm{d}u~ \frac{e^{-u}}{\sqrt{1+\frac{u}{a^2}}}\cr
~=~&\frac{1}{2a}\int_{\mathbb{R_+}} \!\mathrm{d}u\left( 1- \frac{u}{2a^2} +\frac{3u^2}{8a^4}+O(a^{-6}) \right)\cr
~=~&\frac{1}{2a}\left( 1- \frac{1}{2a^2} +\frac{3}{4a^4}+O(a^{-6}) \right).\end{align} \tag{D}$$


TITLE: Element 137 - Theoretical limit
QUESTION [2 upvotes]: I've seen a few sources that mention the speed of an electron in a hydrogen atom is 1/137 the speed of light. This article also mentions what looks like a correlation between atomic number and the numerator meaning gold's electron speed is 79/137 the speed of light.
Does this imply that the highest element possible would be element 137 (or more likely 136)? Not referring to anything stable, just what is possible with physics (and hadrons).

REPLY [4 votes]: No.
There is indeed a correlation between the nuclear charge $Z$ of an element and the kinetic energy of the innermost electrons, but there are two things to keep clear sight of:

The correlation is not linear, so you cannot simply multiply the speed of the innermost electron by $Z$ and hope for things to just work out. The interior of a large atom is a very complex place, with many dynamical effects going on, and even if the kinetic energy of hydrogenic orbitals does increase with $Z^2$ that does not mean that this will remain true when multiple electrons are present. To begin with, even in helium the average kinetic energy of either electron is not four times that of hydrogen - so there's that settled.

More importantly, it is the kinetic energy that increases with $Z$. Once the corresponding electron speed approaches the speed of light, you have to switch over to the relativistic expression of kinetic energy,
$$E_K=\frac{mc^2}{\sqrt{1-v^2/c^2}}-mc^2,$$
and this can increase without bounds while still maintaining $|v|<c$. That is, you can keep adding energy to the electron without fearing that it will break the speed of light, so your fears are unfounded.

Going a bit further, the article you linked to is fairly misleading on several fronts. The presence of relativistic effects inside atoms has been perfectly appreciated, from the fine structure of hydrogen to strongly relativistic effects in heavier atoms like the "anomaly" of mercury being liquid, which are bread and butter to the atomic physicists that study them.
Nowadays, and for the past sixty years or so, having atoms "deviate from their expect chemical behavior predicted by their place on the Periodic Table of Elements" simply means that the calculations were either taken naively or without sufficient care. The behaviour described (the decoupling of the S orbitals of Sg from the valence shell due to relativistic energy considerations, leaving an effective valence of +4 instead of the naive +6) is in no way unprecedented.


TITLE: Prove that the function $\Phi :\mathcal{F}(X,Y)\longrightarrow Y$, is not injective.
QUESTION [4 upvotes]: Given

Let $\mathcal{F}(X,Y)$ be the set of all functions $f:X\longrightarrow Y$, and let $x_0 \in X$. Given:
    $$\Phi :\,\mathcal{F}(X,Y)\longrightarrow Y$$
    Defined by $\Phi (f)=f(x_0)$ for $f\in\mathcal{F}(X,Y)$.

Definition

The function f is injective if $f(x)=f(y)$ implies $x=y$ for
    all $x,y\in Dom(f)$
The function f is surjective if for every $b\in Codom(f)$
    there exists some $a\in A$ such that $f(a)=b$
The function f is bijective if it is both injective and
    surjective

Exercise
Is $\Phi$ surjective, injective, bijective?

My Solution

$\Phi$ is surjective
Proof:  let $y\in Y$. We will show that there exists some
element $x\in X$ such that $\Phi(f)=f(x)=b$. Define $f(x)=y, \forall
    x\in X$, We know $f\in\mathcal{F}(X,Y)$, because $\mathcal{F}(X,Y)$
is the set of all functions from $X$ to $Y$ therefore
$\Phi(f)=f(x_0)=y$, thus $\Phi$ is Surjective
$\hspace{15cm}$ ${\Large ▫}$
$\Phi$ is not injective:
Proof:  Special case If $|X|=1$, then
$\Phi$ is injective.
But when $|X|>1$, $\Phi$ is not injective. Ok here I get stuck. I
cant figure out how to define two functions such that $\Phi$ isn't
injective.

My Question

Is the surjective proof correct?
How can I define two functions in $\mathcal{F}$ such that
  $\Phi$ is not injective?

Can someone give me some hints/tips?

REPLY [4 votes]: Your surjectivity proof is ok.
You observe that $\Phi$ is injective if $|X|=1$. Note that $\Phi$ is also injective if $Y=\emptyset$ or $|Y|=1$. (You should prove injectivity in these three cases)
If $x_1\in X$  and $y_0, y_1\in Y$ with $x_1\ne x_0$, $y_0\ne y_1$, you can define two functions
$f,g\colon X\longrightarrow Y$, namely $f(x)=y_0$ and
$$g(x)=\begin{cases}y_0&\text{if }x=x_0,\\y_1&\text{otherwise.}\end{cases}$$
Then $\Phi(f)=\Phi(g)=y_0$, but $f\ne g$ because $f(x_1)=y_0\ne y_1=g(x_1)$.


text:
\section*{Acknowledgement}

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. This is contribution LLNL-JRNL-810518. This research was supported by the Exascale Computing Project (ECP), project 17-SC-20-SC, a collaborative effort of two U.S. Department of Energy (DOE) organizations - the Office of Science and the National Nuclear Security Administration.

Part of the computations were performed on resources provided by Swedish National Infrastructure for Computing (SNIC) through Uppsala Multidisciplinary Center for Advanced Computational Science (UPPMAX) under Project SNIC 2019/8-263.


TITLE: Explicite calculation of the map $m_g^* : H^0(G, \mathbb{Z}) \to H^0(G, \mathbb{Z})$
QUESTION [2 upvotes]: Let $G$ be a finite group and $g \in G$ be a non identity element. So we can write a map from $G$ to itself induces from $g$ i.e. $m_g : G \to G$  where $m_g(x) = g.x$. Again this $m_g$ induces a map in cohomology $$m_g^* : H^0(G, \mathbb{Z}) \to H^0(G, \mathbb{Z})$$
Question: $m_g^*(n_1, n_2, \cdots , n_{|G|}) = ?$
Note : Action of $G$ on $\mathbb{Z}$ is trivial.
Any help will be appreciated.
Thank you too much.

REPLY [1 votes]: It's true that if $X$ is a finite discrete space, then $H^0(X;\mathbb{Z}) \cong \bigoplus_{|X|} \mathbb{Z}$. However, it's easier for this exercise to back up a bit and see that $H^0(X;\mathbb{Z})$ is isomorphic to the group of maps of sets $X \to \mathbb{Z}$ (with pointwise addition for the group structure), i.e. $H^0(X;\mathbb{Z}) = \operatorname{Map}(X, \mathbb{Z})$. When $X = \{ x_1, \dots, x_n \}$ is finite, this is identified with $\mathbb{Z}^{\oplus n}$ by $\varphi \mapsto (\varphi(x_1), \dots, \varphi(x_n))$.
And more importantly, if $f : X \to Y$ is so map (which is automatically continuous), then the induced map $H^0(Y;\mathbb{Z}) \to H^0(X;\mathbb{Z})$ is given by pre-composition:
$$\varphi \in \operatorname{Map}(Y, \mathbb{Z}) \mapsto f^*\varphi := \varphi \circ f \in \operatorname{Map}(X, \mathbb{Z}).$$
So now for our finite group $G$, $H^0(G;\mathbb{Z}) \cong \operatorname{Map}(G, \mathbb{Z})$ (these are just maps of sets, not group homomorphisms). And if $\varphi : G \to \mathbb{Z}$, then $m_g(\varphi) = \varphi \circ m_g : x \mapsto \varphi(gx)$.
So given an enumeration $\{x_1, \dots, x_n\}$ of $G$ ($n = |G|$), you can view $H^0(X; \mathbb{Z}) \cong \mathbb{Z}^n$. If $\alpha = (\alpha_1, \dots, \alpha_n) \in \mathbb{Z}^n$, then $m_g^*\alpha$ is a permutation of $\alpha$, and you need to use the group law of $G$ to find out which one: $g x_i = x_{\sigma(i)}$ for some permutation $\sigma \in \mathfrak{S}_n$, and $m_g^*\alpha = (\alpha_{\sigma(1)}, \dots, \alpha_{\sigma(n)})$.


TITLE: When convergence in distribution implies stable convergence
QUESTION [6 upvotes]: In a previous post I asked help to clarify a property of stable convergence in distribution:
Definition
Let $X_n$ be a sequence of random variables defined on a probability space $(\Omega,\mathcal{F},\mathbb{P})$
with value in $\mathbb{R}^N$. We say that the sequence $X_n$ converges stably in distribution
with limit $X$, written $X_n\stackrel{\text{st}}{\longrightarrow} X$, if and only
if,  for any bounded continuous function $f:\mathbb{R}^N\to\mathbb{R}$ and for any $\mathcal{F}$-measurable bounded random variable $W$, it happens that:
$$
\lim_{n\rightarrow \infty}\mathbb{E}[f(X_n)\,W]=\mathbb{E}[f(X)\,W].
$$
What I need to prove now is the following:
Assume
$$
(Y_n,Z)\stackrel{\text{d}}{\longrightarrow}(Y,Z),
$$
for all measurable random variable $Z$, then
$$
(Y_n,Z)\stackrel{\text{st}}{\longrightarrow}(Y,Z)
$$
for all measurable random variables $Z$. So I need to prove that, for any bounded continuous function $f$ and for any measurable $Z$ it holds that
$$
\lim_{n\rightarrow \infty}\mathbb{E}[f(Y_n,Z)\,W]=\mathbb{E}[f(Y,Z)\,W]
$$
for all bounded random variables $W$.
I tried unsuccessfully with Portmanteau and Levy continuity theorem…
=================================================================
In practice I am trying to prove this proposition from the paper by Podolskij and Vetter:

I did this reasoning for (1)=>(3), but I am not so sure of its correctness.

REPLY [1 votes]: $\def\dto{\xrightarrow{\mathrm{d}}}\def\stto{\xrightarrow{\mathrm{st}}}\def\mto{\xrightarrow{\mathrm{m}}}$$(3) \Rightarrow (2)$: Trivial.
$(2) \Rightarrow (1)$: For any $g \in C_b(\mathbb{R}^N)$ and bounded $\mathscr{F}$-measurable $W$, suppose $|W| \leqslant M$. Take\begin{align*}
f: \mathbb{R}^N × \mathbb{R} &\longrightarrow \mathbb{R},\\
(y, z) &\longmapsto g(y) · \frac{1}{2} (|z + M| - |z - M|).
\end{align*}
Because $(Y_n, W) \dto (Y, W)$ and $f \in C_b(\mathbb{R}^{N + 1})$, then$$
E(g(Y_n) W) = E(f(Y_n, W)) \to E(f(Y, W)) = E(g(Y) W). \quad n \to \infty
$$
Therefore, $Y_n \stto Y$.
$(1) \Rightarrow (3)$: Suppose $Z$ and $W$ are $\mathscr{F}$-measurable and $W$ is bounded. First, for any $A \in \mathscr{B}(\mathbb{R}^N)$ and $B \in \mathscr{B}(\mathbb{R})$, there exists $\{g_k\} \subseteq C_b(\mathbb{R}^N)$ such that $g_k \mto I_A$, i.e.$$
m(\{ x \in \mathbb{R}^N \mid g_k(x) \neq I_A(x)\}) \to 0. \quad k \to \infty
$$
For any $k \geqslant 1$, because $Y_n \stto Y$ and $I_B(Z) W$ is $\mathscr{F}$-measurable and bounded, then$$
E(g_k(Y_n) I_B(Z) W) \to E(g_k(Y) I_B(Z) W). \quad n \to \infty
$$
Note that $g_k \mto I_A$ and $I_B(Z) W$ is bounded, thus$$
E(I_A(Y_n) I_B(Z) W) \to E(I_A(Y) I_B(Z) W). \quad n \to \infty \tag{1}
$$
Now, for any $C \in \mathscr{B}(\mathbb{R}^{N + 1})$, there exists $\{A_{k, j}\} \subseteq \mathscr{B}(\mathbb{R}^N)$ and $\{B_{k, j}\} \subseteq \mathscr{B}(\mathbb{R})$ such that $\{h_k\}$ defined by\begin{align*}
h_k : \mathbb{R}^N × \mathbb{R} &\longrightarrow \mathbb{R},\\
(y, z) &\longmapsto \sum_{j = 1}^{s_k} I_{A_{k, j}}(y) I_{B_{j, k}}(z)
\end{align*}
satisfies $h_k \mto I_C$. For any $k \geqslant 1$, from (1) there is$$
E(h_k(Y_n, Z) W) \to E(h_k(Y, Z) W). \quad n \to \infty
$$
Because $h_k \mto I_C$ and $W$ is bounded, then$$
E(I_C(Y_n, Z) W) \to E(I_C(Y, Z) W). \quad n \to \infty \tag{2}
$$
Now, for any $f \in C_b(\mathbb{R}^{N + 1})$, there exists a sequence of simple functions $\{f_k\}$ such that $f_k \rightrightarrows f$. For any $k \geqslant 1$, from (2) there is$$
E(f_k(Y_n, Z) W) \to E(f_k(Y, Z) W). \quad n \to \infty
$$
Because $f_k \rightrightarrows f$ and $W$ is bounded, then$$
E(f(Y_n, Z) W) \to E(f(Y, Z) W). \quad n \to \infty
$$
Therefore, $(Y_n, Z) \stto (Y, Z)$.


TITLE: Prove that there exists a right ideal $K \in R$ such that $I \cap K = \{0\}$ and $I + K = J$.
QUESTION [1 upvotes]: Let $I$ and $J$ be right ideals in a ring $R$ with identity such that $I \subset J$ and $\frac JI \cong R$ as right R-module. Prove that there exists a right ideal $K \in R$ such that $I \cap K = \{0\}$
and $I + K = J$.

REPLY [2 votes]: Hint:
$0\to I\to J \to J/I\to 0$ is a split exact sequence since, by your hypothesis, $J/I$ is projective.


TITLE: If $A$ is finite and $B\subset A$, then $B$ is finite
QUESTION [1 upvotes]: I am having trouble completing the proof of this statement.

If $A$ is finite then every $B \subset A$ is finite, where "finite" means "empty or in bijection with $\{1,...,t\}$ for some $t \in N$"

Suppose that $A\neq \emptyset$ and $A$ is not a singleton. It suffices to prove that if $B \subset \{1,...,m\}$ for some $m \ge 2$, then $B$ is in bijection with some $\{1,...,t\}$ for some $t \ge 1$. Let $a=\min B$.
For $x\in B$, consider:
$$S(x) = \begin{cases} \min\{t\in B: t>x\}, \text{ if $x < \max B$} \\ \max B, \text{ otherwise} \end{cases}$$
It's clear that there exists $r\in N$, such that $S^n(a) =S^r(a)$ for all $n\ge r$. Where $S^n(a)$ is $S(S(...S(a)..))$.
Let $t = \min\{r\in N: S^n(a) = S^r(a), \forall n\ge r\}$. We claim that: $B=\{a,...,S^t(a)\}$.
Proof: (here is where I am having trouble)
It is clear that $\{a,...,S^t(a)\}\subset B$. Now if $x\in B$, then if $x=a$ we are done. Suppose that $x > a$. If $L_x:= \{k\in B: a<k<x\}=\emptyset$, then clearly $x = S(a)$.
If $L_x \neq \emptyset$, how to prove that there must be some $r\in \{1,...,t\}$ such that $x = S^r(a)$?
Thanks

REPLY [0 votes]: Prove the following by induction on $n$:
If $A$ is in bijection with $\{1,...,n\}$ then $B$ is finite.

 Step Case:
 Suppose $A$ is in bijection with $\{1,...,n+1\}$. Let $\phi$ be this bijection. Then $A - \{\phi(n+1)\}$ is in bijection with $\{1,...,n\}$. Apply the IH to conclude that $B - \{\phi(n+1)\}$ is in bijection with $\{1,...,t\}$ for some $t$. Then it is easy to show that $B$ must be finite.