Publication: Limits of Dirichlet problems in perforated domains: a new formulation
Date
1994
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Università degli Studi di Trieste. Dipartimento di Scienze Matematiche
Abstract
Sia A un operatore ellittico lineare del secondo ordine con coefficienti
misurabili e limitati su un aperto limitato $\Omega$ di $\mathbf{R}^{\textrm{n}}$
, sia
\[
K*=\{w*\epsilon H_{0}^{1}\left(\Omega\right):A*w*\leq1\, in\,\mathcal{D}'\left(\Omega\right)\qquad,
\]
\[
e,\, w*\geq0\, a.e.\, in\,\Omega\}\qquad,
\]
e sia $\Omega_{h}$ un'arbitraria successione di sottoinsiemi aperti
di $\Omega$. Dimostriamo il seguente risultato di compattezza: esistono
una sottosuccessione, che indichiamo ancora con $\Omega_{h}$ ed una
funzione w{*} $\epsilon$ K{*} tali che, per ogni f $\epsilon L^{\infty}\left(\Omega\right)$
, le soluzioni u$_{h}\epsilon H_{0}^{1}\left(\Omega_{h}\right)$ delle
equazioni Au$_{h}$ = f in $\Omega_{h}$ , estese a zero su $\Omega/\Omega_{h}$,
convergano debolmente in $H_{0}^{1}\left(\Omega\right)$ all'unica
soluzione u del problema.
\[
\left(*\right)\begin{cases}
\begin{array}{c}
u\epsilon H_{0}^{1}\left(\Omega\right)\cap L^{\infty}\left(\Omega\right)\\
\left\langle Au,\, w*\varphi\right\rangle -\left\langle A*w*,\, u\varphi\right\rangle +\left\langle 1,u\varphi\right\rangle =\left\langle f,w*\varphi\right\rangle \:\forall\varphi\epsilon C_{0}^{\infty}\left(\Omega\right)
\end{array}\end{cases}
\]
Studiamo inoltre in maniera sistematica le proprietà delle soluzioni
di tale equazione. Dimostriamo infine il seguente risultato di densità:
per ogni w{*}$\epsilon$K{*} esiste una successione $\Omega_{h}$
di sottoinsiemi aperti di $\Omega$ tali che per ogni f $\epsilon L^{\infty}\left(\Omega\right)$
le soluzioni u$_{h}\epsilon H_{0}^{1}\left(\Omega_{h}\right)$ dell'equazione
Au$_{h}$=f in $\Omega_{h}$, estese a zero $\Omega/\Omega_{h}$ convergano
debolmente in $H_{0}^{1}\left(\Omega\right)$alla soluzione di ({*}).
Let A be a linear elliptic operator of the second order with bounded measurable coefficients on a bounded open set $\Omega$ of $\mathbf{R}^{\textrm{n}}$ , let \[ K*=\{w*\epsilon H_{0}^{1}\left(\Omega\right):A*w*\leq1\, in\,\mathcal{D}'\left(\Omega\right)\qquad, \] \[ e,\, w*\geq0\, a.e.\, in\,\Omega\}\qquad, \] and let $\Omega_{h}$ be an arbitrary sequence of open subsets of $\Omega$. We prove the following compactness result: there exist a subsequence, still denoted by $\Omega_{h}$ and a function w{*} $\epsilon$ K{*} such that, for every f $\epsilon L^{\infty}\left(\Omega\right)$ , the solutions u$_{h}\epsilon H_{0}^{1}\left(\Omega_{h}\right)$ of the equation Au$_{h}$ = f in $\Omega_{h}$ , extended by zero on $\Omega/\Omega_{h}$, converge weakly in $H_{0}^{1}\left(\Omega\right)$ to the unique solution u of the problem. \[ \left(*\right)\begin{cases} \begin{array}{c} u\epsilon H_{0}^{1}\left(\Omega\right)\cap L^{\infty}\left(\Omega\right)\\ \left\langle Au,\, w*\varphi\right\rangle -\left\langle A*w*,\, u\varphi\right\rangle +\left\langle 1,u\varphi\right\rangle =\left\langle f,w*\varphi\right\rangle \:\forall\varphi\epsilon C_{0}^{\infty}\left(\Omega\right) \end{array}\end{cases} \] We provide a self-contained study of the properties of the solutions of ({*}). We prove also the following density result: for any w{*}$\epsilon$K{*} there exists a sequence $\Omega_{h}$ of open subsets of $\Omega$ such that for every f $\epsilon L^{\infty}\left(\Omega\right)$ the solutions u$_{h}\epsilon H_{0}^{1}\left(\Omega_{h}\right)$ of the equation Au$_{h}$=f in $\Omega_{h}$, extended by zero on $\Omega/\Omega_{h}$ converge weakly in $H_{0}^{1}\left(\Omega\right)$to the solution of ({*}).
Let A be a linear elliptic operator of the second order with bounded measurable coefficients on a bounded open set $\Omega$ of $\mathbf{R}^{\textrm{n}}$ , let \[ K*=\{w*\epsilon H_{0}^{1}\left(\Omega\right):A*w*\leq1\, in\,\mathcal{D}'\left(\Omega\right)\qquad, \] \[ e,\, w*\geq0\, a.e.\, in\,\Omega\}\qquad, \] and let $\Omega_{h}$ be an arbitrary sequence of open subsets of $\Omega$. We prove the following compactness result: there exist a subsequence, still denoted by $\Omega_{h}$ and a function w{*} $\epsilon$ K{*} such that, for every f $\epsilon L^{\infty}\left(\Omega\right)$ , the solutions u$_{h}\epsilon H_{0}^{1}\left(\Omega_{h}\right)$ of the equation Au$_{h}$ = f in $\Omega_{h}$ , extended by zero on $\Omega/\Omega_{h}$, converge weakly in $H_{0}^{1}\left(\Omega\right)$ to the unique solution u of the problem. \[ \left(*\right)\begin{cases} \begin{array}{c} u\epsilon H_{0}^{1}\left(\Omega\right)\cap L^{\infty}\left(\Omega\right)\\ \left\langle Au,\, w*\varphi\right\rangle -\left\langle A*w*,\, u\varphi\right\rangle +\left\langle 1,u\varphi\right\rangle =\left\langle f,w*\varphi\right\rangle \:\forall\varphi\epsilon C_{0}^{\infty}\left(\Omega\right) \end{array}\end{cases} \] We provide a self-contained study of the properties of the solutions of ({*}). We prove also the following density result: for any w{*}$\epsilon$K{*} there exists a sequence $\Omega_{h}$ of open subsets of $\Omega$ such that for every f $\epsilon L^{\infty}\left(\Omega\right)$ the solutions u$_{h}\epsilon H_{0}^{1}\left(\Omega_{h}\right)$ of the equation Au$_{h}$=f in $\Omega_{h}$, extended by zero on $\Omega/\Omega_{h}$ converge weakly in $H_{0}^{1}\left(\Omega\right)$to the solution of ({*}).
Description
Keywords
Citation
G. dal Maso and R. Toader, "Limits of Dirichlet problems in perforated domains: a new formulation", in: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics, 26 (1994), pp. 339-360.