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\(\Gamma\) convergence of concentration problems. (English) Zbl 1121.35048

The main purpose of the paper is to describe the asymptotic behaviour, when \(\varepsilon \) goes to 0, of the solution of the semilinear elliptic equation \(\Delta u=\lambda _{\varepsilon }f\left( u\right) \), posed in \(\Omega \), with homogeneous Dirichlet boundary conditions on \(\partial \Omega \). This elliptic problem is in fact obtained when maximizing the quantity \(\varepsilon ^{-2^{*}}\int_{\Omega }F\left( u\right) dx\) under the constraints \(u=0\) on \(\partial \Omega \), and \(\int_{\Omega }\left| \nabla u\right| ^{2}dx\leq \varepsilon ^{2}\). Here \(2^{*}\) is the critical Sobolev exponent equal to \(2N/(N-2)\) and \(F\) is a nonnegative and upper semicontinuous functional which is bounded from above by \(C_{F}\left| t\right| ^{2^{*}}\). In the previous elliptic equation, \(\lambda _{\varepsilon }\) is a Lagrange multiplier associated to the above constraints and \(f\) is the derivative of \(F\).
The first part of the paper describes the asymptotic behaviour of this problem using a variant, called \(\Gamma ^{+}\)-convergence, of the \(\Gamma \)–convergenceintroduced by E. De Giorgi and T. Franzoni in [Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 58, 842–850 (1975; Zbl 0339.49005)] (see also the book by G. Dal Maso [An introduction to \(\Gamma\)-convergence. Progress in Nonlinear Differential Equations and their Applications. 8. Basel: Birkhäuser (1993; Zbl 0816.49001)]). This \(\Gamma ^{+}\)–convergence is indeed devoted to the asymptotic study of maximization problems. The authors first introduce the limit functional space \(X=\{(u,\mu)\in H_{0}^{1}(\Omega )\times \mathcal{M}(\overline{\Omega }):\mu \geq \left| \nabla u\right| ^{2}\), \(\mu (\overline{\Omega })\leq 1\}\) and the convergence notion for sequences \((u_{\varepsilon },\mu _{\varepsilon })_{\varepsilon }\) in the following way: a sequence \((u_{\varepsilon },\mu _{\varepsilon })_{\varepsilon }\) \(\tau \)-converges to \((u,\mu )\) if \( (u_{\varepsilon })_{\varepsilon }\) converges to \(u\) in the weak topology of \( L^{2^{*}}(\Omega )\) and if \((\mu _{\varepsilon })_{\varepsilon }\) converges to \(\mu \) in the weak\(^{*}\)-topology of \(\mathcal {M}(\overline{\Omega })\). Here \(\mathcal {M}(\overline{\Omega })\) is the space of nonnegative Borel measures on \(\overline{\Omega }\).
The first main result of the paper proves that if \(\mathcal {F}_{\varepsilon }(u)=\varepsilon ^{-2^{*}}\int_{\Omega }F(\varepsilon u)\,dx\), under the constraints \(u=0\) on \(\partial \Omega \), and \(\int_{\Omega }\left| \nabla u\right| ^{2}dx\leq 1\), then \((\mathcal {F}_{\varepsilon })_{\varepsilon }\) \(\Gamma ^{+}\)-converges to the functional \(\overline{\mathcal {F}}\) defined on \(X\) through \(\overline{\mathcal {F}}(u,\mu )=F_{0}\int_{\Omega }\left| u\right| ^{2^{\ast }}dx+S^{F}\sum_{i}(\mu _{i})^{2^{*}/2}\), where the \(\mu _{i}\) are associated to the limit \(\mu \) of \(\left| \nabla u_{\varepsilon }\right| ^{2}\): \(\mu =\left| \nabla u_{\varepsilon }\right| ^{2}+ \widetilde{\mu }+\sum_{i}\mu _{i}\delta _{x_{i}}\), with \(\widetilde{\mu }\) non-atomic. Here \(S^{F}\) is the limit of the maximum \(S_{\varepsilon }^{F}\) of \(\mathcal {F}_{\varepsilon }\), under the above constraints. The proof of this result is given with details and the main part of this proof is devoted to the construction of a so-called recovery sequence for the \(\Gamma ^{+}\) -convergence. As a consequence of this result,the authors prove that any maximizing sequence \((u_{\varepsilon },\mu _{\varepsilon })_{\varepsilon }\) concentrates at a unique point \(x_{0}\) of \(\overline{\Omega }\) and that every \(x_{0}\in \overline{\Omega }\) is a concentration point of a maximizing sequence.
The authors then extend this result to a nonhomogeneous case considering the functional \(\mathcal {F}_{\varepsilon }^{A}(u)=\varepsilon ^{-2^{\ast }}\int_{\Omega }A(x)F(\varepsilon u)\,dx\), where \(A\) is a nonnegative function in \(L^{\infty }(\Omega )\) which satisfies further hypotheses which at least ensure the upper semicontinuity of the functional \(\mathcal {F}_{\varepsilon }^{A}\).
In the last part of the paper, the authors first recall the asymptotic behaviour of \(S_{\varepsilon }^{F}\) as exposed in M. Flucher, A. Garroni and S. Müller [Calc. Var. Partial Differ. Equ. 14, No. 4, 483–516 (2002; Zbl 1004.35040)].
The main result of this last part describes the asymptotic behaviour of the functional \(\mathcal {H}_{\varepsilon }\) defined through \(\mathcal {H}_{\varepsilon }(u,\mu )=(\varepsilon ^{-2^{*}}\int_{\Omega }F(\varepsilon u)\,dx-S^{F})/\varepsilon ^{2}\), if \((u,\mu )\in X\) and \(\mu =\left| \nabla u\right| ^{2}\), and \(\mathcal {H}_{\varepsilon }(u,\mu )=-S^{F}/\varepsilon ^{2}\), otherwise. The authors still use the \(\Gamma ^{+} \)-convergence for this study, which is based on Robin function.

MSC:

35J60 Nonlinear elliptic equations
35C20 Asymptotic expansions of solutions to PDEs
49J45 Methods involving semicontinuity and convergence; relaxation
35J20 Variational methods for second-order elliptic equations
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
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References:

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