Asymptotic analysis of periodically-perforated nonlinear media.
Given \(\Omega\) a bounded open set in \({\mathbb R}^n\) (\(n \geqslant 3\)), \(p \in (1,n)\), \(B_i^{\delta}\) the ball of centre \(i\delta\) (\(i \in {\mathbb Z}^n\) and \(\delta > 0\)) and radius \(\delta^{n/(n-p)}\), \(\Omega_{\delta} = \Omega \setminus \bigcup_{i \in {\mathbb Z}^n} \overline{B_i^{\delta}}\), consider a sequence of positive numbers \(\{\delta_j\}_j\) converging to zero and the nonlinear sequence of the minimum problem \[ \inf \Big\{ F_j u = \int_{\Omega_{\delta_j}} f(Du) \text{d}x + \langle \phi, u\rangle \;\Big| \;u \in W^{1,p}_0(\Omega_{\delta_j}; {\mathbb R}^m) \Big\} = m_j \] where \(\phi \in W^{-1,p'}_0(\Omega; {\mathbb R}^m)\) and \(f:M^{m\times n} \to [0,+\infty)\) (\(M^{m\times n}\) the space of \(m \times n\) matrices) is a Borel function satisfying \[ f(0) = 0, \quad c_1 (| A| ^p - 1) \leqslant f(A) \leqslant c_2 (| A| ^p + 1) \quad \text{ for every } A \in M^{m\times n} \] with \(c_1, c_2\) positive constants. Denote by \(Qh\), \(h:M^{m\times n} \to [0,+\infty)\) a Borel function, the \(W^{1,p}\)-quasiconvexification of \(h\) \[ Q h (A) = \inf \Big\{ \int_{(0,1)^n} h(A+Du) \text{d}x \;\Big| \;u \in W^{1,p}_0((0,1)^n; {\mathbb R}^m) \Big\}. \] Under the assumption that there exists the limit \[ g(z) = \lim_j \delta_j^{{np}\over{n-p}} Qf (\delta_j^{-{{np}\over{n-p}}} z) \quad \text{ for every } z \in {\mathbb R}^m \] and define \[ F u = \int_{\Omega} (Qf(Du) + \varphi (u)) \text{d}x + \langle \phi, u\rangle, \quad u \in W^{1,p}_0(\Omega; {\mathbb R}^m) \] where \[ \varphi (z) = \inf \Big\{ \int_{{\mathbb R}^n} g(D \zeta ) \text{d}x \;\Big| \;\zeta - z \in W^{1,p}({\mathbb R}^n; {\mathbb R}^m) \, , \zeta =0 \text{ on } B_1(0) \Big\} \] the authors prove that \[ F_j\;\Gamma \text{-converge to } F \text{ with respect to } L^p(\Omega ; {\mathbb R}^m). \] In particular \(m_j \to m = \min Fu\) and, if \(u_j\) is such that \(F_j u_j = m_j + o(1)\) then, up to a subsequence, \(u_j \to u\) in \(W^{1,p}_0(\Omega; {\mathbb R}^m)\)-weak where \(u\) is a minimum of \(F\).
scientific article; zbMATH DE number 1975940
scientific article