Stratified materials allowing asymptotically prescribed equipotentials
The authors study the asymptotic behaviour of stratified conductors, where the conductivity coefficient \(a_h\) is highly oscillating in one direction. The following sequence of minimization problems is considered: \[ \min\Biggl\{ \int_\Omega {1\over a_h} G_h(x, a_h Du) dx- \int_\Omega f_h u dx: u\in W^{1,p}(\Omega), u= 0\text{ on } \Gamma_0,\;u= 1\text{ on } \Gamma_1\Biggr\}, \] where \(\Omega\) is a domain of \(\mathbb{R}^n\) of annulus type with boundaries \(\Gamma_0\) and \(\Gamma_1\), the coefficients \(a_h\) are measurable and such that \(c\leq a_h(x)\), \(\forall h\in \mathbb{N}\), \(\forall x\in \Omega\) with \(c> 0\), the sequence \(f_h\) tends to \(f\) weakly in \(L^{p'}(\Omega)\), and \(G_h\) are Carathéodory functions, with \(G_h(x, \cdot)\) strictly convex and differentiable, such that \hskip17mm \(c|z|^p\leq G_h(x, z)\leq C(a(x)+ |z|^p)\qquad (a\in L^1(\Omega))\) \hskip17mm \(|DG_h(x, z)|\leq C(1+ |z|^{p- 1})\) \hskip17mm \(G_h(x, z)\to G(x, z)\) \hskip17mm \(DG_h(x, z)\to DG(x, z)\) with \(0< c\leq C\). On the coefficients \(a_h\) it is assumed that \(a_h(x)= \alpha_h(\phi(x))\), where \(1/\alpha_h\to 1/\alpha\) weakly\(^*\) in \(L^\infty(0, 1)\) and \(\int_I \alpha^{p- 1}_h(t) dt\to +\infty\) for every open set \(I\subset [0, 1]\). It is shown that, setting \(a(x)= \alpha(\phi(x))\), the limit minimum problem is \[ \min\Biggl\{ \int_\Omega {1\over a} G(x, a Du) dx- \int_\Omega fu dx: u= v\circ \phi,\;v\in W^{1, p}(0, 1),\;v(0)= 0,\;v(1)= 1\Biggr\}. \]
scientific article; zbMATH DE number 750852