Asymptotic Behavior of Nonlinear Elliptic Systems on Varying Domains
Juan Casado‐Díaz, Adriana Garroni · SIAM Journal on Mathematical Analysis · 2000
We consider a monotone operator of the form Au= - div(a(x,Du)), with $\Omega\subseteq{\bf R}^n$ and $a:\Omega\times{\bf M}^{M \times N} \to {\bf M}^{M \times N}$, acting on $W^{1,p}_0(\Omega, {\bf R}^M)$. For every sequence $(\Omega_h)$ of open subsets of $\Omega$ and for every $f \in W^{-1,p'}(\Omega, {\bf R}^M)$, $1/p+1/p'=1$, we study the asymptotic behavior, as $h\to+\infty$, of the solutions $u_h\in W^1_0(\Omega_h,{\bf R}^M)$ of the systems $Au_h=f$ in $W^{-1,p'}(\Omega_h,{\bf R}^M)$, and we determine the general form of the limit problem.