×

Asymptotic behavior of nonlinear elliptic systems on varying domains. (English) Zbl 0952.35036

The following homogenization problem \(Au_n=f\) in \(\Omega_n, u_n\in W^{1,p}_0(\Omega_n,\mathbb{R}^M)\) ist studied, where \(A:W_0^{1,p}(\Omega,\mathbb{R}^M)\to W^{-1,p}(\Omega,\mathbb{R}^M)\) is a nonlinear monotone operator in divergence form. The sequence of open subsets of a bounded open set \(\Omega\) in \(\mathbb{R}^N\) is arbitrary without assumptions involving the geometry or the capacity of the closed sets \(\Omega\setminus\Omega_n.\) As additional term in the equation for the limit function \(u\) appears \(F(x,u)\mu\) where \(\mu\) is a positive Borel measure with value \(+\infty\) on large families of sets. Analogous results in the scalar case \((M=1)\) were established by G. Dal Maso and F. Murat.

MSC:

35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35B40 Asymptotic behavior of solutions to PDEs
47H05 Monotone operators and generalizations
PDFBibTeX XMLCite
Full Text: DOI