A capacitary method for the asymptotic analysis of Dirichlet problems for monotone operators
Given a nonlinear elliptic operator of the form \(Au=-\text{div} (a(x,Du))\), where \(a:\Omega \times\mathbb{R}^n \to\mathbb{R}^n\) satisfies usual \(p\)-growth and monotonicity assumptions, it is known that for every sequence \((\Omega_j)\) of subdomains of \(\Omega\) and for every right-hand side \(f\) the sequence \((u_j)\) of solutions of the Dirichlet problems \[ Au= f \quad \text{in }\Omega_j, \qquad u\in W_0^{1,p} (\Omega_j) \] always admits a subsequence which converges weakly in \(W_0^{1,p} (\Omega)\) to the solution \(u\) of the limit problem which is formally written as \[ Au+ b(x,u) \mu=f \quad \text{in } \Omega, \qquad u\in W_0^{1,p} (\Omega) \cap L^p_\mu (\Omega). \] The precise meaning of the problem above is explained in the paper, where a method for constructing the function \(b\) and the measure \(\mu\) from the sequence \((\Omega_j)\) is given. This method is related to the notion of \(A\)-capacity of compact sets \(K\subset \Omega\) with respect to a constant \(s\): \[ C_A(K,s) =\int_{\Omega \setminus K} a(x,Du) \cdot Du dx, \] where \(u\) is the solution of the Dirichlet problem \[ Au=0 \quad \text{in } \Omega \setminus K, \qquad u=s \text{ on } \partial K, \quad u=0 \text{ on } \partial \Omega. \]
scientific article; zbMATH DE number 1038449
scientific article