G-convergence of elliptic operators
Let \((\epsilon_ h)\) be a numerical sequence, where \(\epsilon_ h\geq 0\) and \(\epsilon_ h\to 0\), let \(\Omega\) be a bounded domain in \({\mathbb{R}}^ n\), and let \(m>k>0\) be integers. Consider operators of the form \(L_ h=\epsilon_ h^{2(m-k)}B_ h+A_ h\) where \[ B_ h=(-1)^ m\sum_{| \alpha | =| \beta | =m}D^{\alpha}(a^ h_{\alpha \beta}(x)D^{\beta}),\quad A_ h=\sum_{| \alpha |,| \beta | \leq k}(-1)^{| \alpha |}D^{\alpha}(a^ h_{\alpha \beta}(x)D^{\beta}). \] The coefficients \(a^ h_{\alpha \beta}\) are measurable and equi-bounded on \(\Omega\) and satisfy the inequalities \(<B_ hu,u>\geq c\| u\|^ 2_{H^ m(\Omega)},\) \(<A_ hu,u>\geq c\| u\|^ 2_{H^ k(\Omega)}\) for every \(u\in C_ 0^{\infty}(\Omega)\), where c is a fixed positive constant. The sequence \(L_ h\) is said to be G-convergent to \(A=\sum_{| \alpha |,| \beta | \leq k}(-1)^{| \alpha |}D^{\alpha}(a_{\alpha \beta}(x)D^{\beta})\) in \(\Omega\) [see also \textit{S. Spagnolo}, Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 22 (1968), 571-597 (1969; Zbl 0174.421) and \textit{E. De Giorgi} and \textit{S. Spagnolo}, Boll. Unione Mat. Ital., IV. Ser. 8, 391-411 (1973; Zbl 0274.35002)] if for every \(f\in H^{-k}(\Omega)\) we have \(L_ h^{-1}f\to A^{-1}f\) weakly in \(H^ k_ 0(\Omega)\). The sequence \(L_ h\) is said to be strongly G-convergent to A in \(\Omega\) if for every \(u\in H^ k_ 0(\Omega)\) and every sequence \((u_ h)\) converging to u weakly in \(H^ k_ 0(\Omega)\) we have that the ''generalized gradients'' \(\Gamma^ h_{\alpha}(u_ h)=\sum_{| \beta | \leq k}a^ h_{\alpha \beta}(x)D^{\beta}u_ h\) weakly converging in \(L^ 2(\Omega)\) to \(\Gamma_{\alpha}(u)=\sum_{| \beta | \leq k}a_{\alpha \beta}(x)D^{\beta}u.\) The following results are proved. Theorem 1. Every sequence \((L_ h)\) of operators contains a strongly G-convergent subsequence in \(\Omega\). Theorem 2. If \(L_ h\) is strongly G-convergent to A in \(\Omega\), then \(L_ h\) is strongly G-convergent to A in each subdomain \(\Omega\) '\(\subseteq \Omega\). Theorem 3. Let \(L_ h\) be strongly G-convergent to A in \(\Omega\) and let \(u_ h\) be such that \(L_ hu_ h=f\in H^{- k}(\Omega)\); \(u_ h\to u\) weakly in \(H^ k(\Omega)\); \(\epsilon_ h^{2(m-k)}\| u_ h\|^ 2_{H^ m(\Omega)}+\| u_ h\|^ 2_{H^ k(\Omega)}\leq\) constant. Then we have \(Au=f\) and \(\Gamma^ h_{\alpha}(u_ h)\to \Gamma_{\alpha}(u)\) weakly in \(L^ 2(\Omega)\).
scientific article; zbMATH DE number 3884586