G-convergence of elliptic operators in nondivergence form
In the smooth bounded domain \({\mathcal D}\) of the Euclidean space \(E_ d\) a sequence of elliptic operators \(L_ n\), \(n=1,2,...\), of the type \[ L_ n=a_ n^{ij}(x)\partial^ 2/\partial x^ i\partial x^ j+b^ i_ n(x)\partial /\partial x^ i+c_ n(x), \] is given, where \(c_ n\leq 0\), \(a_ n^{ij}\), \(b^ i_ n\), \(c_ n\) are uniformly bounded in n with respect to \({\mathcal D}\), \(a_ n^{ij}=a_ n^{ji}\) and the matrices \((a_ n^{ij})\) are non-degenerate \({\mathcal D}\) with respect to n, x. If the coefficients of \(L_ n\) are smooth in \(\bar {\mathcal D}\), then it is known, that for any function \(f\in {\mathcal L}_ p({\mathcal D})\) there exists a unique function \(u_ n\in \overset\circ W^ 2_ p({\mathcal D})\) such that \(L_ nu_ n=f\). The paper gives a positive solution of the problem to select a subsequence, which converges to some function u with \(Lu=f\), where L is a constructed elliptic operator.
scientific article; zbMATH DE number 3944382