A condition for the complete continuity of operators arising in nonlocal problems for elliptic equations.
The author studies the Dirichlet problem for the uniformly elliptic operator \[ D_i(a_{ij}(x)D_ju)=0, \quad x\in Q,\qquad u|_{\partial Q}=Bu+u_0 \] where \((Bu)(x)=\int_{\overline Q}b(x,y)u(y)\,d\nu_x(y),\) \(x\in \partial Q.\) The domain \(Q\) supposed to be bounded with a smooth boundary and the normal to \(\partial Q\) satisfies the Dini condition, the coefficients are measurable and bounded in \(Q\) and satisfy the Dini condition on the boundary. It is introduced a functional space and is described the basic properties of the solutions of the problem under consideration with a boundary function from \(L^2(\partial Q)\) in terms of inclusion in this space. It is established a condition for the complete continuity of integral operators from the space introduced to \(L^2(\partial Q)\) that are determined by measures close to the Carleson measures. This makes it possible to substantially strengthen results on Fredholmness of a wide class of nonlocal problems.
scientific article; zbMATH DE number 2074619