On the solvability of the first nonlocal boundary value problem for an elliptic equation
The author studies problems of existence, uniqueness and sign-definiteness of the classical solutions of the problem \[ Lu(x)= f(x), \quad x \in D; \qquad u(x)- \beta(x) u(\sigma x)= \psi(x), \quad x \in S, \] where \(D\) is a bounded domain in \(\mathbb R^n\) with boundary \(S\), \(\sigma \) is a single-valued continuous mapping, \(\beta, \psi \in C(S)\). A lot of papers deals with the solvability of boundary value problems for elliptic equations with nonlocal boundary conditions in various function spaces. In this note more general objects are considered, we point out the problem of the existence of solutions in situations in which the support of a nonlocal data meet the boundary of the domain. Under natural assumptions on the smoothness of \(\beta, \sigma\) and the coefficients of \(L\), the treated problem is Fredholm provided that either \(\sigma\) has no attractors on \(S\), or \(\sigma \) generates an attractor on \(S\) and the spectral radius of the ``generalized weighted shift'' operator occurring in the boundary condition \[ (Az)(x)= \beta (x) z(\sigma x), \quad x \in S, \] is less than unity. The author obtains semieffective conditions, in terms of test functions, for the unique solvability of the problem.
scientific article; zbMATH DE number 5611090
scientific article