Singular Integral Neumann Boundary Conditions for Semilinear Elliptic PDEs

Abstract

In this article, we discuss semilinear elliptic partial differential equations with singular integral Neumann boundary conditions. Such boundary value problems occur in applications as mathematical models of nonlocal interaction between interior points and boundary points. Particularly, we are interested in the uniqueness of solutions to such problems. For the sublinear and subcritical case, we calculate, on the one hand, illustrative, rather explicit solutions in the one-dimensional case. On the other hand, we prove in the general case the existence and—via the strong solution of an integro-PDE with a kind of fractional divergence as a lower order term—uniqueness up to a constant.

Conditional stability estimate for an ill-posed elliptic equation by using nonlocal conditions

“…For readers interested in the ill-posedness of elliptical PDEs in parameter estimation, i.e. inverse problems, or in non-local boundary conditions, one can mention the following recent papers: [18,19]. The rest of the paper is organized as follows: In Section 2, notations and some definitions are given, and we also consider the formulation of a solution of problem (P) and its ill-posed property.…”