Neumann and Poincare problems for Poisson's equations with measurable data

TL;DR

This paper proves the existence of solutions to boundary value problems for generalized harmonic functions with arbitrary measurable data, extending classical results to more general domains and boundary conditions.

Contribution

It introduces new existence theorems for boundary value problems with measurable data for generalized harmonic functions in arbitrary Jordan domains.

Findings

Existence of solutions for Hilbert boundary value problem with measurable data.

Existence of solutions for Neumann and Poincare problems with measurable data.

Application of geometric interpretation to boundary value problems.

Abstract

The research of the Dirichlet problem with arbitrary measurable datafor harmonic functions is due to the famous dissertation of Luzin. The present paper is devoted to various theorems on the existence of nonclassical solutions of the Hilbert and Riemann boundary value problems with arbitrary measurable data for generalized analytic functions by Vekua and the corresponding applications to the Neumann and Poincare problems for generalized harmonic functions. Our approach is based on the geometric (theoretic-functional) interpretation of boundary values in comparison with the classical operator approach in PDE. Here it is proved the existence theorems on solutions of the Hilbert boundary value problem with arbitrary measurable data for generalized analytic functions in arbitrary Jordan domains with rectifiable boundaries in terms of the na\-tu\-ral parameter and angular (nontangential) limits, moreover, to arbitrary Jordan domains in terms of harmonic measure and principal asymptoticvalues. Moreover, it is established the existence theorems on solutions for the appropriate boun\-da\-ry value problems of Hilbert and Riemann with arbitrary measurable data along the Bagemihl--Seidel systems of Jordan arcs terminating at the boundary in arbitrary domains whose boundaries consist of finite collections of rectifiable Jordan curves. On this basis, it is established the corresponding existence theorems for the Poincare boundary value problem on the directional derivatives and, in particular, for the Neumann problem with arbitrary measurable data to the Poisson equations.