On Neumann and Poincare problems for Laplace equation

TL;DR

This paper proves the existence of infinitely many nonclassical solutions to Neumann and Poincare problems for harmonic functions in Jordan rectifiable domains with arbitrary boundary data, expanding the understanding of boundary value problems.

Contribution

It establishes the existence of nonclassical solutions with infinite-dimensional solution spaces for Neumann and Poincare problems in general Jordan rectifiable domains.

Findings

Existence of nonclassical solutions for Neumann problems in Jordan rectifiable domains.

Solutions have infinite-dimensional spaces.

Results extend to Poincare problems with directional derivatives.

Abstract

It is proved the existence of nonclassical solutions of the Neumann problem for the harmonic functions in the Jordan rectifiable domains with arbitrary measurable boundary distributions of normal derivatives. The same is stated for the partial case of the Poincare problem on directional derivatives. Moreover, it is shown that the spaces of the found solutions have the infinite dimension.