On Neumann and Poincare problems in mathematical physics

TL;DR

This paper proves the existence of infinite-dimensional spaces of nonclassical solutions to Neumann and Poincare problems for generalized Laplace equations in complex media, with boundary data measurable in logarithmic capacity.

Contribution

It demonstrates the existence and infinite-dimensionality of nonclassical solutions for generalized Laplace equations in complex media with arbitrary boundary data.

Findings

Existence of nonclassical solutions in anisotropic, nonhomogeneous media

Solutions form infinite-dimensional spaces

Applicable to domains with almost smooth boundaries

Abstract

It is proved the existence of nonclassical solutions of the Neumann and Poincare problems for generalizations of the Laplace equation in anisotropic and nonhomogeneous media in almost smooth domains with arbitrary boundary data that are measureable with respect to logarithmic capacity. Moreover, it is shown that the spaces of such solutions have the infinite dimension.