Hilbert and Poincare problems for semi-linear equations in domains with rectifiable boundaries

TL;DR

This paper develops methods to solve boundary-value problems with measurable data for semi-linear equations in domains with rectifiable boundaries, extending classical PDE approaches with geometric interpretations.

Contribution

It introduces continuous operators for nonclassical solutions to boundary problems in complex domains, applying geometric boundary value interpretation to semi-linear equations.

Findings

Existence of solutions for nonlinear Vekua type equations in rectifiable domains.

Solutions to Poincare boundary problems with measurable data for Poisson equations.

Applications to physical phenomena like absorption, plasma states, and combustion.

Abstract

In the last paper \cite{R7}, it was studied Hilbert, Poincare and Neumann boundary-value problems with arbitrary measurable data for generalized analytic functions and generalized harmonic functions with applications to the relevant problems of mathematical physics. The present paper is devoted to the study of the boundary-value problems with arbitrary measurable boundary data in domains with rectifiable boundaries for the corresponding semi-linear equations with suitable nonlinear sources. For this purpose, here it is constructed completely continuous operators generating nonclassical solutions of the Hilbert and Poincare boundary-value problems with arbitrary measurable data for the Vekua type equations and the Poisson equations, respectively. On this base, it is first proved the existence of solutions of the Hilbert boundary-value problem with arbitrary measurable data in any domains with rectifiable boundaries for the nonlinear equations of the Vekua type. It is necessary to note that our approach is based on the geometric interpretation of boundary values as angular (along nontangential path) limits in comparison with the classical variational approach in PDE. The latter makes it is also possible to obtain the theorem on the existence of nonclassical solutions of the Poincare boundary-value problem on the directional derivatives and, in particular, of the Neumann problem with arbitrary measurable data to the Poisson equations with nonlinear sources in Jordan domains with rectifiable boundaries. As consequences, then it is given a series of applications of this theorem to some problems of mathematical physics describing, for instance, such phenomena as physical and chemical absorption with diffusion, plasma states, stationary burning etc.