Toward the theory of semi-linear equations

TL;DR

This paper develops a factorization theorem for semilinear PDEs with divergence form linear parts, linking solutions to isotropic equations and quasiconformal mappings, thus easing boundary regularity constraints.

Contribution

It introduces a novel factorization theorem that represents solutions of semilinear divergence form equations via isotropic solutions and quasiconformal mappings, expanding boundary analysis capabilities.

Findings

Representation of solutions as compositions with quasiconformal maps

Removal of boundary regularity restrictions in boundary value problems

Establishment of a factorization theorem for semilinear PDEs

Abstract

In this paper we study the semilinear partial differential equations in the plane the linear part of which is written in a divergence form. The main result is given as a factorization theorem. This theorem states that every weak solution of such an equation can be represented as a composition of a weak solution of the corresponding isotropic equation in a canonical domain and a quasiconformal mapping agreed with a matrix-valued measurable coefficient appearing in the divergence part of the equation. The latter makes it possible, in particular, to remove the regularity restrictions on the boundary in the study of boundary value problems for such semilinear equations.