Toward the theory of Dirichlet problem for the degenerate Beltrami equations

TL;DR

This paper develops a theoretical framework for solving the Dirichlet problem for degenerate Beltrami equations, establishing existence criteria for regular and multi-valued solutions with applications to potential theory in complex media.

Contribution

It introduces new existence criteria and representations for solutions to degenerate Beltrami equations, extending the theory to multi-valued solutions and potential applications.

Findings

Existence of regular homeomorphic solutions with hydrodynamic normalization.

Criteria for solutions to the Dirichlet problem in simply connected domains.

Application to potential theory in anisotropic media.

Abstract

In this article, first we give a general lemma on the existence of regular homeomorphic solutions $f$ with the hydrodynamic normalization $f(z)=z+o(1)$ as $z\to\infty$ to the degenerate Beltrami equations $\overline{\partial}f=\mu\,\partial f$ in $\mathbb C$ whose complex coefficients $\mu$ have compact supports. On this basis, we establish criteria for existence and representation of regular discrete open solutions for the Dirichlet problem with continuous data to degenerate Beltrami equations in arbitrary simply connected bounded domains $D$ in $\mathbb C$. Moreover, we obtain similar criteria for the existence of multi-valued solutions $f$ in the spirit of the theory of multi-valued analytic functions in arbitrary bounded domains $D$ in $\mathbb C$ with no boundary component degenerated to a single point. Note that the latter request is necessary and that the real parts $u$ of such solutions $f$ are the so-called $A-$harmonic functions, i.e., single-valued continuous weak solutions of elliptice quations ${\rm div}\, (A\cdot\nabla u)=0$ with matrix-valued coefficients $A$ associated with $\mu$. Thus, the results can be applied to potential theory in anisotropic and inhomogeneous media.