On Dirichlet problem for degenerate Beltrami equations with sources

Abstract

The present paper is devoted to the study of the Dirichlet problem $\mathrm{Re}\omega (z)\to \phi (\zeta )$ as $z\to \zeta ,$ $z\in D,\zeta \in \partial D,$ with continuous boundary data $\phi :\partial D\to \mathbb{R}$ for Beltrami equations ${\omega }_{\bar{z}}=\mu (z){\omega }_z+\sigma (z)$, $|\mu (z)|<1$ a.e., with sources $\sigma :D\to \mathbb{C}$ in the case of locally uniform ellipticity. In this case, we establish a series of effective integral criteria of the type of BMO, FMO, Calderon-Zygmund, Lehto and Orlicz on singularities of the equations at the boundary for existence, representation and regularity of solutions in arbitrary bounded domains $D$ of the complex plane $\mathbb{C}$ with no boun\-da\-ry component degenerated to a single point for sources $\sigma$ in $L_p(D)$, $p>2$, with compact support in $D$. Moreover, we prove in such domains existence, representation and regularity of weak solutions of the Dirichlet problem for the Poisson type equation $\mathrm{div}[A(z)\nabla u(z)]=g(z)$ whose source $g\in L_p(D)$, $p>1$, has compact support in $D$ and whose mat\-rix valued coefficient $A(z)$ guarantees its locally uniform ellipticity.

31 pages. arXiv admin note: substantial text overlap with arXiv:2111.10375