The Dirichlet problem for the Beltrami equations with sources

Author:

Gutlyanskii Vladimir1,Ryazanov Vladimir1,Nesmelova Olga1,Yakubov Eduard2

Affiliation:

1. Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Sloviansk Institute of Mathematics of the NAS of Ukraine, Kiev, Ukraine

2. Holon Institute of Technology, Holon, Israel

Abstract

The paper is devoted to the study of the Dirichlet problem ${\rm{Re}}\,\omega(z)\to\varphi(\zeta)$ as $z\to\zeta,$ $z\in D,\zeta\in \partial D,$ with continuous boundary data $\varphi :\partial D\to\mathbb R$ 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 have established a series of effective integral criteria of the BMO, FMO, Calderon-Zygmund, Lehto, and Orlicz types on the singularities of the equations at the boundary for the 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 have proved the existence, representation, and regularity of weak solutions of the Dirichlet problem in such domains for the Poisson-type equation ${\rm div} [A(z)\nabla\,u(z)] = g(z)$, 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.

Publisher

Institute of Applied Mathematics and Mechanics of the National Academy of Sciences of Ukraine

Subject

Ocean Engineering

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