BMO and Dirichlet problem for semi-linear equations in the plane

Abstract

First we study 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 $\varphi \colon \partial D\to\mathbb R$ for semi-linear Beltrami equations $\omega_{\overline{z}}-\mu(z) \omega_z=\sigma (z)q({\rm Re}\,\omega(z))$. We assume here that $D$ is an arbitrary bounded domain of the complex plane $\mathbb C$, which is either simply connected or has no boundary components degenerated to a single point, and that the equations are locally uniform elliptic with possible singularities at the boundary. For $\sigma\in L_p(D)$, $p> 2$, with compact support, and continuous $q\colon \mathbb R\to\mathbb C$, $q(t)/t\to 0$ as $t\to\infty$, we establish a series of effective criteria for existence of solutions of the Dirichlet problem in terms of BMO, FMO, Calderon-Zygmund, Lehto and Orlicz integral means. We also establish representation and regularity of these solutions. Then, we prove existence, representation and regularity of weak solutions of the Dirichlet problem $u(z)\to\varphi(\zeta)$ as $z\to\zeta,$ $z\in D,\zeta\in \partial D,$ to semi-linear Poisson type equations ${\rm div} [A(z)\nabla u(z)] = g(z)Q(u(z))$ for $g\in L_p(D)$, $p> 1$, with compact support, and continuous $Q\colon \mathbb R\to\mathbb R$, $Q(t)/t\to 0$ as $t\to\infty$. We also assume here conditions on the matrix coefficients $A(z)$ guaranteing locally uniform ellipticity of these equations. Finally, we give examples of possible applications of the obtained results to various semi-linear equations of the mathematical physics in anisotropic and inhomogeneous media.