On boundary-value problems for semi-linear equations in the plane

Abstract

The study of the Dirichlet problem with arbitrary measurable data for harmonic functions in the unit disk $\mathbb{D}$ is due to the dissertation of Luzin. Later on, the known monograph of Vekua was devoted to boundary-value problems only with H\"{o}lder continuous data for generalized analytic functions, i.e., continuous complex-valued functions $f(z)$ of the complex variable $z=x+iy$ with generalized first partial derivatives by Sobolev satisfying equations of the form $\partial _{\bar{z}}f\,+\,af\,+\ b{% \overline{f}}\,=\,c\,$, where the complex-valued functions $a,b$, and $c$ are assumed to belong to the class $L^{p}$ with some $p>2$ in smooth enough domains $D$ in $\mathbb{C}$. Our last paper \cite{GNRY} contained theorems on the existence of nonclassical solutions of the Hilbert boundary-value problem with arbitrary measurable data (with respect to logarithmic capacity) for generalized analytic functions $f:D\to\mathbb{C}$ such that $\partial_{\bar z}f = g$ with the real-valued sources. On this basis, the corresponding existence theorems were established for the Poincar\'e problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations $\triangle\, U=G\in L^p, p>2$, with arbitrary measurable boundary data over logarithmic capacity. The present paper is a natural continuation of the article \cite{GNRY} and includes, in particular, theorems on the existence of solutions for the Hilbert boundary-value problem with arbitrary measurable data for the corresponding nonlinear equations of the Vekua type $\partial _{\bar{z}% }f(z)=h(z)q(f(z))$. On this basis, existence theorems were also established for the Poincar\'{e} boundary-value problem and, in particular, for the Neumann problem for the nonlinear Poisson equations of the form $\triangle \,U(z)=H(z)Q(U(z))$ with arbitrary measurable boundary data over logarithmic capacity. The Dirichlet problem was investigated by us for the given equations, too. Our approach is based on the interpretation of boundary values in the sense of angular (along nontangential paths) limits that are a conventional tool of the geometric function theory. As consequences, we give applications to some concrete semi-linear equations of mathematical physics arising from modelling various physical processes. Those results can also be applied to semi-linear equations of mathematical physics in anisotropic and inhomogeneous media.