Dirichlet problem with measurable data for semilinear equations in the plane

Abstract

The study of the Dirichlet problem with arbitrary measurable data for harmonic functions in the unit disk  goes to the known dissertation of Luzin. His result was formulated in terms of angular limits (along nontangent paths) that are a traditional tool for the research of the boundary behavior in the geometric function theory. With a view to further developments of the theory of boundary-value problems for semilinear equations, the present paper is devoted to the Dirichlet problem with arbitrary measurable (over logarithmic capacity) boundary data for quasilinear Poisson equations in such Jordan domains. For this purpose, it is firstly constructed completely continuous operators generating nonclassical solutions of the Dirichlet boundary-value problem with arbitrary measurable data for the Poisson equations ΔU =G over the sources G ∈Lp, p >1. The latter makes it possible to apply the Leray— Schauder approach to the proof of theorems on the existence of regular nonclassical solutions of the measurable Dirichlet problem for quasilinear Poisson equations of the form ΔU (z ) =H (z )⋅Q(U (z )) for multipliers H ∈Lp with p >1 and continuous functions Q: R→R with Q(t ) /t →0 as t →∞. These results can be applied to some specific quasilinear equations of mathematical physics, arising under a modeling of various physical processes such as the diffusion with absorption, plasma states, stationary burning, etc. These results can be also applied to semilinear equations of mathematical physics in anisotropic and inhomogeneous media.

Hilbert problem with measurable data for semilinear equations of the Vekua type

“…Later on, to apply the approach of Leray-Schauder for extending Theorem 1 in [5] to the generalized analytic functions, satisfying nonlinear equations of the Vekua type, we need just the complete continuity of such correspondence. Now, we show that, fixing only one antiderivative Ф for the function φ from Lemma 1 in [2], it is possible to obtain a completely continuous Hilbert operator. For this purpose, let us analyze the construction of solutions of Eq.…”

“…on  form a function :      that is measurable with respect to the logarithmic capacity. By Remark 2 in [2], the second analytic function B can be obtained in the form…”

“…where Ф and Ф g are antiderivatives of the functions of e   and * e   from Lemma 1 in [2], respectively. Note that the analytic functions A and   do not depend on the sources g at all.…”

“…, and all g with large enough p g   . Note also that, by Lemma 1 in [2], we are able always to choose Ф for any φ, including 0   , which is not identically 0 on .…”

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The basic part of definitions and historic comments can be found in papers [1] and [2]. However, let us recall here that a completely continuous mapping from a metric space M 1 into a metric space M 2 is defined as a continuous mapping on M 1 which takes bounded subsets of M 1 into relatively compact ones of M 2 , i.e., with compact closures in M 2 .

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