Quasiconformal Mappings in the Theory of Semi-linear Equations

Abstract

We study the Dirichlet problem with continuous boundary data in simply connected domains D of the complex plane for the semi-linear partial differential equations whose linear part has the divergent form. We prove that if a Jordan domain D satisfies the so-called quasihyperbolic boundary condition, then the problem has regular (continuous) weak solutions whose first generalized derivatives by Sobolev are integrable in the second degree. We give a suitable example of a Jordan domain with the quasihyperbolic boundary condition that fails to satisfy both the well-known (A)-condition and the outer cone condition. We also extend these results to some non-Jordan domains in terms of the prime ends by Caratheodory. The proofs are based on our factorization theorem established earlier. This theorem allows us to represent solutions of the semilinear equations in the form of composition of solutions of the corresponding quasilinear Poisson equation in the unit disk and quasiconformal mapping of D onto the unit disk generated by the measurable matrix function of coefficients. In the end we give applications to relevant problems of mathematical physics in anisotropic inhomogeneous media.