2018/04/16 by Vladimir Gutlyanskiî, Gutlyanskii, Vladimir, Olga Nesmelova +3
Computer Science · Mathematics · #30C62 #30E25 #31A05 #31A20 #31A25 #31B25 #31C05 #34M50 #35F45 #35J61 (Primary) #35Q15 (Secondary) #Advanced Mathematical Modeling in Engineering #Analytic and geometric function theory #Complex Variables (math.CV) #Differential Equations and Boundary Problems #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1804.05875
openalex publication_date 2018/04/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the Dirichlet problem for the semi--linear partial differential equations \rm div (A∇ u)=f(u) in simply connected domains D of the complex plane \mathbb C with continuous boundary data. We prove the existence of the weak solutions u in the class C∩ W1,2\rm loc(D) if a Jordan domain D satisfies the quasihyperbolic boundary condition by Gehring--Martio. An example of such a domain that fails to satisfy the standard (A)--condition by Ladyzhenskaya--Ural'tseva and the known outer cone condition is given. We also extend our results to simply connected non-Jordan domains formulated in terms of the prime ends by Caratheodory. Our approach is based on the theory of the logarithmic potential, singular integrals, the Leray--Schauder technique and a factorization theorem in \citeGNR2017. This theorem allows us to represent u in the form u=U∘ω, where ω(z) stands for a quasiconformal mapping of D onto the unit disk \mathbb D, generated by the measurable matrix function A(z), and U is a solution of the corresponding quasilinear Poisson equation in the unit disk \mathbb D. In the end, we give some applications of these results to various processes of diffusion and absorption in anisotropic and inhomogeneous media.