Linear Elliptic Differential Equations

Abstract

At first, we transform boundary value problems for elliptic differential equations with two independent variables into a Riemann-Hilbert boundary value problem in Section 1. The latter can be solved by the integral equation method due to I. N. Vekua in Section 2 and Section 3. Then, we derive potential-theoretic estimates for the solution of Poisson’s equation in Section 4. For use in Chapter 12 we prove corresponding inequalities for solutions of the inhomogeneous Cauchy-Riemann equation. For elliptic differential equations in n variables we solve the Dirichlet problem by the continuity method in the classical function space $C^{2+\alpha}(\overline{\Omega})$; see Section 5 and Section 6. The necessary Schauder estimates are completely derived in the last paragraph.