In this chapter we study boundary value problems for elliptic partial differential equations. As we have seen in Chapt. 1 such equations are central in both theory and application of partial differential equations; they describe a large number of physical phenomena, particularly modelling stationary situations, and are stationary limits of evolution equations. After some preliminaries in Sect. 3.1 we begin by showing a maximum principle in Sect. 3.2. In the same way as for the two-point boundary value problem in Chapt. 2 this may be used to show uniqueness and continuous dependence on data for boundary value problems. In the following Sect. 3.3 we show the existence of a solution of Dirichlet’s problem for Poisson’s equation in a disc with homogeneous boundary conditions, using an integral representation in terms of Poisson’s kernel. In Sect. 3.4 similar ideas are employed to introduce fundamental solutions of elliptic equations, and we illustrate their use by constructing a Green’s function. Another important approach, presented in Sect. 3.5, is based on a variational formulation of the boundary value problem and simple functional analytic tools. In Sect. 3.6 we discuss briefly the Neumann problem, and in Sect. 3.7 we describe some regularity results.
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