\(L^{p}\) Theory of Elliptic Boundary Value Problems

In this chapter we study the non-homogeneous general Robin problem under the conditions (H.1) and (H.2). Moreover, we assume that the conditions (SC) are satisfied. Section 8.1 is devoted to the study of the classical surface and volume potentials arising in boundary value problems for elliptic differential operators, in terms of pseudo-differential operators. In Sec. 8.2 we consider the Dirichlet problem in the framework of Sobolev spaces of \(L^{p}\) type. This is a modern version of the classical potential approach to the Dirichlet problem. In Sect. 8.3 we formulate elliptic boundary value problems in the framework of \(L^{p}\) Sobolev spaces. The pseudo-differential operator approach to elliptic boundary value problems can be traced back to the pioneering work of Calderón in early 1960s. In Sect. 8.4, by using the Poisson kernel or Poisson operator for the Dirichlet problem and the Green operator for the Neumann problem we show that our general Robin problem can be reduced to the study of a pseudo-differential operator T, called the Fredholm boundary operator. The virtue of this reduction to the boundary is that there is no difficulty in taking adjoints or transposes after restricting the attention to the boundary, whereas boundary value problems in general do not have adjoints or transposes. This allows us to discuss the existence theory more easily.