Dirichlet Problems in Sobolev Spaces

The purpose of this chapter is to formulate the homogeneous Dirichlet problemDirichlet problem in the framework of \(L^{p}\) Sobolev spacesSobolev space. We state interior and global a priori estimates for the Dirichlet problem (Theorems 12.1 and 12.2) that will play an essential role in the proof of the unique solvability theorem for the homogeneous Dirichlet problem (Theorem 15.1) in Chap. 15. The results discussed here are adapted from Chiarenza–Frasca–Longo [34, 35] and Chiarenza [33]. Our approach can be traced back to the pioneering work of J. Schauder [127, 128] on the Dirichlet problem for second order, elliptic differential operators.