Chapter 23 - Appendix on elliptic boundary value problems in Sobolev and Hölder spaces

Abstract

This chapter provides basic notions and results on elliptic boundary value problems (BVPs). A domain Ω satisfies the segment property if for every x ∈ Ω, there exists an open set Ux ϶x and a nonzero vector yx such that if z ∈ Ω ∩Ux then z +tyx ∈ Ω for 0 <t < 1. A domain satisfying the segment property has by necessity (n –1)-dimensional boundary and cannot lie on both sides of its boundary. This chapter discusses concepts related to Sobolev spaces on manifolds without boundary, Sobolev spaces on the torus Tn, and Sobolev spaces on the sphere Sn−l. This chapter elaborates on Holder spaces, Sobolev spaces on manifolds with boundary, and Trace theorem. The general Sobolev-type embedding theorems are discussed. Regular elliptic boundary value problems and regular elliptic boundary value problems in Rn+1 are elaborated. Concepts related to boundary operators, adjoint problem and Green formula are presented. The chapter also discusses boundary operators in neighboring domains, a priori estimates in Sobolev spaces, Fredholm operator, and elliptic boundary value problems in Holder spaces. Schauder's continuous parameter method is also presented in the chapter.