On the Hm-Regularity for Second Order Elliptic Equations

Abstract

Our primary objective in this paper is to discuss the Hm-Regularity for second order elliptic equations over Sobolev Spaces. We here consider the cases m = 1 and m≥ 2 separately. We revisit some elementary concepts in Functional Analysis and Abstract Harmonic Analysis before providing a proper definiton to the notion of weak solution of a Dirichlet Problem. While towards the later stages, we shall classify different types of regularity conditions, the main focus lies upon deducing appropriate ellipticity conditions in support of commenting about the existence and uniqueness of the weak solutions to a given problem. Appropriate references are provided in the bibliography section to facilitate further reading for ardent readers and researchers in this field