On the $H^{m}$-Regularity for Second Order Elliptic Equations
Résumé
This article serves as one of the integral topics of my Masters Thesis under the kind supervision of Prof. K. Sreenadh at IIT Delhi, India. Our primary objective in this paper is to discuss the $H^{m}$-Regularity for second order elliptic equations over Sobolev Spaces. We here consider the cases $m=1$ and $m\geq 2$ separately. We revisit some elementary concepts in \textit{Functional Analysis} and \textit{Abstract Harmonic Analysis} before providing a proper definiton to the notion of weak solution of a Dirichlet Problem.\par
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.
Mots clés
Fourier Transform
Sobolev Space
Schwartz Space
Hilbert Space
Weak Derivative
Trace Theorem
Integration by Parts
Outward Normal
Bounded Domain
Boundary Value Problem
Dirichlet Problem
Weak Solution
Uniqueness
Existence
Interior Regularity
Boundary Regularity
Higher Regularity
Elliptic Equation
Elliptic Operator
Fourier Transform Sobolev Space Schwartz Space Hilbert Space Weak Derivative Trace Theorem Integration by Parts Outward Normal Bounded Domain Boundary Value Problem Dirichlet Problem Weak Solution Uniqueness Existence Interior Regularity Boundary Regularity Higher Regularity Elliptic Equation Elliptic Operator . 2020 MSC: Primary 35A01 35A02 35A15 35J20 35D30 . Secondary 35B65 35A20 35J25 35J35 58J05
Elliptic Operator . 2020 MSC: Primary 35A01
35A02
35A15
35J20
35D30 . Secondary 35B65
35A20
35J25
35J35
58J05
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