Sobolev Spaces and Linear Elliptic Partial Differential Equations

Abstract

The cardinal goal to the study of theory of Partial Differential Equations (PDEs) is to insure or find out properties of solutions of PDE that are not directly at- tainable by direct analytical means. Certain function spaces have certain known properties for which solutions of PDEs can be classified. As a result, this work critically looked into some function spaces and their properties. We consider extensively, Lp − spaces, distribution theory and sobolev spaces. The emphasis is made on sobolev spaces, which permit a modern approach to the study of differential equations, defined as W m,p (Ω) = {u ∈ Lp (Ω), | Dα u ∈ Lp (Ω), for all α ∈ Nn : |α| ≤ m}. (1) it is provided with the norm: u W m,p (Ω) Dα u = (2) Lp (Ω) . |α|≤m for 1 ≤ p < ∞, we have 1 p  u W m,p (Ω) Dα u = p  Lp (Ω) . (3) |α|≤m Finally,we also have u W m,p (Ω) = max{ Dα u Lp (Ω) : |α| ≤ m} (4) This is the key to the entire work, even though others spaces are ingredients. The study is based on variational formulation of some Boundary Value Problems using some known thoerem (Lax-Milgram Theorem) to ascertain the existence and uniqueness of solution to such linear PDEs. v We do not consider all types of PDEs (Parabolic, Hyperbolic and Elliptic).