turkmath.org

A Sobolev space is a vector space of functions equipped with a norm that is a combination of $L^{p}$-norms of the function together with its "derivatives" up to a given order. Intuitively, a Sobolev space is a space of functions possessing sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function. Their importance comes from the fact that weak solutions of some important partial differential equations exist in appropriate Sobolev spaces. We will mainly focus on the $L^{2}$-based Sobolev spaces, known as $H^{s}$ spaces, which are also Hilbert spaces because of the nature of $L^{2}$ spaces. We will also state and prove the Sobolev Embedding theorem in a special case.