Sobolev Space

Sobolev spaces are a fundamental concept in functional analysis and partial differential equations (PDEs). They generalize the idea of differentiability and integrability for functions and provide a natural setting for studying solutions to PDEs.

Definition

Given an open subset $Ω \subset R^{n}$ , a Sobolev space $W^{k, p} (Ω)$ is defined as the set of functions $u \in L^{p} (Ω)$ (Lebesgue space) such that all weak derivatives of $u$ up to order $k$ also belong to $L^{p} (Ω)$ . That is,

W^{k, p} (Ω) = {u \in L^{p} (Ω) : D^{α} u \in L^{p} (Ω) for all | α | \leq k},

where $α$ is a multi-index and $D^{α} u$ denotes the weak derivative.

Sobolev spaces are normed spaces. The norm on $W^{k, p} (Ω)$ is given by:

∥ u ∥_{W^{k, p} (Ω)} = {(\sum_{| α | \leq k} ∥ D^{α} u ∥_{L^{p} (Ω)}^{p})}^{1 / p} for 1 \leq p < \infty,

and by the usual supremum norm when $p = \infty$ . Under this norm, $W^{k, p} (Ω)$ is complete, hence a Banach space.

In the special case where $p = 2$ , the space $W^{k, 2} (Ω)$ , often denoted $H^{k} (Ω)$ , is a Hilbert space due to the inner product:

⟨ u, v ⟩_{H^{k}} = \sum_{| α | \leq k} \int_{Ω} D^{α} u (x) D^{α} v (x) d x .

Special Cases

$W^{1, 2} (Ω) = H^{1} (Ω)$ is a Hilbert space commonly used in the theory of weak solutions to PDEs.
For $p = \infty$ , the space $W^{k, \infty} (Ω)$ consists of functions with essentially bounded derivatives up to order $k$ .

Importance

Sobolev spaces:

Allow analysis of functions with limited smoothness.
Provide the framework for weak (distributional) solutions of PDEs.
Support embedding theorems and compactness results critical in variational methods.