Sobolev Spaces - Key Properties

Sobolev spaces possess embedding, compactness, and density properties that are fundamental for proving existence of weak solutions and regularity theory.

DefinitionSobolev Embedding Theorem

If $kp < n$ , then $W^{k,p}(\mathbb{R}^n) \hookrightarrow L^q(\mathbb{R}^n)$ continuously for $1/q = 1/p - k/n$ . If $kp > n$ , then $W^{k,p}(\mathbb{R}^n) \hookrightarrow C^{m,\alpha}(\mathbb{R}^n)$ where $m + \alpha = k - n/p$ (Hölder spaces).

For $kp = n$ (critical case), $W^{k,p} \hookrightarrow L^q$ for all $q < \infty$ but not $L^\infty$ .

These embeddings quantify how many derivatives (in $L^p$ ) are needed for integrability or continuity, connecting differential and integral properties.

DefinitionRellich-Kondrachov Compactness

If $\Omega$ is bounded with smooth boundary and $kp < n$ , then the embedding: $W^{k,p}(\Omega) \hookrightarrow L^q(\Omega)$ is compact for any $q < p^* = \frac{np}{n-kp}$ (the Sobolev conjugate).

For $kp > n$ , $W^{k,p}(\Omega) \hookrightarrow C^{m,\alpha}(\overline{\Omega})$ is compact for $\alpha < k - n/p - m$ .

Remark

Compactness of Sobolev embeddings is crucial for existence theorems. It allows extracting convergent subsequences from bounded sequences in $W^{k,p}$ , enabling direct methods in the calculus of variations.

Poincaré Inequality: For $u \in H^1_0(\Omega)$ with bounded $\Omega$ : $\|u\|_{L^2(\Omega)} \leq C(\Omega)\|\nabla u\|_{L^2(\Omega)}$

This shows $\|\nabla u\|_{L^2}$ is an equivalent norm on $H^1_0$ , simplifying many variational arguments.

DefinitionTrace Theorem

For $\Omega$ with smooth boundary, there exists a bounded linear trace operator: $\text{Tr}: W^{k,p}(\Omega) \to W^{k-1/p,p}(\partial\Omega)$ generalizing the restriction $u|_{\partial\Omega}$ to Sobolev functions.

Moreover, $H^1_0(\Omega) = \ker(\text{Tr})$ , characterizing functions with zero boundary values.

ExampleFractional Sobolev Spaces on Boundaries

The space $H^{1/2}(\partial\Omega)$ naturally appears as the trace space of $H^1(\Omega)$ . This has exactly "half a derivative" and appears in boundary value problems and domain decomposition methods.

Density: $C^\infty(\overline{\Omega})$ is dense in $W^{k,p}(\Omega)$ for $1 \leq p < \infty$ . For $p = \infty$ , this fails (discontinuous functions cannot be approximated uniformly by smooth functions while preserving bounded derivatives).

These properties make Sobolev spaces the correct setting for weak formulations: they're large enough to contain solutions yet structured enough for functional analysis.