Sobolev Spaces — Mathematics Knowledge System

Overview

Sobolev spaces $W^{k,p}(\Omega)$ are function spaces that combine $L^p$ integrability with control on weak derivatives up to order $k$. They are the natural setting for the variational (weak) formulation of partial differential equations: classical solutions require pointwise smoothness, but Sobolev spaces allow 'solutions' that satisfy an integrated version of the PDE, dramatically expanding existence theory. The Sobolev embedding theorems control how much regularity is gained or preserved as one varies $k$ and $p$, and trace theorems allow boundary conditions to be imposed rigorously on domains with boundary.

Intuition

A function in $W^{1,2}(\Omega)$ need not have a classical derivative—its derivative exists in the sense that integration by parts gives a consistent answer. Think of a piecewise-linear function: the derivative (slope) is defined almost everywhere and is square-integrable even though the function is not smooth. Sobolev spaces quantify how 'rough' a function can be while still having enough structure for PDE analysis. The embedding theorems say: if you have enough derivatives in $L^p$, you automatically gain continuity or $L^q$ integrability for a larger $q$.

Formal Definition

Definition

Let $\Omega \subset \mathbb{R}^n$ be an open set, $k \in \mathbb{N}_0$, $1 \le p \le \infty$. The Sobolev space is $W^{k,p}(\Omega) = \{u \in L^p(\Omega) : D^\alpha u \in L^p(\Omega) \text{ for all } |\alpha| \le k\}$, where $D^\alpha u$ denotes the weak partial derivative of order $\alpha$. The Sobolev norm is $\|u\|_{W^{k,p}} = \left(\sum_{|\alpha|\le k} \|D^\alpha u\|_{L^p}^p\right)^{1/p}$ for $p < \infty$. The special case $p = 2$ yields the Hilbert space $H^k(\Omega) = W^{k,2}(\Omega)$. The space $H_0^1(\Omega)$ is the closure of $C_c^\infty(\Omega)$ in $H^1(\Omega)$ and encodes homogeneous Dirichlet boundary conditions.

W^{k,p}(\Omega) = \left\{ u \in L^p(\Omega) : D^\alpha u \in L^p(\Omega), \; |\alpha| \le k \right\}

Sobolev space definition

\|u\|_{W^{k,p}(\Omega)} = \left( \sum_{|\alpha| \le k} \|D^\alpha u\|_{L^p(\Omega)}^p \right)^{1/p}

Sobolev norm (p < ∞)

H^1_0(\Omega) = \overline{C_c^\infty(\Omega)}^{\,H^1}

H^1_0 as closure

Theorems

Theorem 6.1: Sobolev Embedding Theorem

Let \Omega \subset \mathbb{R}^n be a bounded domain with Lipschitz boundary, 1 \le p < \infty, k \ge 1. If kp < n, then W^{k,p}(\Omega) \hookrightarrow L^{p^*}(\Omega) where 1/p^* = 1/p - k/n. If kp = n (and k < n), then W^{k,p}(\Omega) \hookrightarrow L^q for all 1 \le q < \infty. If kp > n, then W^{k,p}(\Omega) \hookrightarrow C^{0,\gamma}(\overline{\Omega}) for \gamma = k - n/p - \lfloor k - n/p \rfloor.

Theorem 6.2: Rellich–Kondrachov Compactness Theorem

Under the assumptions of Theorem 6.1 with kp < n, the embedding W^{k,p}(\Omega) \hookrightarrow L^q(\Omega) is compact for every 1 \le q < p^*.

Theorem 6.3: Poincaré Inequality

Let \Omega be a bounded connected open set. There exists C = C(\Omega, p) > 0 such that \|u\|_{L^p(\Omega)} \le C \|\nabla u\|_{L^p(\Omega)} for all u \in W^{1,p}_0(\Omega).

Theorem 6.4: Trace Theorem

Let \Omega be a bounded domain with C^1 boundary. There exists a bounded linear operator \gamma_0: H^1(\Omega) \to L^2(\partial\Omega) (the trace operator) with \gamma_0(u) = u|_{\partial\Omega} for u \in C^1(\overline{\Omega}), and \|\gamma_0 u\|_{L^2(\partial\Omega)} \le C\|u\|_{H^1(\Omega)}.

Worked Examples

Check $u \in L^2(B_1)$: $\int_{B_1} |x|^{2\alpha}\,dx = c_n \int_0^1 r^{2\alpha} r^{n-1}\,dr < \infty$ iff $2\alpha + n - 1 > -1$, i.e., $\alpha > -n/2$.
$\int_0^1 r^{2\alpha + n - 1}\,dr < \infty \iff 2\alpha + n > 0$
Compute weak derivative: $\partial_{x_i} |x|^\alpha = \alpha x_i |x|^{\alpha - 2}$. Check $|\nabla u|^2 = \alpha^2 |x|^{2\alpha - 2}$.
$|\nabla u(x)|^2 = \alpha^2 |x|^{2(\alpha-1)}$
Check $\nabla u \in L^2(B_1)$: $\int_{B_1} |x|^{2\alpha - 2}\,dx < \infty$ iff $2\alpha - 2 + n > 0$, i.e., $\alpha > 1 - n/2$.
$\alpha > 1 - n/2$
The condition $\alpha > 1 - n/2$ is more restrictive than $\alpha > -n/2$ (for $n \ge 2$), so $u \in H^1(B_1)$ iff $\alpha > 1 - n/2$.

Answer: $|x|^\alpha \in H^1(B_1)$ if and only if $\alpha > 1 - n/2$.

Practice Problems

Difficulty 8/10

Verify that $W^{k,p}(\Omega)$ is a Banach space (complete normed space).

Difficulty 9/10

State and explain the Sobolev critical exponent $p^* = np/(n-kp)$ and explain why $W^{1,2}(\mathbb{R}^3) \hookrightarrow L^6(\mathbb{R}^3)$ but not $L^\infty(\mathbb{R}^3)$.

Difficulty 9/10

Use the Rellich–Kondrachov theorem to prove that the embedding $H^1_0(\Omega) \hookrightarrow L^2(\Omega)$ is compact for bounded $\Omega \subset \mathbb{R}^n$ with $n \ge 1$.

Common Mistakes

Common Mistake

Every function in $H^1(\Omega)$ is continuous

In dimension $n \ge 2$, $H^1$ functions need not be continuous. Continuity (via Morrey's inequality) requires $W^{1,p}$ with $p > n$. In $\mathbb{R}^2$, $\ln|\ln|x||$ is in $H^1(B_{1/2})$ but is unbounded.

Common Mistake

The Sobolev embedding into $L^{p^*}$ is compact

The embedding $W^{1,p} \hookrightarrow L^{p^*}$ is continuous but NOT compact. Compactness holds only for strictly subcritical exponents $q < p^*$ (Rellich–Kondrachov) on bounded domains.

Quiz

Summary

$W^{k,p}(\Omega)$ consists of $L^p$ functions whose weak derivatives up to order $k$ are also in $L^p$; it is a Banach space (Hilbert for $p=2$).
The Sobolev embedding theorem: if $kp < n$ then $W^{k,p} \hookrightarrow L^{p^*}$ with $p^* = np/(n-kp)$; if $kp > n$, functions are Hölder continuous.
Rellich–Kondrachov: the embedding is compact for $L^q$ with $q < p^*$ on bounded domains—crucial for PDE compactness arguments.
The Poincaré inequality on $H_0^1(\Omega)$ (bounded $\Omega$): $\|u\|_{L^2} \le C\|\nabla u\|_{L^2}$; this makes $\|\nabla u\|_{L^2}$ an equivalent norm.
Sobolev spaces are the natural setting for weak formulations of elliptic PDEs; the Lax–Milgram theorem on $H_0^1$ gives existence and uniqueness of weak solutions.

References

BookEvans, L. C. Partial Differential Equations. 2nd ed., AMS, 2010. Chapter 5.
BookBrezis, H. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, 2011. Chapters 8–9.
WebsiteWikipedia — Sobolev space