Sobolev Spaces

The idea of a Sobolev space is to differentiate weakly. A function $u \in L^2$ need not have a classical derivative, but there may be a $v \in L^2$ with

and if such $v$ exists it is the weak derivative of $u$, written $u' = v$. Integration by parts is the definition rather than a theorem. The Sobolev space $W^{k,p}$ collects the functions whose weak derivatives up to order $k$ are in $L^p$, and for $p = 2$ the space $H^k = W^{k,2}$ is a Hilbert space.

The reason PDE theory lives here is that PDEs are equations in derivatives, and the natural objects on which derivatives behave well are these spaces. Weak formulations turn differential equations into variational ones, the Lax–Milgram theorem; proved in the fundamental theorems page; gives existence and uniqueness, and elliptic regularity then promotes weak solutions to classical ones. Sobolev spaces are the language in which that promotion is stated.

The weak derivative is the derivative that integration by parts can see. Consider the function $u(x) = |x|$ on $(-1, 1)$: it has no derivative at $0$, but its distributional derivative is the function $v = \operatorname{sign}(x)$, because $\int_{-1}^{1} |x|\varphi'(x)\,dx = -\int_{-1}^{1}\operatorname{sign}(x)\varphi(x)\,dx$ for every test function; the kink integrates out. And $v$ is itself weakly differentiable, with $v' = 2\delta_0$; the Dirac delta, not a function. So $|x| \in H^1$ but $|x| \notin H^2$.

That example carries the whole subject: each derivative costs one order of regularity, and the Sobolev scale is exactly the bookkeeping of how much smoothness is left after differentiating. Whether a boundary value problem has a classical solution becomes the question of whether enough orders of regularity survive, and the embedding theorems are the tool that answers it.

Definition. For $k \in \mathbb{N}$ and $1 \leq p \leq \infty$,

where $D^\alpha u$ is the weak derivative and $\alpha$ ranges over multi-indices. For $p = 2$ write $H^k$; the norm comes from the inner product $\langle u, v\rangle_{H^k} = \sum_{|\alpha| \le k} \langle D^\alpha u, D^\alpha v\rangle$.

Theorem (completeness). $W^{k,p}(\Omega)$ is a Banach space, and $H^k(\Omega)$ is a Hilbert space.

Proof. The map $u \mapsto (D^\alpha u)_{|\alpha| \leq k}$ embeds $W^{k,p}$ isometrically into the product $(L^p)^{N}$, $N$ the number of multi-indices. The product is Banach, so it suffices to show the image is closed. Let $u_n \to u$ in $W^{k,p}$; then $D^\alpha u_n \to v_\alpha$ in $L^p$ for each $\alpha$. For any test function $\varphi$, the definition of the weak derivative and $L^p$ convergence give

so $v_\alpha$ is the weak derivative $D^\alpha u$, and $u \in W^{k,p}$ with $D^\alpha u = v_\alpha$. Hence the image is closed. $\blacksquare$

The only analysis in the proof is that $L^p$ limits commute with integration against a fixed test function, which is Hölder’s inequality. Completeness is otherwise pure functional analysis, and it is what licenses using the Banach and Hilbert space theorems; the open mapping theorem, uniform boundedness, Lax–Milgram; on spaces of functions.

$C_c^\infty$ is dense. In $W^{k,p}(\Omega)$ for $1 \leq p < \infty$, the smooth compactly supported functions are dense (Meyers–Serrin). This is worth stating because its failure at $p = \infty$ and its dependence on $\Omega$ at the boundary are the two places the theory surprises people. It means "$W^{k,p}$" is not a bigger, stranger space than smooth functions; it is their completion, and every element is approximable.

Theorem (Poincaré). Let $\Omega \subset \mathbb{R}^n$ be bounded in some direction, and $u \in W_0^{1,2}(\Omega)$; the closure of $C_c^\infty(\Omega)$ in $H^1$. Then

Proof for $\Omega = (0, L)$ in one dimension. For $u \in C_c^\infty(0, L)$ and $x \in (0,L)$,

since $u$ vanishes at the endpoints. Squaring and applying Cauchy–Schwarz,

Integrating over $x \in (0,L)$: $\|u\|_{L^2}^2 \leq L^2\|u'\|_{L^2}^2$, so $C_\Omega = L$ works. The case of general $u \in W_0^{1,2}$ follows by density. $\blacksquare$

The proof is two lines, and it is worth noticing what the two hypotheses do. Vanishing at the boundary is what made the fundamental theorem applicable; for the constant function $u = 1$, which has $\|\nabla u\| = 0$, the inequality is false, and constants are exactly the elements of $H^1$ removed by the $W_0^{1,2}$ condition. And boundedness of $\Omega$ is what made $L$ finite; Poincaré fails on unbounded domains for the same reason, and its repair there requires a weight or a spectral gap.

Poincaré is the Poincaré inequality of the spectral theory of the Laplacian: applied to $-\Delta$ with Dirichlet conditions, it says the first eigenvalue $\lambda_1 \geq C_\Omega^{-2} > 0$, which is why the Laplacian’s spectrum is discrete and why solutions of the heat equation decay.

Theorem (Sobolev embedding). Let $\Omega \subset \mathbb{R}^n$ be bounded with smooth boundary. Then

and if $\frac{1}{p} - \frac{k}{n} < 0$, then $W^{k,p} \hookrightarrow C^{0,\alpha}$ with $\alpha = k - n/p$.

Proof for the sharpest case in one dimension, $W^{1,2}(0,L) \hookrightarrow C^0$. For $u \in C^\infty$ and $x, y \in (0,L)$,

by Cauchy–Schwarz. So $u$ is Hölder continuous with exponent $1/2$ and constant $\|u'\|_{L^2}$. For the sup norm, fix any $x_0$:

and averaging over $x_0$ gives $|u(x_0)| \leq L^{-1/2}\|u\|_{L^2}$ for some choice, so

Density of $C^\infty$ in $H^1$ extends the inequality, and a uniform limit of continuous functions is continuous, giving the embedding. $\blacksquare$

The two-dimensional statement $\frac{1}{q} = \frac{1}{2} - \frac{1}{2} = 0$ is the borderline, and the theorem correctly refuses to say $H^1(\mathbb{R}^2) \hookrightarrow L^\infty$: it embeds into every $L^q$ for $q < \infty$ and not into $L^\infty$. The standard counterexample family is $u_\epsilon(x) = \log\log(|x|^{-1})$ truncated near $0$, whose $H^1$ norm stays bounded while its supremum tends to infinity. That borderline is exactly where $\Phi^4_2$‘s renormalisation lives, as in the stochastic PDE page: the critical dimension is the one where the embedding into $L^\infty$ fails.

Theorem (Rellich–Kondrachov). With $\Omega$ bounded and smooth, $W^{1,2}(\Omega) \hookrightarrow\hookrightarrow L^2(\Omega)$: the embedding is compact, meaning bounded sets in $H^1$ have subsequences converging in $L^2$.

Proof idea. Boundedness in $H^1$ bounds the $L^2$ norms of both $u$ and $\nabla u$. The gradient bound gives an equicontinuity-type estimate after averaging: convolution with a mollifier $\rho_\delta$ satisfies

uniformly for $u$ in a bounded set, while $\{u * \rho_\delta\}$ is bounded and equi-uniformly-continuous in $L^\infty$ for fixed $\delta$. The Arzelà–Ascoli theorem then gives a uniformly convergent subsequence of the mollified family, and letting $\delta \to 0$ along a diagonal subsequence extracts an $L^2$- convergent subsequence of the original. $\blacksquare$

This is the only source of compactness in the theory, and it is what makes the Dirichlet Laplacian have a discrete spectrum: an operator with compact resolvent has eigenvalues marching to infinity, and Rellich–Kondrachov is the compactness of the resolvent. The heat kernel’s decay, the convergence of spectral approximations, and the existence of eigenfunction bases all trace back here.

The pipeline that solves a PDE is now complete:

1. Weak solution. Lax–Milgram gives $u \in H_0^1(\Omega)$ with $a(u, v) = \langle f, v\rangle$ for all $v$; existence, uniqueness, continuous dependence.
2. Improved integrability. Sobolev embedding turns $H^1$ control into $L^q$ control, so the solution is a function with integrable powers.
3. Interior regularity. Difference quotients, controlled uniformly using the equation, show $u \in H^2_{\mathrm{loc}}$ and by iteration $u \in H^m_{\mathrm{loc}}$ for every $m$ when $f$ is smooth.
4. Classical solution. Sobolev embedding with $m$ large enough makes $u$ continuous, then $C^1$, then $C^2$; at which point the weak equation holds pointwise and $u$ is a classical solution.

Each arrow is a theorem; none is optional. Step 3 is the elliptic regularity that the Hodge theory page uses as a black box to put smooth representatives on de Rham classes, and step 4’s threshold $m > n/2 + j$ for $C^j$ regularity is the Sobolev embedding theorem again, with $n$ the dimension.

• Confusing weak and classical derivatives. A weak derivative is an equivalence class defined by integration by parts; it may fail to exist even where a pointwise derivative exists, and it may exist where no classical derivative does. $|x|$ is the standard example in both directions.
• Applying Poincaré without the boundary condition. Constants have zero gradient. On $H^1$ rather than $W_0^{1,2}$, add $\|u\|_{L^2}$ to the right side, or quotient out constants.
• Reading the embedding theorem at the borderline. When $\frac{1}{p} = \frac{k}{n}$ the embedding into $L^\infty$ fails; into every finite $q$ it holds. The failure is not a weakness of the method but the critical case, and it is where renormalisation appears.
• Assuming $C_c^\infty$ is dense in every Sobolev space on every domain. For $W^{k,p}$ with $p = \infty$ it is false, and for domains with bad boundaries the closure can be a strict subspace ($W_0^{1,p}$ rather than $W^{1,p}$), which is a boundary condition in disguise.
• Expecting Sobolev functions to be continuous. $H^1$ functions in one dimension are continuous; that is the embedding; but in two dimensions they need not be, and in $n$ dimensions a $W^{k,p}$ function is continuous only when $k > n/p$. The whole subject is the bookkeeping of that threshold.

• The Fundamental Theorems; Lax–Milgram, the source of weak solutions, and the PDE applications section.
• Normed Spaces and Banach Spaces; completeness, which is Theorem 1 above in the abstract.
• Spectral Theory of Self-Adjoint Operators; the Dirichlet Laplacian, whose discrete spectrum Poincaré initiates.
• Hodge Theory; elliptic regularity as the black box, and harmonic forms as the kernel of $\Delta$.
• Stochastic PDEs and Regularity Structures; the critical dimension where the $H^1 \to L^\infty$ embedding fails.
• Applications of Differential Geometry; where Sobolev spaces enter geometric PDE.

• Applications
• Bounded Linear Operators
• Compact Operators
• Historical Context