Course 4 Book 4A: Function Spaces and Sobolev SpacesChapter 9

Sobolev Spaces

Functions with derivatives in Lᵖ, approximation, and the Poincaré inequality.

22 min read · Updated Oct 2, 2026

Read with Brezis, chapter 8 (Sobolev spaces in one dimension, which shows every idea with the least machinery) and the first half of chapter 9 (definitions, approximation, extension, traces, $W^{1,p}_0$ ); Evans §5.2–5.5 is a good second voice.

In this chapter · 6 sections

9.1Computing with hat functions
9.2Definitions
9.2.1Which functions are in W1,pW^{1,p}W1,p?
9.3Approximation, extension, traces
9.4The Poincaré inequality
9.5History
9.6Exercises

A Sobolev space consists of the functions whose weak derivatives (4A.8 Distributions and Weak Derivatives), up to some order, are in $L^p$ . The definition looks technical, and the reason for it is practical. The natural energies of physics and geometry, $\int|\nabla u|^2$ for a membrane, $\int(R + |\nabla f|^2)e^{-f}$ for Perelman, are integrals of squares of derivatives. To minimise them, or to solve the equations they produce, one needs a complete space in which those integrals are the norm, and $C^1$ functions with that norm are not complete. Completing them gives the Sobolev space $H^1$ , just as completing the rationals gave the reals (2A.4 The Real Numbers) and completing continuous functions gave $L^1$ (3A.3 The Lebesgue Integral).

This chapter defines the spaces, shows they are complete and that smooth functions are dense in them, explains how boundary values make sense for functions that are only defined almost everywhere, and proves the most-used inequality about them, the Poincaré inequality, which says that a function vanishing on the boundary is controlled by its gradient. The anchor is the finite element method, which is Sobolev space theory turned into the main computational tool of engineering.

By the end of this chapter you will be able to:

define $W^{k,p}$ and $H^k$ , prove that they are complete, and decide whether a given function belongs to them;
state the density theorems (smooth functions are dense) and the trace and extension theorems;
define $H^1_0$ and explain in what sense its functions vanish on the boundary;
prove the Poincaré inequality and relate its constant to the lowest frequency of a drum;
rewrite Perelman's $\mathcal{F}$ -functional as a quadratic form on $H^1$ .

Computing with hat functions

In the world In use The finite element method

How does a bridge deck sag under load, a turbine blade deform, or heat spread through an engine block? Engineering software answers such questions by the finite element method. The region is divided into a mesh of small triangles or tetrahedra. On the mesh, a hat function is attached to each vertex: the piecewise-linear function equal to $1$ at that vertex, $0$ at all the others, and linear on each triangle (Figure 9.1). The unknown (a displacement, a temperature) is approximated by a combination $u_h = \sum_iU_i\varphi_i$ of hat functions, and the coefficients $U_i$ are found by requiring the weak form of the equation (4A.4 Hilbert Spaces and Lax–Milgram) to hold for every hat function as test function. That gives a large, sparse, symmetric linear system, the "stiffness matrix" of structural engineering.

Hat functions are not differentiable at the edges of the mesh, so $u_h$ is not a classical solution of anything. But it has weak derivatives in $L^2$ : it lies in $H^1$ , the space in which the problem is posed. The method's convergence theory is Sobolev-space theory. Céa's lemma (Jean Céa, 1964; 4A.4 Hilbert Spaces and Lax–Milgram's rehearsal) says the error in the energy norm is at most a constant times the best possible approximation of the true solution by piecewise-linear functions, and interpolation estimates in Sobolev norms bound that by a multiple of the mesh size. The method's roots are in a 1943 paper by Richard Courant using piecewise-linear functions on triangles, and in aircraft structural analysis at Boeing, published by Turner, Clough, Martin and Topp in 1956; Ray Clough named it the finite element method in 1960.

Figure 9.1. Top: hat functions on a one-dimensional mesh, and the piecewise-linear interpolant

\sum_iu(x_i)\varphi_i

of a smooth function (computed). Bottom: a triangular mesh, with the hat function of one vertex shaded:

1

at that vertex,

0

at the others, linear on each triangle. Such functions are continuous with corners, so they lie in

H^1

but not in

C^1

Definitions

Let $\Omega \subseteq \mathbb{R}^n$ be open, $1 \leq p \leq \infty$ and $k \geq 0$ an integer.

Definition 9.1 Sobolev spaces

$W^{k,p}(\Omega)$ is the space of $u \in L^p(\Omega)$ whose weak derivatives $\partial^\alpha u$ exist and lie in $L^p(\Omega)$ for every multi-index $|\alpha| \leq k$ , with the norm

\|u\|_{W^{k,p}} = \Big(\sum_{|\alpha|\leq k}\|\partial^\alpha u\|_p^p\Big)^{1/p} \quad (p < \infty), \qquad \|u\|_{W^{k,\infty}} = \max_{|\alpha|\leq k}\|\partial^\alpha u\|_\infty.

For $p = 2$ we write $H^k(\Omega) = W^{k,2}(\Omega)$ , a Hilbert space with $\langle u, v\rangle = \sum_{|\alpha|\leq k}\int\partial^\alpha u\,\partial^\alpha v$ .

The space used most is $H^1$ , with $\|u\|_{H^1}^2 = \int(u^2 + |\nabla u|^2)$ . On $\mathbb{R}^n$ it is the space $H^1$ of 4A.5 The Fourier Transform, defined there by the Fourier transform ( $\int|\nabla u|^2 = 4\pi^2\int|\xi|^2|\hat u|^2$ ).

Theorem 9.2 Completeness

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

Proof. Let $(u_m)$ be Cauchy in $W^{k,p}$ . Then each $(\partial^\alpha u_m)$ is Cauchy in $L^p$ , so by Riesz–Fischer (3A.7 Lᵖ Spaces and Jensen’s Inequality) $u_m \to u$ and $\partial^\alpha u_m \to u_\alpha$ in $L^p$ . For a test function $\phi$ ,

\int u\,\partial^\alpha\phi = \lim_m\int u_m\,\partial^\alpha\phi = \lim_m(-1)^{|\alpha|}\int\partial^\alpha u_m\,\phi = (-1)^{|\alpha|}\int u_\alpha\,\phi,

the limits being justified by Hölder's inequality. So $u_\alpha$ is the weak derivative $\partial^\alpha u$ , and $u_m \to u$ in $W^{k,p}$ .

The proof shows the key property of weak derivatives: they pass to $L^p$ limits. Classical derivatives don't (2B.5 Uniform Convergence and Arzelà–Ascoli), which is exactly why $C^1$ with the norm $\|u\|_p + \|\nabla u\|_p$ is incomplete and $W^{1,p}$ is its completion.

Which functions are in $W^{1,p}$ ?

In one dimension, a function in $W^{1,p}(a, b)$ has a continuous representative, with $u(y) - u(x) = \int_x^yu'$ , and by Hölder $|u(y) - u(x)| \leq |y - x|^{1 - 1/p}\|u'\|_p$ (Exercise 9.8). So in one dimension, Sobolev functions are continuous, even Hölder continuous (4A.11 Hölder Spaces).

In higher dimensions they need not be. The test case is a power of the distance to a point.

Example 9.3 Powers of

|x|

On the unit ball $B \subseteq \mathbb{R}^n$ , let $u(x) = |x|^\alpha$ . Away from $0$ , $|\nabla u| = |\alpha|\,|x|^{\alpha-1}$ , and in polar coordinates (3A.5 Product Measures and Change of Variables)

\int_B|\nabla u|^p = |\alpha|^p|S^{n-1}|\int_0^1r^{(\alpha-1)p}r^{n-1}\,dr < \infty \iff (\alpha - 1)p + n > 0 \iff \alpha > 1 - \frac np.

(One also checks that the classical gradient away from $0$ is the weak gradient on all of $B$ , which holds when it is integrable, as in 4A.8 Distributions and Weak Derivatives's exercise on the distance function.) So $u \in W^{1,p}(B)$ iff $\alpha > 1 - \frac np$ . For $p < n$ this allows negative $\alpha$ : the function $|x|^{-1/4}$ is in $H^1$ of the unit ball in $\mathbb{R}^3$ although it is unbounded. In $\mathbb{R}^2$ with $p = 2$ the threshold is $\alpha > 0$ , and the borderline unbounded example is $\log\log(1 + 1/|x|)$ (4A.5 The Fourier Transform).

The number $1 - \frac np$ measures the trade between derivatives and dimension. It is the exponent that 4A.10 Sobolev Embeddings and Critical Exponents makes the organising principle of the Sobolev embedding theorems: $W^{1,p}$ functions are continuous when $p > n$ , but can be unbounded when $p \leq n$ .

Approximation, extension, traces

Theorem 9.4 Smooth functions are dense

For $1 \leq p < \infty$ :

$C_c^\infty(\mathbb{R}^n)$ is dense in $W^{k,p}(\mathbb{R}^n)$ ;
(Meyers–Serrin) $C^\infty(\Omega) \cap W^{k,p}(\Omega)$ is dense in $W^{k,p}(\Omega)$ for every open $\Omega$ ;
if $\Omega$ is bounded with $C^1$ (or Lipschitz) boundary, functions smooth up to the boundary, $C^\infty(\bar\Omega)$ , are dense in $W^{k,p}(\Omega)$ .

Proof architecture. (1) Cut off at large radius and mollify: weak derivatives commute with mollification (4A.8 Distributions and Weak Derivatives), and mollifications converge in $L^p$ (3A.8 Convolution and Mollifiers), so $u * \phi_\varepsilon \to u$ in $W^{k,p}$ . (2) Mollify at a scale that shrinks near the boundary, using a partition of unity subordinate to an exhaustion of $\Omega$ by compact sets. (3) Near the boundary, translate the function slightly inwards before mollifying, which needs some regularity of $\partial\Omega$ . (Brezis, chapter 9; Evans §5.3.)

So the Sobolev space is the completion of smooth functions in the Sobolev norm: Meyers and Serrin's 1964 paper was titled " $H = W$ ", recording that the two possible definitions, by completion ( $H$ ) and by weak derivatives ( $W$ ), agree. In practice: prove inequalities for smooth functions, then extend them by density. That is how almost every result in this chapter and the next is proved.

Two further theorems, stated without proof (Brezis, chapter 9; Evans §5.4–5.5), let functions on a domain be treated like functions on $\mathbb{R}^n$ and give them boundary values.

Extension. If $\Omega$ is bounded with $C^1$ boundary, there is a bounded linear extension operator $E : W^{1,p}(\Omega) \to W^{1,p}(\mathbb{R}^n)$ with $Eu = u$ on $\Omega$ .
Trace. A function in $W^{1,p}(\Omega)$ is defined only almost everywhere, and $\partial\Omega$ has measure zero, so "the value on the boundary" doesn't make sense directly. But if $\partial\Omega$ is $C^1$ there is a bounded linear trace operator $T : W^{1,p}(\Omega) \to L^p(\partial\Omega)$ with $Tu = u|_{\partial\Omega}$ whenever $u$ is continuous up to the boundary. Boundary values of Sobolev functions are defined by continuity from smooth functions.

Definition 9.5 Zero boundary values

$W^{1,p}_0(\Omega)$ is the closure of $C_c^\infty(\Omega)$ in $W^{1,p}(\Omega)$ , and $H^1_0(\Omega) = W^{1,2}_0(\Omega)$ . For bounded $C^1$ domains it is exactly the kernel of the trace: the Sobolev functions that vanish on the boundary.

$H^1_0(\Omega)$ is the space of the Dirichlet problem (4A.4 Hilbert Spaces and Lax–Milgram, 4A.7 Compact Operators and Spectra): a clamped membrane, a drum fixed at its rim.

The Poincaré inequality

On a bounded domain, a function that vanishes on the boundary can't be large unless its gradient is.

Theorem 9.6 Poincaré inequality

Let $\Omega$ lie between two parallel hyperplanes at distance $d$ . Then for every $u \in W^{1,p}_0(\Omega)$ , $1 \leq p < \infty$ ,

\int_\Omega|u|^p \leq d^p\int_\Omega|\nabla u|^p.

Proof. By density it suffices to take $u \in C_c^\infty(\Omega)$ , extended by $0$ . Choose coordinates with $\Omega \subseteq \{0 < x_1 < d\}$ . For each $x = (x_1, x')$ , since $u(0, x') = 0$ ,

|u(x_1, x')| = \Big|\int_0^{x_1}\partial_1u(s, x')\,ds\Big| \leq \int_0^d|\nabla u(s, x')|\,ds \leq d^{1 - 1/p}\Big(\int_0^d|\nabla u(s, x')|^p\,ds\Big)^{1/p}

by Hölder. Raise to the power $p$ , integrate over $x_1 \in (0, d)$ (giving a factor $d$ ), then over $x'$ (Fubini, 3A.5 Product Measures and Change of Variables).

The proof slices the domain into segments and uses the fundamental theorem of calculus on each, with the boundary value $0$ to start. Without the boundary condition the inequality fails (constants have zero gradient). The version for functions with mean zero on a bounded connected domain (or on a closed manifold),

\int_\Omega|u - \bar u|^p \leq C\int_\Omega|\nabla u|^p, \qquad \bar u = \frac{1}{|\Omega|}\int_\Omega u,

is also true, but its proof is a contradiction–compactness argument that needs Rellich's theorem (4A.10 Sobolev Embeddings and Critical Exponents, Exercise 9.11); on the circle it is Wirtinger's inequality (2B.7 Fourier Series and the First Heat Equation).

In the world Model The lowest note of a drum

For a membrane under tension $T$ with area density $\rho$ , clamped on the boundary of $\Omega$ , the strain energy of a small displacement $u$ is $\frac T2\int|\nabla u|^2$ , an $H^1$ seminorm, and the kinetic energy is $\frac\rho2\int u_t^2$ . The frequencies of vibration are $\frac{1}{2\pi}\sqrt{T\lambda_k/\rho}$ , where $\lambda_k$ are the Dirichlet eigenvalues (4A.7 Compact Operators and Spectra). The best constant in the $p = 2$ Poincaré inequality is exactly $1/\lambda_1$ : $\int u^2 \leq \frac{1}{\lambda_1}\int|\nabla u|^2$ , with equality for the fundamental mode. So the Poincaré inequality is the statement that a clamped drum has a lowest note, strictly above zero; the slicing proof gives $\lambda_1 \geq \frac{1}{d^2}$ for a drum of width $d$ (the true value for a strip is $\frac{\pi^2}{d^2}$ ). A wider drum can have a lower note; an infinitely long thin one still has a lowest note fixed by its width.

Where this goes Perelman's

\mathcal{F}

is an

H^1

quadratic form

On a closed Riemannian manifold, Perelman's functional is

\mathcal{F}(g, f) = \int_M\big(R + |\nabla f|^2\big)\,e^{-f}\,dV.

With $w = e^{-f/2}$ , so that $|\nabla f|^2e^{-f} = 4|\nabla w|^2$ (3A.4 Measures, Probability and Weights), it becomes

\mathcal{F} = \int_M\big(4|\nabla w|^2 + Rw^2\big)\,dV,

a quadratic form in $w$ whose natural domain is the Sobolev space $H^1(M)$ : it is finite exactly when $w \in H^1$ . The constraint $\int e^{-f}dV = 1$ becomes $\int w^2 = 1$ , the unit sphere of $L^2$ . Minimising gives $\lambda(g)$ , the bottom of the spectrum of $-4\Delta + R$ (4A.7 Compact Operators and Spectra, 12A.2 Ricci Flow as a Gradient Flow); the minimiser exists because $H^1(M) \hookrightarrow L^2(M)$ is compact on a closed manifold (4A.10 Sobolev Embeddings and Critical Exponents). When the definition appears in 12A.2 Ricci Flow as a Gradient Flow, it should look familiar.

History

Sergei Sobolev introduced the spaces in the 1930s, in work on wave equations and on the Dirichlet problem (his book Some Applications of Functional Analysis in Mathematical Physics appeared in 1950). Kurt Friedrichs studied the relation between weak and strong derivatives in 1944, and Norman Meyers and James Serrin proved " $H = W$ " in 1964. Henri Poincaré proved an inequality of this type in 1890, in work on the Dirichlet problem. Emilio Gagliardo developed the theory of traces in 1957. The finite element method grew from Courant's 1943 paper, aircraft structural analysis in the 1950s (Turner, Clough, Martin and Topp, 1956), and Clough's naming of it in 1960; its convergence theory from Céa's 1964 thesis.

Recall Where we stand

$W^{k,p}$ consists of $L^p$ functions with weak derivatives in $L^p$ up to order $k$ ; it is complete because weak derivatives pass to $L^p$ limits, and smooth functions are dense in it, so it is the completion of smooth functions in the Sobolev norm. In one dimension Sobolev functions are continuous; in higher dimensions $|x|^\alpha \in W^{1,p}$ iff $\alpha > 1 - \frac np$ , so they can be unbounded when $p \leq n$ . Extension and trace theorems handle domains and boundary values; $H^1_0$ is the space of functions vanishing on the boundary. The Poincaré inequality bounds a function vanishing on the boundary by its gradient, with best constant $1/\lambda_1$ for $p = 2$ . Perelman's $\mathcal{F}$ is an $H^1$ quadratic form. 4A.10 Sobolev Embeddings and Critical Exponents finds exactly which $L^q$ and Hölder spaces $W^{1,p}$ embeds in, and when the embedding is compact.

Exercises

Exercise 9.7 Powers of

|x|

, in detail

Show that $|x|^\alpha \in H^1(B)$ for the unit ball $B \subseteq \mathbb{R}^n$ iff $\alpha > 1 - \frac n2$ (and $\alpha$ is not required to be positive when $n \geq 3$ ). Check the cases $n = 1$ (must be continuous), $n = 2$ ( $\alpha > 0$ ) and $n = 3$ ( $\alpha > -\frac12$ ).

Exercise 9.8 One-dimensional Sobolev functions

Let $u \in C^1([a, b])$ . Show $|u(y) - u(x)| \leq |y - x|^{1 - 1/p}\|u'\|_{L^p}$ for $p > 1$ , and $\sup|u| \leq \frac{1}{b - a}\int|u| + \int|u'|$ . Deduce, by density, that every $u \in W^{1,p}(a, b)$ has a continuous representative satisfying the same bounds.

Solution

$|u(y) - u(x)| = |\int_x^yu'| \leq |y - x|^{1/q}\|u'\|_p$ by Hölder, with $\frac1q = 1 - \frac1p$ . For the sup bound, $u(x) = u(y) + \int_y^xu'$ ; average over $y$ . A Cauchy sequence of smooth functions in $W^{1,p}$ is then uniformly Cauchy, so its limit is continuous.

Exercise 9.9 The product rule

Show that if $u \in W^{1,p}(\Omega)$ and $v \in C^1(\bar\Omega)$ with bounded derivatives, then $uv \in W^{1,p}$ with $\nabla(uv) = v\nabla u + u\nabla v$ . (Approximate $u$ by smooth functions, using Meyers–Serrin.)

Exercise 9.10 Poincaré on an interval, sharply

Show that for $u \in H^1_0(0, L)$ , $\int_0^Lu^2 \leq \frac{L^2}{\pi^2}\int_0^Lu'^2$ , using the sine expansion of 4A.7 Compact Operators and Spectra's exercise, and that the constant is attained. Compare with the constant $L^2$ from the slicing proof.

Exercise 9.11 Rehearsal: Poincaré for mean-zero functions by contradiction–compactness

Let $\Omega$ be bounded, connected, with $C^1$ boundary. Prove that there is $C$ with $\|u - \bar u\|_{L^2} \leq C\|\nabla u\|_{L^2}$ for all $u \in H^1(\Omega)$ , assuming Rellich's theorem (4A.10 Sobolev Embeddings and Critical Exponents: bounded sequences in $H^1(\Omega)$ have subsequences converging in $L^2(\Omega)$ ). Run the template of 2B.3 Compactness: suppose $u_k$ with $\bar u_k = 0$ , $\|u_k\|_2 = 1$ (normalise) and $\|\nabla u_k\|_2 < \frac1k$ ; extract a subsequence converging in $L^2$ and weakly in $H^1$ ; show the limit has $\nabla u = 0$ , so is constant (connectedness), has mean $0$ and norm $1$ . Contradiction. This proof gives no value for $C$ , which is typical of [CC] arguments; on a closed manifold the same proof works, and the best constant is $1/\lambda_1$ for the first non-zero eigenvalue of the Laplacian.

Exercise 9.12 Rehearsal: two forms of

\mathcal{F}

On a closed manifold, show $\int\Delta f\,e^{-f}\,dV = \int|\nabla f|^2e^{-f}\,dV$ (integrate by parts: $\int\Delta f\,e^{-f} = -\int\langle\nabla f, \nabla e^{-f}\rangle$ , 2A.11 The Riemann Integral, 8A.8 Differential Forms and Stokes’ Theorem). Deduce that

\mathcal{F}(g, f) = \int(R + |\nabla f|^2)e^{-f}\,dV = \int(R + 2\Delta f - |\nabla f|^2)e^{-f}\,dV.

Perelman uses the second form because $R + 2\Delta f - |\nabla f|^2$ is the quantity whose evolution is computed pointwise in 12A.2 Ricci Flow as a Gradient Flow, and the identity between the two forms is exactly the kind of integration by parts that the monotonicity formula consists of.

Solution

$\nabla e^{-f} = -e^{-f}\nabla f$ , so $-\int\langle\nabla f, \nabla e^{-f}\rangle = \int|\nabla f|^2e^{-f}$ . Then $\int(2\Delta f - |\nabla f|^2)e^{-f} = \int(2|\nabla f|^2 - |\nabla f|^2)e^{-f} = \int|\nabla f|^2e^{-f}$ .

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9.1Computing with hat functions

9.2Definitions

9.2.1Which functions are in W1,pW^{1,p}W1,p?

9.3Approximation, extension, traces

9.4The Poincaré inequality

9.5History

9.6Exercises