Sobolev Embeddings and Critical Exponents

A Sobolev function has derivatives in $L^p$. What does that say about the function itself? This chapter's theorems answer exactly. If $p < n$, the function is in $L^{p^*}$ for a larger exponent $p^* = \frac{np}{n - p}$ (Gagliardo–Nirenberg–Sobolev). If $p > n$, it is Hölder continuous (Morrey). And on a bounded domain or a closed manifold, the embedding into $L^q$ is compact for every $q < p^*$ (Rellich–Kondrachov), which is the theorem that finally restores compactness to the analysis of PDE, completing the spectral theorem of 4A.7 Compact Operators and Spectra and the Poincaré inequality of 4A.9 Sobolev Spaces.

All the exponents can be predicted before anything is proved, by dimensional analysis: an inequality that holds for all functions on $\mathbb{R}^n$ must be unchanged when the functions are rescaled, and that pins down the exponent. This is thread S at its most effective, and it also explains the chapter's last theme. At the critical exponent $p^*$ the inequality is scale-invariant, and compactness fails, because a function can concentrate into a tall narrow bubble without changing either side. Loss of compactness by concentration, repaired by rescaling, is the mechanism of singularity analysis in Ricci flow.

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

• derive the Sobolev exponents by scaling, and place any space $W^{k,p}$ on the "Sobolev number line" $k - n/p$;
• state and use the Gagliardo–Nirenberg–Sobolev and Morrey inequalities, and prove the former in the plane;
• explain the equivalence between the $L^1$ Sobolev inequality and the isoperimetric inequality;
• prove the Rellich–Kondrachov theorem for $H^1_0$, and use it to finish the Dirichlet spectral theorem;
• show that compactness fails at the critical exponent by concentration, and explain how rescaling repairs it.

10.1Why bubbles are round

10.2Scaling first

Suppose an inequality $\|u\|_{L^q(\mathbb{R}^n)} \leq C\|\nabla u\|_{L^p(\mathbb{R}^n)}$ held for all smooth compactly supported $u$. Replace $u$ by $u_\lambda(x) = u(\lambda x)$. By the scaling rules of 3A.5 Product Measures and Change of Variables,

So the inequality says $\lambda^{-n/q}\|u\|_q \leq C\lambda^{1-n/p}\|\nabla u\|_p$ for every $\lambda > 0$. Unless the powers of $\lambda$ agree, letting $\lambda \to 0$ or $\infty$ forces $u = 0$. So the only possible exponent is given by

For $p = 2$ in dimension $3$, $p^* = 6$; in dimension $n$, $2^* = \frac{2n}{n-2}$. The exponent comes from dimensions alone, before any analysis.

The same bookkeeping organises everything: assign to $W^{k,p}$ the number $k - \frac np$ ("derivatives minus dimension over integrability"), and to the Hölder space $C^{m,\alpha}$ the number $m + \alpha$. Sobolev's theorems say that a space embeds in the spaces with smaller numbers: $W^{1,p}$ (number $1 - \frac np$) in $L^{p^*}$ (number $-\frac{n}{p^*} = 1 - \frac np$, equal), and in $C^{0, 1 - n/p}$ when $1 - \frac np > 0$ (Figure 10.1). Embeddings into spaces with strictly smaller numbers are compact on bounded domains; the borderline ones are not.

10.3The Gagliardo–Nirenberg–Sobolev inequality

We prove the key case $p = 1$ in the plane, where the idea is visible, and then explain the rest.

On a bounded domain, the same holds for $u \in W^{1,p}_0(\Omega)$ (extend by zero), and for all of $W^{1,p}(\Omega)$ on a $C^1$ domain with $\|u\|_{W^{1,p}}$ on the right (extend, 4A.9 Sobolev Spaces). Since $\Omega$ has finite measure, $L^{p^*}(\Omega) \subseteq L^q(\Omega)$ for every $q \leq p^*$ (3A.7 Lᵖ Spaces and Jensen’s Inequality), so $W^{1,p}(\Omega) \hookrightarrow L^q(\Omega)$ for $1 \leq q \leq p^*$.

10.3.1Sobolev inequalities are isoperimetric inequalities

The converse also holds, with the same constant, via the coarea formula, which writes $\int|\nabla u|$ as the integral over $t$ of the areas of the level sets $\{u = t\}$ (Federer–Fleming; Maz'ya). The best constant is the isoperimetric one, $C = \big(n\omega_n^{1/n}\big)^{-1}$, attained (in the limit) by balls. This equivalence is the cleanest instance of a principle that recurs in geometry: an analytic inequality for all functions encodes a geometric inequality for all regions. On a Riemannian manifold, Sobolev constants are controlled by isoperimetric constants, and both are controlled by curvature and volume (9B.2 Volume Comparison).

10.4Morrey's inequality

When $p > n$ the number $1 - \frac np$ is positive, and the functions are continuous.

Idea of the proof (Evans §5.6.2). For smooth $u$, the average of $|u(y) - u(x)|$ over a ball $B(x, r)$ is bounded by $C\int_{B(x,r)}\frac{|\nabla u(y)|}{|x - y|^{n-1}}\,dy$ (integrate $\nabla u$ along rays from $x$). By Hölder, that is at most $C\|\nabla u\|_p\big(\int_{B(x,r)}|x - y|^{-(n-1)q}dy\big)^{1/q}$, which is finite exactly when $(n - 1)q < n$, that is $p > n$, and then it equals $Cr^{1 - n/p}\|\nabla u\|_p$. Comparing $u(x)$ and $u(y)$ through the average over a ball containing both, of radius $|x - y|$, gives the Hölder estimate. In one dimension it is 4A.9 Sobolev Spaces's Hölder bound.

At the borderline $p = n$, functions need not be bounded ($\log\log(1 + 1/|x|)$ in $H^1(\mathbb{R}^2)$, 4A.5 The Fourier Transform); they are exponentially integrable instead (Trudinger, 1967), and they lie in every $L^q_{\mathrm{loc}}$ with $q < \infty$.

For higher derivatives the inequalities iterate. $W^{k,p} \hookrightarrow L^q$ with $\frac1q = \frac1p - \frac kn$ when $kp < n$, and $W^{k,p} \hookrightarrow C^{m,\alpha}$ when $k - \frac np > m + \alpha$ with $0 < \alpha < 1$ (and in borderline cases with care). This is what lets smoothness be proved one derivative at a time in $L^2$ and converted to classical derivatives at the end: a function in $H^k(\mathbb{R}^n)$ for every $k$ is smooth.

10.5Rellich–Kondrachov: compactness restored

Here is a complete proof in the case used most, $H^1_0(\Omega) \hookrightarrow L^2(\Omega)$, which is Arzelà–Ascoli (2B.5 Uniform Convergence and Arzelà–Ascoli) in disguise.

Step 1 is where the derivative bound is used: it says that, uniformly over the family, the functions are close to their own smoothed versions, so they can't oscillate on scales below $\varepsilon$. That is exactly the oscillation escape of 2B.3 Compactness being ruled out by a derivative bound. Escape to infinity is ruled out by the bounded domain. And concentration is ruled out by working below the critical exponent: Exercise 10.8 shows each of the three hypotheses is needed.

Now the deferred proofs can be completed. The solution operator $G$ of the Dirichlet problem maps $L^2$ boundedly into $H^1_0$, which embeds compactly in $L^2$; so $G$ is compact, and the spectral theorem of 4A.7 Compact Operators and Spectra gives the Dirichlet eigenfunctions. The mean-zero Poincaré inequality of 4A.9 Sobolev Spaces follows by the contradiction–compactness argument of its rehearsal. And on a closed manifold, Perelman's $\lambda(g)$ is attained.

10.6Failure at the critical exponent

At $q = p^*$ the embedding $W^{1,p} \hookrightarrow L^{p^*}$ is continuous but never compact, even on a bounded domain. Fix $u \in C_c^\infty(B_1)$ with $u \neq 0$, take $p = 2$ and $n \geq 3$, and let

By the scaling rules of 3A.5 Product Measures and Change of Variables (Exercise 10.9), $\|\nabla u_k\|_2 = \|\nabla u\|_2$ and $\|u_k\|_{2^*} = \|u\|_{2^*}$ for every $k$: the critical norms don't notice the rescaling. The functions are supported in $B_{1/k}$, so $u_k \to 0$ almost everywhere and weakly in $H^1_0(B_1)$ (4A.6 Weak Convergence and the Direct Method). If a subsequence converged in $L^{2^*}$, its limit would have to be $0$, contradicting $\|u_k\|_{2^*} = \|u\|_{2^*} \neq 0$. The mass concentrates into a bubble at the origin: escape to vertical infinity (3A.3 The Lebesgue Integral), now in a form that is invisible to the critical norms.

The remedy is the one the template of 2B.3 Compactness always uses: normalise by rescaling. If a sequence concentrates at a point at scale $\frac1k$, undo the concentration by $v_k(y) = k^{-(n-2)/2}u_k(y/k)$; the rescaled sequence no longer concentrates, and a limit can be extracted from it. Pierre-Louis Lions's concentration–compactness principle (1984) makes this systematic: a bounded sequence in a critical space either converges, or loses mass in finitely many concentrating bubbles, each of which, rescaled, converges to a non-trivial limit, or escapes to infinity. Thierry Aubin and Richard Schoen used these ideas to solve the Yamabe problem (finding a conformal metric of constant scalar curvature), whose difficulty is exactly that it lives at the critical Sobolev exponent.

10.7History

Sergei Sobolev proved his embedding theorem in 1938, by potential estimates. Emilio Gagliardo (1958) and Louis Nirenberg (1959) gave the elementary proof by integration along lines used above, which also handles $p = 1$. Charles Morrey proved his Hölder estimate in 1938–40. Franz Rellich proved the compactness theorem in 1930 and Vladimir Kondrachov extended it in 1945. The equivalence with the isoperimetric inequality is due to Federer and Fleming and to Maz'ya, both in 1960; the sharp constants for $p > 1$ were found by Thierry Aubin and Giorgio Talenti in 1976. Neil Trudinger's exponential integrability at $p = n$ dates from 1967, Lions's concentration–compactness from 1984, and the solution of the Yamabe problem was completed by Schoen in 1984.

10.8Exercises

Solution

(a) $-\frac nq = 2 - \frac n2$, so $q = \frac{2n}{n-4}$ for $n > 4$. (b) $0 = \theta(1 - \frac np) + (1 - \theta)(-\frac nq)$, so $\theta = \frac{n/q}{1 - n/p + n/q}$. (c) Under $u(\lambda x)$: left $\lambda^{-\frac n2(2 + \frac4n)} = \lambda^{-n-2}$; right $\lambda^{2-n}\lambda^{-n\cdot\frac4n} = \lambda^{-n-2}$.

Solution

Disc $2\sqrt\pi \approx 3.545$; square $4$; triangle $\approx 4.559$. Ball $(36\pi)^{1/3} \approx 4.836$; cube $6$.

Solution

(a) Escape to infinity: the translates are bounded in $H^1$, converge weakly to $0$, and stay at $L^2$ distance $\sqrt2\|u\|_2$ apart. (b) Bad boundaries let functions concentrate at the cusp tip (a missing-point phenomenon); this is why some boundary regularity is assumed. (c) Concentration at a point.

Solution

$\|u\|_{2^*}^2 \geq \big((\frac r2)^{2^*}V(\frac r2)\big)^{2/2^*} = (\frac r2)^2V(\frac r2)^{(n-2)/n}$, and $\int|\nabla u|^2 \leq V(r)$. Setting $V(r) = a_r r^n$, the inequality becomes $a_r \geq c\,a_{r/2}^{(n-2)/n}$ with $c = \frac{1}{4A}\cdot2^{-(n-2)}$; iterating, $a_r \geq c^{1 + \theta + \theta^2 + \cdots}a_{r/2^m}^{\theta^m}$ with $\theta = \frac{n-2}{n} < 1$, and $a_{r/2^m}^{\theta^m} \to 1$ because $a_{r/2^m} \to \omega_n > 0$. So $a_r \geq c^{1/(1-\theta)} = c^{n/2}$, a positive constant depending only on $n$ and $A$.