De Giorgi–Nash–Moser

Every estimate so far needed some regularity of the coefficients: Hölder continuity for Schauder (6B.7 Schauder Theory and the Continuity Method), continuity for $L^p$ (6B.8 Lᵖ Theory and Strong Solutions). This chapter proves estimates that need none. Consider a divergence-form equation

where the matrix $A(x)$ is only assumed measurable, bounded and uniformly elliptic: it may jump wildly from point to point, as the conductivity of a composite material does. Ennio De Giorgi (1957) and John Nash (1958), independently and by completely different methods, proved that every weak solution is nevertheless Hölder continuous, with an exponent and a constant that depend only on the dimension and the ellipticity ratio. Jürgen Moser (1960–61) then gave a third proof, by an iteration that is now the standard tool, and with it the Harnack inequality for such equations.

Why does such a theorem matter, when equations with rough coefficients are rare in geometry? Because of nonlinearity. If $u$ minimises $\int F(\nabla u)$ with $F$ smooth and uniformly convex, then each derivative $\partial_ku$ satisfies $\operatorname{div}(F''(\nabla u)\nabla\partial_ku) = 0$: an equation of exactly this form, whose coefficients $F''(\nabla u)$ are only as good as $\nabla u$, which is what we are trying to control. De Giorgi–Nash makes $\nabla u$ Hölder, Schauder then bootstraps to $C^\infty$, and Hilbert's nineteenth problem, whether minimisers of such "regular" variational problems are smooth, is solved. The same circle of ideas runs through the analysis of geometric flows: the Moser iteration is how $L^2$ or $L^p$ bounds on curvature are upgraded to pointwise bounds (11B.3 Compactness of Ricci Flows), and the parabolic Harnack inequality is the template for Li–Yau and Hamilton (6A.12 Entropy, Information and Diffusion, 11B.2 Ancient Solutions and the Harnack Inequality).

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

• define weak sub- and supersolutions of divergence-form equations with measurable coefficients, and prove Caccioppoli's inequality;
• prove local boundedness of subsolutions by Moser iteration, combining Caccioppoli with the Sobolev inequality;
• prove the logarithmic estimate for positive supersolutions, and explain how the John–Nirenberg lemma turns it into the weak Harnack inequality;
• deduce Hölder continuity from the Harnack inequality by oscillation decay;
• explain how the theorem solves Hilbert's nineteenth problem, what fails for systems, and what the parabolic version says.

9.1Heat in a composite

9.2Setting: weak sub- and supersolutions

Let $\Omega \subseteq \mathbb{R}^n$ be open and $A = (a^{ij})$ a measurable matrix function on $\Omega$, symmetric (for simplicity), with

for constants $0 < \lambda \leq \Lambda$. A function $u \in H^1_{\mathrm{loc}}(\Omega)$ is a weak subsolution of $\operatorname{div}(A\nabla u) = 0$ if

a weak supersolution if the inequality is reversed, and a weak solution if it is an equality for all such $\varphi$ (of any sign). For a smooth $u$ and smooth $A$, integrating by parts shows that subsolution means $\operatorname{div}(A\nabla u) \geq 0$, the analogue of subharmonic (6B.6 Perron’s Method and Barriers): "the equation pushes $u$ up".

Two facts about Sobolev functions are used without comment (Gilbarg–Trudinger, Lemma 7.6): if $u \in H^1$, then $u^+ = \max(u, 0)$ and $(u - k)^+$ are in $H^1$, with $\nabla(u - k)^+ = \nabla u$ on $\{u > k\}$ and $0$ elsewhere (almost everywhere); and products of $H^1$ functions with bounded Lipschitz functions are in $H^1$. In the proofs we apply the equation to test functions such as $\zeta^2(u^+)^\beta$; when $u$ is unbounded these must first be truncated at a height $N$ and the estimates proved uniformly in $N$, a routine step we note once and then suppress.

9.3Caccioppoli's inequality

The basic estimate is a reverse Poincaré inequality: for subsolutions, the gradient is controlled by the function itself on a slightly larger ball.

For a harmonic function this says that $\int_{B_{r/2}}|\nabla u|^2 \leq \frac{C}{r^2}\int_{B_r}u^2$: the energy on a ball is bounded by the size of the function on a ball twice as large, with the scaling $r^{-2}$ of 6A.2 Harmonic Functions's derivative estimates. Unlike those, it needs no mean value property, only the equation in weak form and ellipticity, and so it holds for every equation in our class.

9.4Local boundedness: Moser iteration

Caccioppoli trades a derivative for a cut-off. The Sobolev inequality (4A.11 Sobolev Embeddings and Critical Exponents) trades a derivative for integrability: for $v \in H^1_0(\mathbb{R}^n)$, $n \geq 3$,

(For $n = 2$, any $\chi > 1$ works, with a constant depending on $\chi$ and on the size of the support.) Together they give a gain of integrability with no loss: if $u$ is a subsolution in $L^q$ on a ball, then it is in $L^{\chi q}$ on a smaller ball. Iterating infinitely often, with the balls shrinking to a limit ball and the exponents growing to infinity, gives a bound in $L^\infty$. The art is to keep the constants under control.

The iteration is a machine: Caccioppoli + Sobolev = reverse Hölder with a gain $\chi > 1$, and an infinite product that converges because the gains compound geometrically while the losses (powers of $4^k$ from the shrinking cut-offs) are only geometric in $k$ inside exponents that shrink like $\chi^{-k}$. The same machine works for subsolutions of parabolic equations, on manifolds that satisfy a Sobolev inequality, and for nonlinear quantities that satisfy a differential inequality (Exercise 9.13). Running it with $q_0$ any positive number (not only $2$) and an interpolation argument gives $\sup_{B_1}u^+ \leq C_p\big(\int_{B_2}(u^+)^p\big)^{1/p}$ for every $p > 0$ (Gilbarg–Trudinger, Theorem 8.17).

9.5Positive supersolutions: the logarithm and the crossover

For the Harnack inequality we need a lower bound for positive supersolutions. Running the iteration with negative exponents $\beta < 0$ in the test function $\zeta^2u^\beta$ gives, by the same computation, a bound

for every $p > 0$ (Gilbarg–Trudinger, Theorem 8.18; Jost, section 14.1). This bounds the infimum by a negative power average. To connect it with a positive power average, which is what the upper bound uses, there is a crossover step: we need, for some small $p_0 > 0$,

That says $u$ is not much larger than $1/(1/u)$ on average, a statement about the oscillation of $\log u$. Moser's insight was that the logarithm of a positive supersolution is controlled by the equation.

A function whose mean oscillation over every ball is bounded is said to have bounded mean oscillation, written $v \in \mathrm{BMO}$. Bounded functions are in BMO, but so is $\log|x|$, which is unbounded. The point of BMO is the following theorem.

Fritz John and Louis Nirenberg proved it in 1961; the proof is a Calderón–Zygmund decomposition (6B.8 Lᵖ Theory and Strong Solutions) of $|v - v_B|$ at height proportional to $K$, iterated: the set where $|v - v_B|$ exceeds $jK$ has measure decaying geometrically in $j$ (Gilbarg–Trudinger, Theorem 7.21; Exercise 9.10 does a simple case). Applied to $v = \log u$,

which is the crossover. Combined with the negative-power iteration, this gives the weak Harnack inequality for positive supersolutions,

and, combined with local boundedness for $p = p_0$, the full theorem.

It is the Harnack inequality of 6A.2 Harmonic Functions for harmonic functions, now for every equation in the class, with a constant that does not see the coefficients beyond their ellipticity. The proof for harmonic functions used the mean value property; this one uses nothing but Caccioppoli, Sobolev and John–Nirenberg.

9.6From Harnack to Hölder continuity

The proof shows where the exponent comes from: each fourfold shrinking of the ball reduces the oscillation by the factor $\gamma$, which depends on the Harnack constant, which depends only on $n$ and $\Lambda/\lambda$. With a large contrast, $\gamma$ is close to $1$ and $\alpha$ is small. That is not an artefact of the proof: there are equations in the class whose solutions are Hölder with no better exponent than $\alpha \to 0$ as $\Lambda/\lambda \to \infty$ (Exercise 9.9).

De Giorgi's method, the original proof, works with level sets instead of powers: Caccioppoli's inequality for $(u - k)^+$ at every level $k$ (a function satisfying it for all $k$ belongs to a De Giorgi class), combined with an isoperimetric-type lemma saying that a function cannot pass from a low level to a high level on a set of small measure without paying in gradient. It proves boundedness by an iteration over levels and Hölder continuity by a "shrinking measure" argument, and it extends to quasi-minimisers of variational problems, where there is no equation at all (Giaquinta; Jost, section 14.4). Nash's method used the fundamental solution and an entropy argument (6A.12 Entropy, Information and Diffusion).

9.7Hilbert's nineteenth problem

In his 1900 list of problems, David Hilbert asked whether the solutions of "regular" variational problems are always analytic. In the simplest form: if $u \in H^1(\Omega)$ minimises $E[u] = \int_\Omega F(\nabla u)\,dx$, where $F$ is smooth (or analytic) and uniformly convex, $\lambda|\xi|^2 \leq F_{p_ip_j}(p)\xi_i\xi_j \leq \Lambda|\xi|^2$, is $u$ smooth?

The difficulty is circular. The Euler–Lagrange equation $\operatorname{div}(F_p(\nabla u)) = 0$ (6A.11 Calculus of Variations and Gradient Flows) is nonlinear. Formally differentiating it in $x_k$ gives, for $w = \partial_ku$,

a linear equation for $w$ with coefficients $a^{ij}(x) = F_{p_ip_j}(\nabla u(x))$. Schauder theory would make $w$ Hölder if the coefficients were Hölder, that is, if $\nabla u$ were Hölder: what we want to prove. But the coefficients are certainly measurable and uniformly elliptic, because of the convexity of $F$, whatever $\nabla u$ is. So De Giorgi–Nash–Moser applies: $w = \partial_ku$ is Hölder, for each $k$. The circle is broken. Now the coefficients $F''(\nabla u)$ are Hölder, Schauder gives $\nabla u \in C^{1,\alpha}$, and bootstrapping gives $C^\infty$ (and analyticity, by a further argument of Morrey and of Friedman). Making the formal differentiation rigorous uses difference quotients, as in 6A.6 Weak Solutions and Elliptic Regularity: the minimiser is first shown to be in $H^2_{\mathrm{loc}}$, so that $\partial_ku \in H^1_{\mathrm{loc}}$ is a weak solution of the differentiated equation (Jost, section 14.4; Exercise 9.12).

Systems are different. For vector-valued minimisers $u : \Omega \to \mathbb{R}^m$, the differentiated equations form an elliptic system with measurable coefficients, and there De Giorgi–Nash–Moser fails. De Giorgi himself gave a counterexample in 1968: a linear elliptic system with bounded measurable coefficients in dimension $n \geq 3$ with an unbounded solution, $u(x) = |x|^{-\gamma}x$ for a suitable $\gamma$. Minimisers of variational problems for maps can be singular, and the regularity theory for systems is partial: smooth outside a closed set of small dimension. This is the situation for harmonic maps (6C.7 Harmonic Maps and Their Heat Flow) and for Yang–Mills fields, and it is why their singularities are a subject of their own.

9.8The parabolic theory

Nash's 1958 paper was about parabolic equations $u_t = \operatorname{div}(A(x, t)\nabla u)$ with measurable coefficients, and proved Hölder continuity for them too. Moser proved the parabolic Harnack inequality in 1964: if $u \geq 0$ solves such an equation in $Q = B_2\times(0, 4)$, then

The supremum is taken at earlier times than the infimum, with a gap between. The gap cannot be removed: for the heat kernel $\Phi(x, t)$ the ratio $\frac{\Phi(x, t)}{\Phi(y, t)} = e^{(|y|^2 - |x|^2)/4t}$ at a single time is unbounded as $t \to 0$ (Exercise 9.11). Heat must have time to arrive. The Li–Yau inequality (6A.12 Entropy, Information and Diffusion) is the sharp form of this Harnack inequality for smooth coefficients on manifolds with $\operatorname{Ric} \geq 0$, and Hamilton's Harnack inequality for the Ricci flow (11B.2 Ancient Solutions and the Harnack Inequality) is its geometric descendant.

9.9History

David Hilbert posed his nineteenth problem in 1900; Sergei Bernstein (1904) and others solved it under extra smoothness assumptions on the minimiser. Ennio De Giorgi's 1957 paper solved it in general, and John Nash's 1958 paper, written independently and published shortly afterwards, proved the parabolic theorem by another route; Nash learned of De Giorgi's priority after his work was done. Jürgen Moser gave his iteration proof of the elliptic theorem in 1960, the elliptic Harnack inequality in 1961, and the parabolic Harnack inequality in 1964. Fritz John and Louis Nirenberg introduced BMO and proved their lemma in 1961. De Giorgi's counterexample for systems dates from 1968. Krylov and Safonov proved the non-divergence Harnack inequality in 1979–80.

9.10Exercises

Solution

(a) Testing $\int\nabla u\cdot\nabla(\zeta^2u) = 0$: $\int\zeta^2|\nabla u|^2 = -2\int\zeta u\nabla u\cdot\nabla\zeta \leq \frac12\int\zeta^2|\nabla u|^2 + 2\int u^2|\nabla\zeta|^2$, so $\int\zeta^2|\nabla u|^2 \leq 4\int u^2|\nabla\zeta|^2 \leq \frac{4}{r^2}\int_{B_{2r}}u^2$ with $|\nabla\zeta| \leq \frac1r$. (b) For $u = x_1$, the left side is $|B_r| = \alpha(n)r^n$, and $\int_{B_{2r}}x_1^2 = \frac{1}{n}\int_{B_{2r}}|x|^2 = \frac{\alpha(n)(2r)^{n+2}}{n + 2}$, so the right side is $\frac{C\alpha(n)2^{n+2}r^n}{n + 2}$: both are multiples of $r^n$. Under $u(x) \mapsto u(\lambda x)$ the left side scales like $\lambda^{2-n}$ and $\int u^2$ like $\lambda^{-n}$, so the factor must be $r^{-2}$.

Solution

With $\theta = \frac1\chi < 1$: $\sum\frac{\theta^k}{2} = \frac{1}{2(1 - \theta)}$ and $\sum\frac{(k + 1)\theta^k}{2} = \frac{1}{2(1 - \theta)^2}$. For $n = 3$, $\chi = 3$, $\theta = \frac13$: the sums are $\frac34$ and $\frac98$, so $C \leq C_2^{3/4}4^{9/8}$.

Hint

Compute $A\nabla u$ in polar coordinates: $\nabla u = \alpha r^{\alpha-1}\cos\theta\,e_r - r^{\alpha-1}\sin\theta\,e_\theta$, so $A\nabla u = \alpha r^{\alpha-1}\cos\theta\,e_r - \alpha^2r^{\alpha-1}\sin\theta\,e_\theta$. Then use $\operatorname{div}(fe_r + ge_\theta) = \frac1r\partial_r(rf) + \frac1r\partial_\theta g$.

Solution

$\operatorname{div}(A\nabla u) = \frac1r\partial_r(\alpha r^\alpha\cos\theta) + \frac1r\partial_\theta(-\alpha^2r^{\alpha-1}\sin\theta) = \alpha^2r^{\alpha-2}\cos\theta - \alpha^2r^{\alpha-2}\cos\theta = 0$ away from the origin. Near the origin $|\nabla u|^2 \sim r^{2\alpha-2}$, integrable in the plane, and the boundary terms on small circles vanish, so $u$ is a weak solution in the whole disc. It is Hölder with exponent exactly $\alpha$ at the origin, and $\alpha = \sqrt{\lambda/\Lambda}$.

Solution

(a) By scaling ($v(\lambda x) = v(x) + \log\lambda$, and subtracting the average removes constants), it suffices to treat intervals of length $1$ at distance $d$ from $0$: if $d \geq 1$, $|v'| \leq 1$ on $I$ and the oscillation is at most $1$; if $d < 1$, $I \subset (-2, 2)$ and $\int_{-2}^2|\log|x|| + |v_I| \leq C$. (b) $e^{p|\log|x||} = |x|^{-p}$ near $0$, integrable iff $p < 1$.

Solution

$\frac{\Phi(0, t)}{\Phi(1, t)} = e^{1/4t} \to \infty$. Any constant $C$ would be violated by the heat kernel (shifted in time so it is positive on the cylinder) at small $t$. With a lag, Li–Yau gives $\Phi(0, t_1) \leq \Phi(1, t_2)\big(\frac{t_2}{t_1}\big)^{1/2}e^{1/4(t_2 - t_1)}$; for $t_2 = 2t_1$ this is $\sqrt2e^{1/4t_1}\Phi(1, t_2)$, bounded by $\sqrt2e^{1/4}\Phi(1, t_2)$ for $t_1 \geq 1$. The lag must be comparable to the square of the distance for a uniform constant: parabolic scaling.

Solution

(a) $\partial_k\partial_i(F_{p_i}(\nabla u)) = \partial_i\big(F_{p_ip_j}(\nabla u)\partial_j\partial_ku\big) = 0$. (b) $F_{p_i} = \frac{p_i}{\sqrt{1 + |p|^2}}$ and $F_{p_ip_j} = \frac{1}{\sqrt{1 + |p|^2}}\big(\delta_{ij} - \frac{p_ip_j}{1 + |p|^2}\big)$, with eigenvalues $(1 + |p|^2)^{-1/2}$ (on vectors orthogonal to $p$) and $(1 + |p|^2)^{-3/2}$ (on $p$). The ratio $\Lambda/\lambda = 1 + |p|^2$ is bounded only when $|\nabla u|$ is. So De Giorgi–Nash–Moser applies to the minimal surface equation once a gradient bound is known, and the regularity theory for minimal graphs reduces to proving one.

Solution

(a) Caccioppoli uses only the weak form of $\Delta u \geq 0$ and integration by parts with a cut-off, valid on any manifold; the Sobolev step uses the assumed inequality, now with averages, and the constants $r^2$ from $|\nabla\zeta| \leq \frac{2}{r}$ cancel the $r^2$ in the Sobolev inequality, so the iteration constants are scale-free. (b) The Sobolev inequality on balls with a uniform constant; on manifolds it follows from a lower Ricci bound plus a volume (noncollapsing) lower bound. It is used once per iteration step. (c) If a quantity $Q \geq 0$ satisfies $(\partial_t - \Delta)Q \leq CQ$ (a subsolution inequality), the parabolic version of the iteration with a uniform Sobolev constant gives $\sup Q$ on a parabolic cylinder in terms of an $L^2$ (or $L^p$) average on a larger one. A uniform Sobolev inequality along the flow is exactly the ingredient that makes the constants uniform in time.