Quasilinear Elliptic Equations and Minimal Surfaces

A quasilinear equation is linear in the highest derivatives, with coefficients that depend on the solution and its gradient:

Most equations of geometry have this form: the minimal surface equation, the equation of prescribed mean curvature, the equations of capillary surfaces, and (as systems) the harmonic map and Ricci flow equations. The difficulty is that the ellipticity, and every linear estimate, depends on $\nabla u$, which is part of the unknown. If $\nabla u$ is controlled, the equation is a linear one with known Hölder coefficients and the theory of 6B.7 Schauder Theory and the Continuity Method takes over; if not, nothing works. So the theory of quasilinear equations is, to a remarkable extent, the theory of gradient bounds.

This chapter follows that theory through its most important example, the minimal surface equation

which says that the graph of $u$ has zero mean curvature. We prove existence on uniformly convex domains by the Leray–Schauder method of 6C.5 Fixed Points, Monotonicity and Sub- and Supersolutions, following the classical sequence of four a priori estimates; show by an explicit example that existence fails on an annulus; and prove Sergei Bernstein's theorem that an entire minimal graph over the plane is a plane.

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

• derive the minimal surface equation from the area functional, and compute how its ellipticity degenerates as $|\nabla u| \to \infty$;
• prove the comparison principle for quasilinear equations in divergence form, and uniqueness for the Dirichlet problem;
• prove a global gradient bound from the bounded slope condition, by comparison with planes and with translates of the solution;
• assemble the four a priori estimates into an existence proof by Schaefer's theorem;
• prove that the Dirichlet problem on an annulus has no solution for large boundary data, and prove Bernstein's theorem in dimension two.

6.1Soap films and roofs

6.2The minimal surface equation

The area of the graph of $u : U \to \mathbb{R}$ is

Its first variation in a direction $v \in C_c^\infty(U)$ is $\frac{d}{ds}\mathcal A(u + sv)\big|_{s=0} = \int_U\frac{\nabla u\cdot\nabla v}{W}\,dx$, so the Euler–Lagrange equation (6A.11 Calculus of Variations and Gradient Flows) is

Geometrically, $\operatorname{div}(\nabla u/W)$ is the mean curvature $H$ of the graph (the sum of the principal curvatures, with the sign convention that makes it positive for graphs that curve upward, like $u = \frac12|x|^2$; 9A.8 Submanifolds and Minimal Surfaces), so (MSE) says $H = 0$. Expanding the divergence (Exercise 6.10) gives the non-divergence form

It is convenient to write (MSE) as $\operatorname{div}A(\nabla u) = 0$ with $A(p) = p/\sqrt{1 + |p|^2}$. The matrix $DA(p)$, with entries $\partial A^i/\partial p_j = \frac{1}{W}\big(\delta_{ij} - \frac{p_ip_j}{W^2}\big)$, has eigenvalue $\frac1W$ on vectors perpendicular to $p$ and $\frac{1}{W^3}$ in the direction of $p$ (Exercise 6.11). So the equation is elliptic, but the ratio of the largest to the smallest eigenvalue is $W^2 = 1 + |p|^2$, unbounded as the gradient grows. On a set where $|\nabla u| \leq K$, it is uniformly elliptic, with constants depending on $K$. A gradient bound is exactly what turns (MSE) into an equation that the linear theory can handle.

Two features of (MSE) will be used constantly. It has no explicit dependence on $x$ or $u$, so translates $u(x + \tau)$ and vertical shifts $u + c$ of solutions are solutions. And affine functions $L(x) = a\cdot x + c$ are solutions: their graphs are planes.

6.3Comparison and uniqueness

The proof shows a general principle, already met in 6C.3 Symmetry, Noether’s Theorem and Moving Planes: the difference of two solutions of a nonlinear equation solves a linear equation, whose coefficients are averages of the linearisation along the segment between them. Comparison, uniqueness and stability for nonlinear equations all go through this device.

6.4The gradient estimate from the bounded slope condition

Since affine functions solve (MSE), they can be compared with the solution. This gives a boundary gradient bound whenever the boundary data can be squeezed between planes.

The condition forces $U$ to be convex (unless $\phi$ is affine on flat parts of the boundary), and it holds when $\partial U$ is $C^2$ and uniformly convex and $\phi$ is the restriction of a $C^2$ function on $\overline U$, with $K$ depending on $\|\phi\|_{C^2}$ and the lower bound for the curvature of $\partial U$ (a lemma of Hartman, Stampacchia and Miranda; the ball is Exercise 6.13).

The argument needed only three properties: comparison, invariance under translations and vertical shifts, and planes as solutions. It works for any equation $\operatorname{div}A(\nabla u) = 0$ with $A$ elliptic, and it is the reason the minimal surface problem was first solved on convex domains (by Haar and Radó in the 1920s and 1930s, in the setting of the direct method).

Interior estimates without boundary data. For (MSE) there is also a purely interior gradient bound, much deeper: $|\nabla u(x_0)| \leq C_1\exp\big(C_2\frac{\sup_{B_r(x_0)}(u - u(x_0))}{r}\big)$ for a solution on $B_r(x_0)$, proved by Bombieri, De Giorgi and Miranda (1969) in all dimensions, with a later proof by Korevaar using the maximum principle. It underlies the compactness theory of minimal graphs; we only state it.

6.5The four a priori estimates and existence

The general scheme for the quasilinear Dirichlet problem $Qu = 0$ in $U$, $u = \phi$ on $\partial U$, goes back to Bernstein (around 1910) and was completed by Ladyzhenskaya and Ural'tseva in the 1960s. Embed the problem in the family $Q_\sigma$ ($0 \leq \sigma \leq 1$) with boundary values $\sigma\phi$ and estimate, uniformly in $\sigma$, any solution:

1. $\sup|u|$, usually by the maximum principle;
2. $\sup_{\partial U}|\nabla u|$, by barriers at the boundary;
3. $\sup_U|\nabla u|$, by a maximum principle for the gradient (or, here, by translation);
4. $[\nabla u]_{C^\beta(\overline U)}$, a Hölder bound for the gradient, which is the step that uses the deep linear theory.

Once 1–4 hold, the coefficients $a^{ij}(x, u, \nabla u)$ are Hölder continuous with known bounds, and the Schauder theory gives everything else. Here is step 4 for equations in divergence form.

The interior part of the proof is the idea that solved Hilbert's nineteenth problem (6B.9 De Giorgi–Nash–Moser). Differentiate the equation in $x_k$: $v = \partial_ku$ satisfies

Nothing is known about the regularity of these coefficients, but they are measurable, bounded and uniformly elliptic, with constants depending only on $K$ (for (MSE), between $(1 + K^2)^{-3/2}$ and $1$). De Giorgi–Nash–Moser applies to $v$ and gives a Hölder bound for $\nabla u$ on interior balls. Near the boundary, tangential derivatives are handled the same way and the normal derivative through the equation itself (Gilbarg–Trudinger, section 13.4, and Ladyzhenskaya and Ural'tseva's book).

Mean convexity is the sharp condition. Uniform convexity was used only to get the bounded slope condition. Howard Jenkins and James Serrin proved in 1968 that for a bounded $C^{2,\alpha}$ domain the Dirichlet problem for (MSE) is solvable for every smooth boundary function exactly when the boundary has non-negative mean curvature (with respect to the inner normal) at every point. In two dimensions the condition is convexity of the domain. When the boundary curves the wrong way somewhere, there is data for which no solution exists, and the next example shows the mechanism.

6.6Bernstein's theorem

A solution of (MSE) on all of $\mathbb{R}^n$ is an entire minimal graph. Planes are examples. Are there others? For harmonic functions there are many (any harmonic polynomial), but the nonlinearity of (MSE) changes the answer.

We give the proof of Johannes Nitsche (1957), which passes through the Monge–Ampère equation. Write $p = u_x$, $q = u_y$ and $W = \sqrt{1 + p^2 + q^2}$, all $C^1$ functions.

Higher dimensions. Bernstein's theorem is true for entire minimal graphs over $\mathbb{R}^n$ with $n \leq 7$: Ennio De Giorgi proved it for $n = 3$ (1965), Frederick Almgren for $n = 4$ (1966) and James Simons for $n \leq 7$ (1968), all by reducing it to the non-existence of singular area-minimising cones in one dimension lower. Simons also showed that the cone $\{x_1^2 + \cdots + x_4^2 = x_5^2 + \cdots + x_8^2\}$ in $\mathbb{R}^8$ is stable, and Enrico Bombieri, De Giorgi and Enrico Giusti proved in 1969 that it is area minimising and that there is an entire non-affine minimal graph over $\mathbb{R}^8$ (a hypersurface in $\mathbb{R}^9$). So Bernstein's theorem holds exactly for $n \leq 7$. The dimension $8$ is the same threshold as for singularities of area-minimising hypersurfaces, which can occur only in ambient dimension $8$ and above.

6.7History

Lagrange derived the minimal surface equation in 1762, and Meusnier identified its geometric meaning (zero mean curvature) in 1776. Sergei Bernstein proved his theorem around 1915–17 (his proof had a gap, filled by Hopf and Mickle in 1950), and he also introduced the program of a priori estimates for quasilinear equations. Alfred Haar and Tibor Radó solved the Dirichlet problem on convex domains in the late 1920s and early 1930s. Konrad Jörgens proved his theorem on $\det D^2F = 1$ in 1954, and Johannes Nitsche used it for his short proof of Bernstein's theorem in 1957. Olga Ladyzhenskaya and Nina Ural'tseva completed the general theory of quasilinear elliptic equations in the 1960s (their book appeared in 1964). Jenkins and Serrin's mean-convexity criterion is from 1968, and the Bombieri–De Giorgi–Giusti counterexample from 1969.

6.8Exercises

Solution

$\partial_i\big(\frac{\partial_iu}{W}\big) = \frac{\partial_{ii}u}{W} - \frac{\partial_iu\,\partial_iW}{W^2}$, and $\partial_iW = \frac{\partial_ju\,\partial_{ij}u}{W}$. Summing over $i$ and multiplying by $W$ gives $\Delta u - \frac{\partial_iu\,\partial_ju\,\partial_{ij}u}{W^2}$.

Solution

$\partial_jA^i = \frac{\delta_{ij}}{W} - \frac{p_ip_j}{W^3}$. For $v \perp p$: $DA(p)v = \frac vW$. For $v = p$: $DA(p)p = \frac pW - \frac{|p|^2p}{W^3} = \frac{W^2 - |p|^2}{W^3}p = \frac{p}{W^3}$.

Solution

$p = \tan x$, $q = -\tan y$, $u_{xx} = \sec^2x$, $u_{yy} = -\sec^2y$, $u_{xy} = 0$. Then $(1 + q^2)u_{xx} + (1 + p^2)u_{yy} - 2pqu_{xy} = \sec^2y\sec^2x - \sec^2x\sec^2y = 0$. The solution tends to $+\infty$ on the sides $x = \pm\frac\pi2$ and to $-\infty$ on the sides $y = \pm\frac\pi2$. The data are infinite, not $C^{2,\alpha}$, and the square is convex but not uniformly convex. Jenkins and Serrin (1966) characterised exactly when such infinite boundary data can be prescribed on the sides of a polygon.

Solution

For $|x| = 1$: $x_0\cdot(x - x_0) = x_0\cdot x - 1 = -\frac12|x - x_0|^2$. By Taylor, $|\phi(x) - \phi(x_0) - \nabla\phi(x_0)\cdot(x - x_0)| \leq \frac{K_2}{2}|x - x_0|^2$. So $L^+(x) - \phi(x) = \frac{K_2}{2}|x - x_0|^2 - \big(\phi(x) - \phi(x_0) - \nabla\phi(x_0)\cdot(x - x_0)\big) \geq 0$, and similarly $\phi - L^- \geq 0$. The slopes are $|\nabla\phi(x_0) \mp K_2x_0| \leq K_1 + K_2$.

Solution

(a) $h_{\max} = \cosh^{-1}2 = \log(2 + \sqrt3) \approx 1.317$. (b) The condition is $g(c) = 1$. As $c \to 0$, $g(c) \to \infty$ (the exponential wins), and as $c \to \infty$, $g(c) \to \infty$ too; $g$ is continuous and positive, so it has a positive minimum $m(d)$, which grows with $d$ (for fixed $c$, $g$ increases with $d$). By scaling, $m(d) = d\cdot\min_s s\cosh\frac{1}{2s}$, proportional to $d$. When $m(d) > 1$, the equation $g(c) = 1$ has no solution and no catenoid spans the rings; numerically $\min_s s\cosh\frac1{2s} \approx 0.754$, so this happens for $d > 1/0.754 \approx 1.33$. Physically, the film then breaks into two discs.

Solution

Use $W_x = \frac{pp_x + qq_x}{W}$, $W_y = \frac{pp_y + qq_y}{W}$, and $p_y = q_x = u_{xy}$. Then $\partial_x\frac{1 + q^2}{W} = \frac{2qq_x}{W} - \frac{(1 + q^2)(pp_x + qq_x)}{W^3}$ and $\partial_y\frac{pq}{W} = \frac{p_yq + pq_y}{W} - \frac{pq(pp_y + qq_y)}{W^3}$. Multiply the difference by $W^3$ and use $W^2 = 1 + p^2 + q^2$: with $u_{xx} = p_x$, $u_{xy} = q_x = p_y$, $u_{yy} = q_y$,

$$W^3(\text{LHS} - \text{RHS}) = 2qu_{xy}W^2 - (1 + q^2)(pu_{xx} + qu_{xy}) - (qu_{xy} + pu_{yy})W^2 + pq(pu_{xy} + qu_{yy}).$$

Collecting terms: the $u_{xx}$ coefficient is $-p(1 + q^2)$; the $u_{yy}$ coefficient is $-pW^2 + pq^2 = -p(1 + p^2)$; the $u_{xy}$ coefficient is $2qW^2 - q(1 + q^2) - qW^2 + p^2q = q(W^2 - 1 - q^2 + p^2) = 2p^2q$. So the total is $-p\big[(1 + q^2)u_{xx} - 2pqu_{xy} + (1 + p^2)u_{yy}\big] = 0$. (The identity was also checked numerically on Scherk's surface.)

Solution

Convexity gives $F(y) \geq F(x) + \nabla F(x)\cdot(y - x)$ and $F(x) \geq F(y) + \nabla F(y)\cdot(x - y)$. Adding, $0 \geq \nabla F(x)\cdot(y - x) + \nabla F(y)\cdot(x - y) = -\big(\nabla F(x) - \nabla F(y)\big)\cdot(x - y)$.

Solution

$\Psi = U + iV$ with $U = \Phi^1$, $V = -\Phi^2$. Then $U_{\xi^1} = a$, $U_{\xi^2} = b$, $V_{\xi^1} = -b$, $V_{\xi^2} = a$. The Cauchy–Riemann equations $U_{\xi^1} = V_{\xi^2}$ and $U_{\xi^2} = -V_{\xi^1}$ hold, and $\Psi' = U_{\xi^1} + iV_{\xi^1} = a - ib$, of modulus $\sqrt{a^2 + b^2}$.

Solution

(a) Differentiate in $x_k$: $\partial_tv = a^{ij}(\nabla u)\partial_{ij}v + \frac{\partial a^{ij}}{\partial p_l}(\nabla u)\,\partial_l\partial_ku\,\partial_{ij}u$, and $\partial_l\partial_ku = \partial_lv$. (b) The equation for $v$ is linear in $v$, uniformly parabolic (for the smooth solution, $|\nabla u|$ is bounded on the compact torus for $t$ in compact intervals), with a first-order term and no zero-order term. By the weak maximum principle on the torus (no boundary), $\max_xv(\cdot, t)$ is non-increasing and $\min_xv(\cdot, t)$ is non-decreasing, so $\max_x|v(\cdot, t)| \leq \max_x|v(\cdot, 0)|$. Then $|\nabla u|^2 = \sum_k(\partial_ku)^2 \leq n\max_k\max_x|\partial_ku(\cdot, 0)|^2$.