### The Minkowski problem and the Monge-Ampere equation

So as it has been already noticed I haven’t posted much at all in a couple of months,  Sean has been doing a good job with the posting! Now that is summer and I have more free time I can get back to blogging. I won’t continue with my previous posts on minimal surfaces (at least for now), but rather talk for a couple of posts about something less heavy than geometric measure theory.

I would like to talk about the Minkowski problem. It is not only an  interesting and historically important problem in geometry but it also has deep connections with the theory of non linear elliptic equations.

The Minkowski problem goes as follows:

Given a strictly positive real function $f$ defined on $S^2$, find a strictly convex compact surface $\Sigma \subset \mathbb{R}^3$ such that the Gauss curvature of $\Sigma$ at the point $x$ equals $f(n(x))$, where $n(x)$ denotes the normal to $\Sigma$ at $x$.

The simplest example would be when $f(x)$ is identically equal to the constant $\frac{1}{R^2}$, in that case a solution to the Minkowski problem would be the sphere of radius $R$ (it also turns out that the Minkowski problem has a unique solution for a given $f$, so the sphere is THE solution to the Minkowski problem for $f \equiv R^{-2}$).

It is my understanding (although I only know very little about the historical background) that this problem was studied intensively (among many, many other people) by A.D. Alexksandrov and  A. Pogorelov. Aleksandrov developed a theory of weak solutions and showed existence of weak solutions. Years later, via the work of many people (particularly people working in non-linear PDE) it was shown that these weak solutions were actually smooth, classical solutions, thus proving that the problem always has a unique solution.

How do PDE’s get in the picture? I will do a computation that will make this matter clear, but first let me point out the analogy with the theory of minimal surfaces and the prescription of the mean curvature of a surface: Minimal surfaces are surfaces for which the mean curvature vanishes, thus if the graph of a function happens to be a minimal surface this function solves a PDE (“the minimal surface equation”), which happens to be a quasilinear PDE. The problem of showing the existence of classical solutions for the minimal surface equation and other problems from the calculus of variations motivated much of the work in PDE through part of the 20th century (read for instance, about the work of Ennio De Giorgi on Hilbert’s 19th problem). For the Minkowski problem what we are prescribing is not the average of the principal curvatures of the surface (mean curvature) but the product of the principal curvatures (Gauss curvature), the corresponding PDE is not a quasilinear PDE, but a fully non linear PDE closely related to the Monge-Ampere equation.

To explain this further, allow me to recall a classical computation for the Gauss curvature.

Let us consider a portion of a surface given by the graph of a function u(x,y) (any small patch can be written this way if we rotate and translate our coordinate system), it is well known that one can express the Gauss curvature of this surface in terms of partial derivatives of $u$. I am going to assume that if you are reading this you are familiar with geometry of surfaces, if not, there are a couple of good wikipedia articles on this, such as the one on the Weingarten map (aka the Shape operator).

To compute the the Gauss curvature, I am going to use the fact that the Gauss curvature equals the determinant of the differential of the Gauss map. The Gauss map is the map that associates to each point on the surface the unique unit vector parallel to the unit normal at that point), this defines a map from  the surface $\Sigma$ to the sphere $S^2$. Then its differential its just a map from the tangent plane at a point in $\Sigma$ to a tangent plane to $S^2$.

In the case when the surface is the graph of $u$, in the coordinates $(x,y) \to (x,y,u(x,y))$ the Gauss map is given by

$N_u(x)=\left ( \frac{-\nabla u}{\sqrt{1+\nabla u^2}}, \frac{1}{\sqrt{1+\nabla u^2}} \right )$

Then the Weingarten map (or second fundamental form, or shape operator) its the differential of this map, its determinant is the Gauss curvature, and computing the differential (I am cheating and won’t write the differential here) we see that the Gauss curvature equals

$\frac{det(D^2u)}{\left ( 1+|\nabla u|^2\right )^{\frac{n+2}{2}}}$

Then, if the surface $u$ parametrizes is a solution of the Minkowski problem, then

$\frac{det(D^2u)}{\left ( 1+|\nabla u|^2\right )^{\frac{n+2}{2}}}=f(N_u(x)) = g(\nabla u)$

where $g$ is the obvious function. Thus we have arrived at a PDE for $u$

$det (D^2u) = g(\nabla u) \left ( 1+|\nabla u|^2 \right)^{\frac{n+2}{2}}$

this is an equation of Monge-Ampere type.

There are several methods to construct solutions to these equations (weak solutions, approximation via polygonal surfaces, continuity method), some of these were developed independently in works of some people like A. Alexandroff, A. Pogorelov, P.L. Lions and S.Y. Cheng and S.T, Yau. One of these methods consists in building first solutions in a “weak” sense, known as Alexandroff solutions, which a priori might not be twice differentiable, so they are not solutions of the equation in the classical sense, THEN one proves that these weak solutions must actually be smooth (this is is usually called a regularity theory, and its considered the hardest part).  I will talk a bit about the existence of classical solutions on a future post on the continuity method.