Charles K. Smart

An easy proof of Jensen's theorem on the uniqueness of infinity harmonic functions

We present a new, easy, and elementary proof of Jensen’s Theorem on the uniqueness of infinity harmonic functions. The idea is to pass to a finite difference equation by taking maximums and minimums over small balls.

Quantitative stochastic homogenization of elliptic equations in nondivergence form

We introduce a new method for studying stochastic homogenization of elliptic equations in nondivergence form. The main application is an algebraic error estimate, asserting that deviations from the homogenized limit are at most proportional to a power of the microscopic length scale, assuming a finite range of dependence. The results are new even for linear equations. The arguments rely on a new geometric quantity which is controlled in part by adapting elements of the regularity theory for the Monge–Ampère equation.

Convergence of the Abelian sandpile

The Abelian sandpile growth model is a diffusion process for configurations of chips placed on vertices of the integer lattice

Singular Solutions of Fully Nonlinear Elliptic Equations and Applications

\mathbb{Z}^d

An infinity Laplace equation with gradient term and mixed boundary conditions

, in which sites with at least 2d chips {\em topple}, distributing 1 chip to each of their neighbors in the lattice, until no more topplings are possible. From an initial configuration consisting of

Regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity

n

Convexity Criteria and Uniqueness of Absolutely Minimizing Functions

chips placed at a single vertex, the rescaled stable configuration seems to converge to a particular fractal pattern as

A Gaussian upper bound for martingale small-ball probabilities

n\to \infty

A Free Boundary Problem with Facets

. However, little has been proved about the appearance of the stable configurations. We use PDE techniques to prove that the rescaled stable configurations do indeed converge to a unique limit as

Quantitative stochastic homogenization of convex integral functionals

n \to \infty