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.
Mathematics Subject Classification (2000) Primary 35J70In this short article, we present a new proof of the famous result of Jensen [7], which establishes the uniqueness of viscosity solutions of the infinity Laplace equation
We show that an infinity harmonic function, that is, a viscosity solution of the nonlinear PDE − ∞ u = −u x i u x j u x i x j = 0, is everywhere differentiable. Our new innovation is proving the uniqueness of appropriately rescaled blow-up limits around an arbitrary point.
Mathematics Subject Classification (2000)Primary 49N60 · Secondary 35J20 · 35J65
Abstract. We present a modified version of the two-player "tug-of-war" game introduced by Peres, Schramm, Sheffield, and Wilson (2009). This new tugof-war game is identical to the original except near the boundary of the domain ∂Ω, but its associated value functions are more regular. The dynamic programming principle implies that the value functions satisfy a certain finite difference equation. By studying this difference equation directly and adapting techniques from viscosity solution theory, we prove a number of new results.We show that the finite difference equation has unique maximal and minimal solutions, which are identified as the value functions for the two tug-of-war players. We demonstrate uniqueness, and hence the existence of a value for the game, in the case that the running payoff function is nonnegative. We also show that uniqueness holds in certain cases for sign-changing running payoff functions which are sufficiently small. In the limit ε → 0, we obtain the convergence of the value functions to a viscosity solution of the normalized infinity Laplace equation.We also obtain several new results for the normalized infinity Laplace equation −Δ ∞ u = f . In particular, we demonstrate the existence of solutions to the Dirichlet problem for any bounded continuous f , and continuous boundary data, as well as the uniqueness of solutions to this problem in the generic case. We present a new elementary proof of uniqueness in the case that f > 0, f < 0, or f ≡ 0. The stability of the solutions with respect to f is also studied, and an explicit continuous dependence estimate from f ≡ 0 is obtained.
Abstract. We prove that a viscosity solution of a uniformly elliptic, fully nonlinear equation is C 2,α on the compliment of a closed set of Hausdorff dimension at most ε less than the dimension. The equation is assumed to be C 1 , and the constant ε > 0 depends only on the dimension and the ellipticity constants. The argument combines the W 2,ε estimates of Lin with a result of Savin on the C 2,α regularity of viscosity solutions which are close to quadratic polynomials.
Abstract. 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.
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