Peiyong Wang

A PDE Perspective of the Normalized Infinity Laplacian

The inhomogeneous normalized infinity Laplace equation was derived from the tug-of-war game in [21] with the positive right-hand-side as a running payoff. The existence, uniqueness and comparison with polar quadratic functions were proved in [21] by the game theory. In this paper, the normalized infinity Laplacian, formally written as , is defined in a canonical way with the second derivatives in the local maximum and minimum directions, and understood analytically by a dichotomy. A comparison with polar quadratic polynomials property, the counterpart of the comparison with cones property, is proved to characterize the viscosity solutions of the inhomogeneous normalized infinity Laplace equation. We also prove that there is exactly one viscosity solution of the boundary value problem for the infinity Laplace equation in a bounded open subset of R n . The stability of the inhomogeneous infinity Laplace equation with strictly positive f and of the homogeneous equation by small perturbation of the right-hand-side and the boundary data is established in the last part of the work. Our PDE method approach is quite different from those in [21].

Uniqueness of ∞-Harmonic Functions and the Eikonal Equation

Comparison results are obtained between infinity subharmonic and infinity superharmonic functions defined on unbounded domains. The primary new tool employed is an approximation of infinity subharmonic functions that allows one to assume that gradients are bounded away from zero. This approximation also demystifies the proof in the case of a bounded domain. A second, quite different, topic is also taken up. This is the uniqueness of absolutely minimizing functions with respect to other norms besides the Euclidean, norms that correspond to comparison results for partial differential equations which are quite discontinuous.

REGULARITY OF FREE BOUNDARIES OF TWO-PHASE PROBLEMS FOR FULLY NONLINEAR ELLIPTIC EQUATIONS OF SECOND ORDER. II. FLAT FREE BOUNDARIES ARE LIPSCHITZ

ABSTRACT In this second paper, we continue our study on the regularity of free boundaries for some fully nonlinear elliptic equations. Our result is if the free boundary is trapped in a sufficiently narrow strip formed by two Lipschitz graphs, then it is also a Lipschitz graph. Combining with the results in Part 1 (see Ref. [Wang]), the free boundary is C 1,α.

Another way to say caloric

AbstractThis paper offers characterizations of subsolutions of the heat equation

Inhomogeneous infinity Laplace equation

u_t - \Delta u = 0

Liouville-type theorems and decay estimates for solutions to higher order elliptic equations ✩

(the subcaloric functions) and the infinity heat equation

On the uniqueness of a solution of a two-phase free boundary problem

u_t - \Delta_{\infty} u = 0