On the stochastic p-Laplace equation

ENCLOSURE METHOD FOR THE p-LAPLACE EQUATION

We study the enclosure method for the p-Calderón problem, which is a nonlinear generalization of the inverse conductivity problem due to Calderón that involves the p-Laplace equation. The method allows one to reconstruct the convex hull of an inclusion in the nonlinear model by using exponentially growing solutions introduced by Wolff. We justify this method for the penetrable obstacle case, wh...

A stochastic differential game for the inhomogeneous -Laplace equation

Given a bounded C domain G ⊂ R and functions g ∈ C(∂G,R) and h ∈ C(Ḡ,R \ {0}), let u denote the unique viscosity solution to the equation −2∆∞u = h in G with boundary data g. We provide a representation for u as the value of a two-player zero-sum stochastic differential game. AMS 2000 subject classifications: 91A15, 91A23, 35J70, 49L20

A space mapping approach for the p-Laplace equation

VISCOSITY SUPERSOLUTIONS OF THE EVOLUTIONARY p-LAPLACE EQUATION

has this character. Even obvious results for this equation may require advanced estimates in the proofs. We refer to the books [DB] and [WZYL] about this equation, which is called the “evolutionary p-Laplacian equation,” the “p-parabolic equation” or even the “non-Newtonian equation of filtration.”. Our objective is to study the regularity of the viscosity supersolutions and their spatial gradi...

The Laplace Equation

Definition 1. Among the most important and ubiquitous of all partial differential equations is Laplace’s Equation: ∆u = 0, where the Laplacian operator ∆ acts on the function u : U → R (U is open in R) by taking the sum of the unmixed partial derivatives. For example: n = 1: ∆u = ∂ 2u ∂x2 = u = 0 In this simple case, the solution u = ax + b is found by integrating twice. n = 2: ∆u = ∂ 2u ∂x1 + ...