Nonlinear Parabolic Capacity and Polar Sets of Superparabolic Functions
نویسندگان
چکیده
We extend the theory of the thermal capacity for the heat equation to nonlinear parabolic equations of the p-Laplacian type. We study definitions and properties of the nonlinear parabolic capacity and show that the capacity of a compact set can be represented via a capacitary potential. As an application, we show that polar sets of superparabolic functions are of zero capacity. The main technical tools used include estimates for equations with measure data and obstacle problems.
منابع مشابه
An Existence Result for Superparabolic Functions
We study superparabolic functions related to nonlinear parabolic equations. They are de ned by means of a parabolic comparison principle with respect to solutions. We show that every superparabolic function satis es the equation with a positive Radon measure on the right-hand side, and conversely, for every nite positive Radon measure there exists a superparabolic function which is solution to ...
متن کاملOn the Equivalence of Viscosity Solutions and Weak Solutions for a Quasi-Linear Equation
We discuss and compare various notions of weak solution for the p-Laplace equation −div(|∇u|p−2∇u) = 0 and its parabolic counterpart ut − div(|∇u|p−2∇u) = 0. In addition to the usual Sobolev weak solutions based on integration by parts, we consider the p-superharmonic (or p-superparabolic) functions from nonlinear potential theory and the viscosity solutions based on generalized pointwise deriv...
متن کاملCharacterizing sub-topical functions
In this paper, we first give a characterization of sub-topical functions with respect to their lower level sets and epigraph. Next, by using two different classes of elementary functions, we present a characterization of sub-topical functions with respect to their polar functions, and investigate the relation between polar functions and support sets of this class of functions. Finally, we obtai...
متن کاملParabolic Capacity and soft measures for nonlinear equations
We first introduce, using a functional approach, the notion of capacity related to the parabolic p–laplace operator. Then we prove a decomposition theorem for measures (in space and time) that do not charge the sets of null capacity. We apply this result to prove existence and uniqueness of renormalized solutions for nonlinear parabolic initial boundary value problems with such measures as righ...
متن کاملOn a Problem of Dynkin
Consider an (L, α)-superdiffusion X on Rd, where L is an uniformly elliptic differential operator in Rd, and 1 < α ≤ 2. The G-polar sets for X are subsets of R×Rd which have no intersection with the graph G of X, and they are related to the removable singularities for a corresponding nonlinear parabolic partial differential equation. Dynkin characterized the G-polarity of a general analytic set...
متن کامل