Search results for "Potential theory"

Superharmonic functions are locally renormalized solutions

2011

Abstract We show that different notions of solutions to measure data problems involving p-Laplace type operators and nonnegative source measures are locally essentially equivalent. As an application we characterize singular solutions of multidimensional Riccati type partial differential equations.

Polar Sets in a Nonlinear Potential Theory

1988

In this lecture we discuss nonlinear potential theory based on “A-super-harmonic functions”; the theory can be viewed as a (nonlinear) extension of the classical study of superharmonic functions in ℝn.

Fundamentals of Gravity, Elements of Potential Theory

2009

Equilibrium measures for uniformly quasiregular dynamics

2012

We establish the existence and fundamental properties of the equilibrium measure in uniformly quasiregular dynamics. We show that a uniformly quasiregular endomorphism $f$ of degree at least 2 on a closed Riemannian manifold admits an equilibrium measure $\mu_f$, which is balanced and invariant under $f$ and non-atomic, and whose support agrees with the Julia set of $f$. Furthermore we show that $f$ is strongly mixing with respect to the measure $\mu_f$. We also characterize the measure $\mu_f$ using an approximation property by iterated pullbacks of points under $f$ up to a set of exceptional initial points of Hausdorff dimension at most $n-1$. These dynamical mixing and approximation resu…

Dyadic Norm Besov-Type Spaces as Trace Spaces on Regular Trees

2019

In this paper, we study function spaces defined via dyadic energies on the boundaries of regular trees. We show that correct choices of dyadic energies result in Besov-type spaces that are trace spaces of (weighted) first order Sobolev spaces.

Nonlinear balayage on metric spaces

2009

We develop a theory of balayage on complete doubling metric measure spaces supporting a Poincaré inequality. In particular, we are interested in continuity and p-harmonicity of the balayage. We also study connections to the obstacle problem. As applications, we characterize regular boundary points and polar sets in terms of balayage. Original Publication:Anders Björn, Jana Björn, Tero Mäkäläinen and Mikko Parviainen, Nonlinear balayage on metric spaces, 2009, Nonlinear Analysis, (71), 5-6, 2153-2171.http://dx.doi.org/10.1016/j.na.2009.01.051Copyright: Elsevier Science B.V., Amsterdam.http://www.elsevier.com/

A Density Result for Homogeneous Sobolev Spaces on Planar Domains

2018

We show that in a bounded simply connected planar domain $\Omega$ the smooth Sobolev functions $W^{k,\infty}(\Omega)\cap C^\infty(\Omega)$ are dense in the homogeneous Sobolev spaces $L^{k,p}(\Omega)$.

Equivalence of viscosity and weak solutions for the $p(x)$-Laplacian

2010

We consider different notions of solutions to the $p(x)$-Laplace equation $-\div(\abs{Du(x)}^{p(x)-2}Du(x))=0$ with $ 1<p(x)<\infty$. We show by proving a comparison principle that viscosity supersolutions and $p(x)$-superharmonic functions of nonlinear potential theory coincide. This implies that weak and viscosity solutions are the same class of functions, and that viscosity solutions to Dirichlet problems are unique. As an application, we prove a Rad\'o type removability theorem.

Sobolev Extension on Lp-quasidisks

2021

AbstractIn this paper, we study the Sobolev extension property of Lp-quasidisks which are the generalizations of classical quasidisks. After that, we also find some applications of this property.

$C^{1,��}$ regularity for the normalized $p$-Poisson problem

2017

We consider the normalized $p$-Poisson problem $$-��^N_p u=f \qquad \text{in}\quad ��.$$ The normalized $p$-Laplacian $��_p^{N}u:=|D u|^{2-p}��_p u$ is in non-divergence form and arises for example from stochastic games. We prove $C^{1,��}_{loc}$ regularity with nearly optimal $��$ for viscosity solutions of this problem. In the case $f\in L^{\infty}\cap C$ and $p&gt;1$ we use methods both from viscosity and weak theory, whereas in the case $f\in L^q\cap C$, $q&gt;\max(n,\frac p2,2)$, and $p&gt;2$ we rely on the tools of nonlinear potential theory.