Differential and Integral Equations

On the Dirichlet problem for solutions of a restricted nonlinear mean value property

Angel Arroyo and José G. Llorente

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Let $\Omega \subset \mathbb R^d$ be a bounded domain and suppose that for each $x\in \Omega$ a radius $r = r(x)$ is given so that the ball $B_x = B(x,r)$ is contained in $\Omega$. For $0 \leq \alpha < 1 $, we consider the following operator in $\mathcal{C}(\overline{\Omega})$ $$ T_{\alpha}u(x) = \frac{\alpha}{2}\big ( \sup_{B_x} u + \inf_{B_x} u \big ) + (1-\alpha ) \int_{B_x} u, $$ and show that, under certain assumptions on $\Omega$ and the radius function $r(x)$, the Dirichlet problem with continuous boundary data has a unique solution $u\in \mathcal{C}(\overline{\Omega})$ satisfying $T_{\alpha}u = u$. The motivation comes from the study of so called $p$-harmonious functions and certain stochastic games.

Article information

Source
Differential Integral Equations Volume 29, Number 1/2 (2016), 151-166.

Dates
First available in Project Euclid: 24 November 2015

Permanent link to this document
https://projecteuclid.org/euclid.die/1448323257

Mathematical Reviews number (MathSciNet)
MR3450753

Zentralblatt MATH identifier
1349.31003

Subjects
Primary: 31C05: Harmonic, subharmonic, superharmonic functions 35B60: Continuation and prolongation of solutions [See also 58A15, 58A17, 58Hxx] 31C45: Other generalizations (nonlinear potential theory, etc.)

Citation

Arroyo, Angel; Llorente, José G. On the Dirichlet problem for solutions of a restricted nonlinear mean value property. Differential Integral Equations 29 (2016), no. 1/2, 151--166. https://projecteuclid.org/euclid.die/1448323257.


Export citation