## Differential and Integral Equations

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

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

Angel Arroyo and José G. Llorente

#### 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