Differential and Integral Equations

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

Angel Arroyo and José G. Llorente

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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

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

First available in Project Euclid: 24 November 2015

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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.

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