\(p\)-harmonic functions by way of intrinsic mean value properties. (English) Zbl 1511.35193
Authors’ abstract: Let \(\Omega\) be a bounded domain in \(\mathbb{R}^n\). Under appropriate conditions on \(\Omega\), we prove existence and uniqueness of continuous functions solving the Dirichlet problem associated to certain nonlinear mean value properties in \(\Omega\) with respect to balls of variable radius. We also show that, when properly normalized, such functions converge to the \(p\)-harmonic solution of the Dirichlet problem in \(\Omega\) for \(p\geq 2\). Existence is obtained via iteration, a fundamental tool being the construction of explicit universal barriers in \(\Omega\).
Reviewer: Patrick Winkert (Berlin)
MSC:
| 35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |
| 35J25 | Boundary value problems for second-order elliptic equations |
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