Abstract
In a recent paper, the last three authors showed that a game-theoretic p-harmonic function v is characterized by an asymptotic mean value property with respect to a kind of mean value \(\nu _p^r[v](x)\) defined variationally on balls \(B_r(x)\). In this paper, in a domain \(\Omega \subset \mathbb {R}^N\), \(N\ge 2\), we consider the operator \(\mu _p^\varepsilon \), acting on continuous functions on \(\overline{\Omega }\), defined by the formula \(\mu _p^\varepsilon [v](x)=\nu ^{r_\varepsilon (x)}_p[v](x)\), where \(r_\varepsilon (x)=\min [\varepsilon ,\mathop {\mathrm {dist}}(x,\Gamma )]\) and \(\Gamma \) denotes the boundary of \(\Omega \). We first derive various properties of \(\mu ^\varepsilon _p\) such as continuity and monotonicity. Then, we prove the existence and uniqueness of a function \(u^\varepsilon \in C(\overline{\Omega })\) satisfying the Dirichlet-type problem:
for any given function \(g\in C(\Gamma )\). This result holds, if we assume the existence of a suitable notion of barrier for all points in \(\Gamma \). That \(u^\varepsilon \) is what we call the variational p-harmonious function with Dirichlet boundary data g, and is obtained by means of a Perron-type method based on a comparison principle. We then show that the family \(\{ u^\varepsilon \}_{\varepsilon >0}\) gives an approximation for the viscosity solution \(u\in C(\overline{\Omega })\) of
where \(\Delta _p^G\) is the so-called game-theoretic (or homogeneous) p-Laplace operator. In fact, we prove that \(u^\varepsilon \) converges to u, uniformly on \(\overline{\Omega }\) as \(\varepsilon \rightarrow 0\).
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Acknowledgements
The third author was partially supported by the Gruppo Nazionale di Analisi Matematica, Probabilità e Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Part of the research related to this paper was carried out during a his visit to Kanazawa University and Osaka University. He wishes to thank those institutions for their warm hospitality.
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Chandra, E.W., Ishiwata, M., Magnanini, R. et al. Variational p-harmonious functions: existence and convergence to p-harmonic functions. Nonlinear Differ. Equ. Appl. 28, 51 (2021). https://doi.org/10.1007/s00030-021-00714-7
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DOI: https://doi.org/10.1007/s00030-021-00714-7