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• W3119541887 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 $OmsubsetRR^N$, $Nge 2$, we consider the operator $mu_p^ve$, acting on continuous functions on $ol{Om}$, defined by the formula $mu_p^ve[v](x)=nu^{r_ve(x)}_p[v](x)$, where $r_ve(x)=min[ve,dist(x,Ga)]$ and $Ga$ denotes the boundary of $Omega$. We first derive various properties of $mu^ve_p$ such as continuity and monotonicity. Then, we prove the existence and uniqueness of a function $u^vein C(ol{Om})$ satisfying the Dirichlet-type problem: $$ u(x)=mu_p^ve[u](x) mbox{ for every } xinOm,quad u=g mbox{ on } Ga, $$ for any given function $gin C(Ga)$. This result holds, if we assume the existence of a suitable notion of barrier for all points in $Ga$. That $u^ve$ is what we call the textit{variational} $p$-harmonious function with Dirichlet boundary data $g$, and is obtained by means of a Perron-type method based on a comparison principle. par We then show that the family ${ u^ve}_{ve>0}$ gives an approximation scheme for the viscosity solution $uin C(ol{Om})$ of $$ De_p^G u=0 mbox{ in }Om, quad u=g mbox{ on } Ga, $$ where $De_p^G$ is the so-called game-theoretic (or homogeneous) $p$-Laplace operator. In fact, we prove that $u^ve$ converges to $u$, uniformly on $ol{Om}$ as $veto 0$." @default.
• W3119541887 created "2021-01-18" @default.
• W3119541887 creator A5031357946 @default.
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• W3119541887 date "2021-01-07" @default.
• W3119541887 modified "2023-09-27" @default.
• W3119541887 title "Variational $p$-harmonious functions: existence and convergence to $p$-harmonic functions" @default.
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• W3119541887 hasPublicationYear "2021" @default.
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