Variational p-harmonious functions : existence and convergence to p-harmonic functions
Variational p-harmonious functions : existence and convergence to p-harmonic functions
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变分p调和函数：p调和函数的存在性和收敛性
DOI：
10.1007/s00030-021-00714-7
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发表时间：
2021
期刊：
影响因子：
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通讯作者：
Hidemitsu Wadade
中科院分区：
文献类型：
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作者：
Evan William Chandra;Michinori Ishiwata;Rolando Magnanini;Hidemitsu Wadade
在最近的一篇文章中，后三位作者证明了对策论p-调和函数关于球上变分定义的一类平均值具有渐近中值性质。在这篇文章中，我们考虑作用于上的连续函数的算子，它由公式定义，其中表示的边界。我们首先得到了它的各种性质，如连续性和单调性。然后，我们证明了满足Dirichlet型问题的函数的存在唯一性：对于任意给定的函数，$$\Begin{Align}u(X)=\MU_p^\varepsilon[u](X)\\Text{for}\Text{Every}\x\in\Omega，\quad u=g\\HBox{on}\\Gamma，\end{Align}$$。如果我们假设中的所有点都存在一个合适的势垒概念，那么这个结果是成立的。这就是我们所说的带有Dirichlet边界数据的变分Alp-调和函数，它是用基于比较原理的Perron型方法得到的。然后，我们证明了这个族给出了粘性解的一个近似解：$$\Begin{Align}\Delta_p^Gu=0\\Text{In}\Omega，\quad u=g\\Hbox{on}\\Gamma，\end{Align}$其中是所谓的博弈论(或齐次)p-Laplace算子。事实上，我们证明了它一致地收敛于你。
In a recent paper, the last three authors showed that a game-theoreticp-harmonic functionvis characterized by an asymptotic mean value property with respect to a kind of mean valuedefined variationally on balls. In this paper, in a domain,, we consider the operator, acting on continuous functions on, defined by the formula, whereanddenotes the boundary of. We first derive various properties ofsuch as continuity and monotonicity. Then, we prove the existence and uniqueness of a functionsatisfying the Dirichlet-type problem: $$\begin{aligned} u(x)=\mu _p^\varepsilon [u](x) \ \text{ for } \text{ every } \ x\in \Omega ,\quad u=g \ \hbox { on } \ \Gamma , \end{aligned}$$for any given function. This result holds, if we assume the existence of a suitable notion of barrier for all points in. Thatis what we call thevariationalp-harmonious function with Dirichlet boundary datag, and is obtained by means of a Perron-type method based on a comparison principle. We then show that the familygives an approximation for the viscosity solutionof $$\begin{aligned} \Delta _p^G u=0 \ \text{ in } \Omega , \quad u=g \ \hbox { on } \ \Gamma , \end{aligned}$$whereis the so-called game-theoretic (or homogeneous)p-Laplace operator. In fact, we prove thatconverges tou, uniformly onas.