Existence and uniqueness of $$\infty $$ ∞ -harmonic functions under assumption of $$\infty $$ ∞ -Poincaré inequality
Existence and uniqueness of $$\infty $$ ∞ -harmonic functions under assumption of $$\infty $$ ∞ -Poincaré inequality
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$$infty $$ 假设下 $$infty $$ 的存在性和唯一性 - 调和函数 $$infty $$ - 庞加莱不等式
DOI：
10.1007/s00208-018-1747-z
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发表时间：
2019
影响因子：
1.4
通讯作者：
Shanmugalingam, Nageswari
中科院分区：
文献类型：
--
作者：
Durand-Cartagena, Estibalitz;Jaramillo, Jesús A.;Shanmugalingam, Nageswari
给出了一个度量加倍且支持An-Poincaré不等式的完备度量空间，以及该空间中的一个有界域和一个Lipschitz函数，证明了A-调和扩张的存在唯一性。为此，我们证明了存在与原始度量等价的双Lipschitz度量，使得对于这个新度量，度量空间满足-弱Fubini性质，并且原始度量中的-调和函数一定也是关于新度量的-调和函数。我们还证明了如果度量空间上的度量满足-弱Fubini性质，则-调和函数的概念与Aronsson提出的amles的概念一致。-调和性的概念通常不同于在克兰德尔等人中发现的强绝对最小化Lipschitz扩张的概念。(Calc Var Partial Difference EQU 13：123-139,2001)，Juutinen(Ann Acad Sci Fenn Math 27(1)：57-67,2002)，Juutinen and Shanmugalingam(Math Nachr 279(9-10)：1083-1098,2006)，但当度量空间支持某个有限的AP-Poincaré不等式时，它们是重合的。
Given a complete metric measure space whose measure is doubling and supports an-Poincaré inequality, and a bounded domainin such a space together with a Lipschitz function, we show the existence and uniqueness of an-harmonic extension offto. To do so, we show that there is a metric that is bi-Lipschitz equivalent to the original metric, such that with respect to this new metric the metric space satisfies an-weak Fubini property and that a function which is-harmonic in the original metric must also be-harmonic with respect to the new metric. We also show that if the metric on the metric space satisfies an-weak Fubini property, then the notion of-harmonic functions coincide with the notion of AMLEs proposed by Aronsson. The notion of-harmonicity is in general distinct from the notion of strongly absolutely minimizing Lipschitz extensions found in Crandall et al. (Calc Var Partial Differ Equ 13: 123–139, 2001), Juutinen (Ann Acad Sci Fenn Math 27(1):57–67, 2002), Juutinen and Shanmugalingam (Math Nachr 279(9–10):1083–1098, 2006), but coincides when the metric space supports ap-Poincaré inequality for some finite.