Geometry of Prime End Boundary and the Dirichlet Problem for Bounded Domains in Metric Measure Spaces
Geometry of Prime End Boundary and the Dirichlet Problem for Bounded Domains in Metric Measure Spaces
复制标题
度量测度空间中素端边界的几何和有界域的狄利克雷问题
DOI：
10.1007/s11118-014-9436-3
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发表时间：
2015
影响因子：
1.1
通讯作者：
N. Shanmugalingam
中科院分区：
文献类型：
--
作者：
D. Estep;N. Shanmugalingam
在本文中，我们研究了与配备支持庞加莱不等式的加倍测度的完整度量测度空间中有界域的质端边界版本相关的狄利克雷问题。我们展示了在素数端边界上连续的函数的分辨率，并且当限制于印象为单例集的所有素数端的子集时，这些函数是 Lipschitz 正则的。我们还考虑了适应素数端边界的新容量概念，并表明此类函数对容量为零的素数端边界子集的有界扰动是可解析的，并且它们的 Perron 解与原始函数的 Perron 解一致。我们还描述了一些例子，证明素数端边界方法在获得新结果方面的有效性，甚至对于某些欧几里德域的经典狄利克雷问题也是如此。
In this note we study the Dirichlet problem associated with a version of prime end boundary of a bounded domain in a complete metric measure space equipped with a doubling measure supporting a Poincaré inequality. We show the resolutivity of functions that are continuous on the prime end boundary and are Lipschitz regular when restricted to the subset of all prime ends whose impressions are singleton sets. We also consider a new notion of capacity adapted to the prime end boundary, and show that bounded perturbations of such functions on subsets of the prime end boundary with zero capacity are resolutive and that their Perron solutions coincide with the Perron solution of the original functions. We also describe some examples which demonstrate the efficacy of the prime end boundary approach in obtaining new results even for the classical Dirichlet problem for some Euclidean domains.