The Fractional Relative Capacity and the Fractional Laplacian with Neumann and Robin Boundary Conditions on Open Sets
The Fractional Relative Capacity and the Fractional Laplacian with Neumann and Robin Boundary Conditions on Open Sets
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DOI：
10.1007/s11118-014-9443-4
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发表时间：
2015-02
影响因子：
1.1
通讯作者：
M. Warma
中科院分区：
文献类型：
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作者：
M. Warma
设 是一个具有边界 ∂Ω 的任意开集。 Letts∈(0,1)。在本文的第一部分中，我们给出了分数阶 Sobolev 空间的一些有用的属性。我们在分数阶 Sobolev 空间上定义了相对 (s,p) 容量，给出了它的性质及其与经典 Bessel (s,p) 容量和 Hausdorff 测度的联系。我们还使用相对容量来完全表征零迹分数阶 Sobolev 空间。在本文的第二部分中，我们阐明了与开子集上的分数拉普拉斯算子相关的诺伊曼和罗宾边界条件。与经典拉普拉斯算子相反，事实证明狄利克雷、诺伊曼和罗宾边界条件对于有界域上的分数拉普拉斯算子可能是一致的。在本文的最后部分，我们考虑一些与具有 Neumann 和 Robin 类型边界条件的分数拉普拉斯相关的非局部椭圆问题。我们展示了非光滑域上弱解的一些存在性和规律性结果。
Letbe an arbitrary open set with boundary∂Ω. Letands∈(0,1). In the first part of the article we give some useful properties of the fractional order Sobolev spaces. We define a relative (s,p)-capacity onwith the fractional order Sobolev spaces, give its properties and its connection with the classical Bessel (s,p)-capacity and the Hausdorff measure. We also use the relative capacity to characterize completely the zero trace fractional order Sobolev spaces. In the second part of the article, we clarify the Neumann and Robin boundary conditions associated with the fractional Laplace operator on open subsets of. Contrary to the classical Laplace operator, it turns out that Dirichlet, Neumann and Robin boundary conditions may coincide for the fractional Laplacian on bounded domains. In the last part of the article we consider some nonlocal elliptic problems associated with the fractional Laplacian with Neumann and Robin type boundary conditions. We show some existence and regularity results of weak solutions on non smooth domains.