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Potential Analysis

, Volume 42, Issue 2, pp 499–547 | Cite as

The Fractional Relative Capacity and the Fractional Laplacian with Neumann and Robin Boundary Conditions on Open Sets

  • Mahamadi WarmaEmail author
Article

Abstract

Let \(\Omega \subset \mathbb {R}^{N}\) be an arbitrary open set with boundary Ω. Let \(p\in [1,\infty )\) and s∈(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 on \(\overline {\Omega }\) with 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 \(\mathbb {R}^{N}\). 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.

Keywords

Fractional order Sobolev spaces Arbitrary domains Capacity Fractional Laplacian Neumann and Robin boundary conditions Elliptic boundary value problems Existence and regularity of weak solutions Semigroup Ultracontractivity 

Mathematics Subject Classifications (2010)

31A15 35J25 35B65 46E39 34B10 

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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Faculty of Natural Sciences, Department of MathematicsUniversity of Puerto RicoSan JuanUSA

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