The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary
Xavier Ros-Oton,Joaquim Serra +1 more
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In this article, the Pohozaev identity up to the boundary of the Dirichlet problem for the fractional Laplacian was shown to hold for the case of ( − Δ ) s u = g in Ω, u ≡ 0 in R n \ Ω, for some s ∈ ( 0, 1 ) and g ∈ L ∞ ( Ω ), then u is C s ( R n ) and u / δ s | Ω is C α up to boundary ∂Ω for some α ∈( 0About:
This article is published in Journal de Mathématiques Pures et Appliquées.The article was published on 2014-03-01 and is currently open access. It has received 804 citations till now.read more
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Variational Methods for Nonlocal Fractional Problems
TL;DR: A thorough introduction to the variational analysis of nonlinear problems described by nonlocal operators can be found in this paper, where the authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of equations, plus their application to various processes arising in the applied sciences.
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Fractional Laplacians on domains, a development of Hörmander's theory of μ-transmission pseudodifferential operators
TL;DR: In this paper, the authors develop the theory in L p Sobolev spaces (1 p ∞ ) in a modern setting. But they do not cover complex powers of the Laplacian ( − Δ ) μ with μ ∉ Z, which are not covered in this paper.
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On the spectrum of two different fractional operators
TL;DR: In this article, the integral definition of the fractional Laplacian given by c(n, s) is a positive normalizing constant, and another fractional operator obtained via a spectral definition, that is, where ei, λi are the eigenfunctions of the Laplace operator −Δ in Ω with homogeneous Dirichlet boundary data, while ai represents the projection of u on the direction ei.
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The Pohozaev identity for the fractional Laplacian
Xavier Ros-Oton,Joaquim Serra +1 more
TL;DR: In this article, the Pohozaev identity for the semilinear Dirichlet problem has been proved for a non-local version of the problem with a boundary term (an integral over ∂Ω) which is completely local.
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What is the fractional Laplacian? A comparative review with new results
Anna Lischke,Guofei Pang,Mamikon Gulian,Fangying Song,Christian A. Glusa,Xiaoning Zheng,Zhiping Mao,Wei Cai,Mark M. Meerschaert,Mark Ainsworth,George Em Karniadakis +10 more
TL;DR: A comparison of several commonly used definitions of the fractional Laplacian theoretically, through their stochastic interpretations as well as their analytical properties, and a collection of benchmark problems to compare different definitions on bounded domains using a sample of state-of-the-art methods.
References
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Elliptic Partial Differential Equations of Second Order
David Gilbarg,Neil S. Trudinger +1 more
TL;DR: In this article, Leray-Schauder and Harnack this article considered the Dirichlet Problem for Poisson's Equation and showed that it is a special case of Divergence Form Operators.
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Measure theory and fine properties of functions
TL;DR: In this article, the authors define and define elementary properties of BV functions, including the following: Sobolev Inequalities Compactness Capacity Quasicontinuity Precise Representations of Soboleve Functions Differentiability on Lines BV Function Differentiability and Structure Theorem Approximation and Compactness Traces Extensions Coarea Formula for BV Functions isoperimetric inequalities The Reduced Boundary The Measure Theoretic Boundary Gauss-Green Theorem Pointwise Properties this article.
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An Extension Problem Related to the Fractional Laplacian
TL;DR: In this article, the square root of the Laplacian (−△) 1/2 operator was obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition.
Book
Foundations of Modern Potential Theory
TL;DR: In this paper, the authors define the notion of potentials and their basic properties, including the capacity and capacity of a compact set, the properties of a set of irregular points, and the stability of the Dirichlet problem.
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Fully Nonlinear Elliptic Equations
Luis A. Caffarelli,Xavier Cabré +1 more
TL;DR: The Dirichlet problem for concave equations has been studied in this article, where Alexandroff estimate and maximum principle Harnack inequality uniqueness of solutions Concave equations $W^{2,p}$ regularity Holder regularity