Regularity of the obstacle problem for a fractional power of the laplace operator

By Luis Silvestre. 2007

1,370 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/127e5fe64fb38094a69c30ff324bdef7377c3797

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

A fractional Yamabe flow and some applications (2011, background): “ringing the references [7, 36] to our attention. 2 A strong maximum principle and a Hopf lemma for nonlocal parabolic equations Let x = (x′,xn) ∈ Rn, Rn + = {x : xn > 0,x ∈ Rn}. Recall (see, e.g., [71]) that for σ ∈ (0,1), if u is bounded in Rn and C2 near x, then(−∆)σu is continuous near x, and ”
The Brezis–Nirenberg type problem involving the square root of the Laplacian (2011, methodology): “[9] L. Caffarelli, S. Salsa, L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent.”
Local and global minimizers for a variational energy involving a fractional norm (2011, positive): “Their proofs can be found in [23, Chapter 2], which presents some general properties of the (−∆) operators and provides characterization of its supersolutions (see also [24] and [12]).”
Nonhomogeneous boundary conditions for the spectral fractional Laplacian (2015, positive): “[19]) that ‖u‖C2s+α(RN ) ≤ C‖f‖Cα(RN ), for a constant C depending only on s, α,N and the measure of the support of f .”
Hitchhiker's guide to the fractional Sobolev spaces (2011, background): “Don’t panic, instead, see also [86, 87] for further motivation.”
Overdetermined problems with fractional Laplacian (2013, neutral): “Finally we want to note that if u ∈ C 1 , 1 ( Ω ) , for some open set Ω ⊂ R N , we have that ( − ∆ ) s u is continuous on Ω (see e.g. [39, Proposition 2.5]).”
Fractional elliptic equations, Caccioppoli estimates and regularity (2014, positive): “This obstacle problem for L = −∆ and Ω = R was studied in [5, 24].”
Nonlocal Diffusion and Applications (2015, methodology): “We refer to [87] for a refinement of the space of definition.”
The Calderón problem for variable coefficients nonlocal elliptic operators (2017, background): “For A(x) being an identity matrix, the operator L becomes the well-known fractional Laplacian operator (−∆), and the detailed study about the (−∆) is available in [3, 6, 7, 8, 38, 39, 41, 42, 43].”
Equations involving fractional Laplacian operator: Compactness and application (2015, methodology): “onents, though the methods used in [12, 13, 26] to obtain the estimates are diﬀerent from those in [14]. Problems with the fractional Laplacian have been extensively studied recently. See for example [3, 4, 6, 7, 8, 10, 11, 16, 20, 22, 23, 24]. In particular, the Br´ezis-Nirenberg type problem was d”
Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems (2020, background): “See [28, 80] for more details and a characterization of the dual (Ss(R)).”
Gamma convergence of an energy functional related to the fractional Laplacian (2008, background): “The classical reference for the fractional Laplacian (−∆)s is the book by Landkof [16], although a good reference is the PhD thesis by Luis Silvestre [25].”
From the long jump random walk to the fractional Laplacian (2009, background): “It is known [ Sil05 ] that the singular integral in (7) is related to the fractional Laplacian (−�) α/2 . This”
Fractional Adams–Moser–Trudinger type inequalities (2015, methodology): “Since Ω is not necessarily bounded and might have rough boundary, we must be careful, particularly in using maximum principles (we will use a simple “variational” maximum principle instead of the one of Silvestre [44]).”