Fractional powers of second order partial differential operators: extension problem and regularity theory

47 citations recorded by the source corpus.

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

Unique Continuation for Fractional Orders of Elliptic Equations (2016, methodology): “The case of bounded domains For a bounded Lipschitz domain Ω ⊂ R, we can still define fractional order of elliptic operators using spectral theory [27][28][29].”
Fractional elliptic equations, Caccioppoli estimates and regularity (2014, methodology): “It was shown in [27, 28] that if u = ∑∞ k=0 ukφk then the solution to this problem is given by”
What is the fractional Laplacian? A comparative review with new results (2018, methodology): “We discretize the spectral fractional Laplacian directly using the spectral element method (SEM) of [63], and the heat semigroup 385 approach [23, 24] and Silvestre-Caffarelli extension [20] are also discussed for completeness.”
Fractional Laplacian on the torus (2012, background): “Hölder estimates for the fractional harmonic oscillator (−∆+|x|2)σ/2 can be found in [22, 24].”
Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications (2016, unknown): “For 0 s 1, we deﬁne the fractional powers of the discrete Laplacian ( − ∆ h ) s u on Z h with the semigroup method (see [28, 29]) as Here w j ( t ) = e t ∆ h u j is the solution to the semidiscrete heat equation (1.4) ∂ t w j = ∆ h w j , in Z h × (0 , ∞ ) , w j (0) = u j , on Z h , (see Section…”