On Fractional Laplacians

By R. Musina, Alexander I. Nazarov. 2013

139 citations recorded by the source corpus.

We compare two natural types of fractional Laplacians (− Δ) s , namely, the “Navier” and the “Dirichlet” ones. We show that for 0 < s < 1 their difference is positive definite and positivity preserving. Then we prove the coincidence of the Sobolev constants for these two fractional Laplacians.

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

View Paper: https://www.semanticscholar.org/paper/383ce576774424ac2b02b32b8a7e71c07e1bc632

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion (2017, background): “The difference between RFL and SFL seems to have been well-known to probabilists, it was discussed later in PDEs, see Servadei-Valdinoci [217], Bonforte and the author [51], and Musina-Nazarov [196].”
Nonlocal Diffusion and Applications (2015, background): “[26, 85, 73]), and all of them come from interesting problems in pure or/and applied mathematics.”
A Fractional Laplace Equation: Regularity of Solutions and Finite Element Approximations (2015, background): “This operator is different to the spectral fractional Laplacian; for example, their difference is positive definite and positivity preserving [36].”
What is the fractional Laplacian? A comparative review with new results (2018, background): “ the spectral solutions. This behavior is consistent with the proof that the dierence of the Riesz and spectral fractional Laplacians (with zero Dirichlet boundary conditions) is a positive operator [12]. When the problem has nonzero boundary conditions, this behavior can be reversed, as discussed i”