Extension problem and fractional operators: semigroups and wave equations

By J. Galé, Pedro J. Miana, Pablo Raúl Stinga. 2012

75 citations recorded by the source corpus.

We extend results of Caffarelli–Silvestre and Stinga–Torrea regarding a characterization of fractional powers of differential operators via an extension problem. Our results apply to generators of integrated families of operators, in particular to infinitesimal generators of bounded C0 semigroups and operators with purely imaginary symbol. We give integral representations to the extension problem in terms of solutions to the heat equation and the wave equation.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/2754bc3019a961fa94681e9cc09d41d932708820

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In Applied Mathematics

In Computational Theory and Mathematics

In Modeling and Simulation

Citations in the corpus

Fractional powers of sectorial operators via the Dirichlet-to-Neumann operator (2016, methodology): “We also obtain C-regularity making use of a Poisson formula discovered by Stinga and Torrea in [15] on Hilbert spaces, then extended to Banach spaces in [7].”
Ten equivalent definitions of the fractional laplace operator (2015, background): “Fractional powers of more general operators on arbitrary Banach spaces can be studied in a similar way, see [25, 48].”
Weyl and Marchaud Derivatives: A Forgotten History (2017, background): “[5] for facing the case of the semigroup approach in its abstract generality and then for recalling the contribution given in [58], for fractional Laplace operators, and then recalling [30] for several generalizations in an abstract approach, including in principle: Riemannian manifolds, Lie groups,”
Fractional Laplacian on the torus (2012, background): “1 of [13] applies for generators of semigroups in Banach spaces like L, or operators with complex spectrum.”
Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications (2016, unknown): “…has been a substantial revision of a big amount of problems in diﬀerential equations where the Laplacian is replaced by the fractional Laplacian or more general integro-diﬀerential operators, see for example [3, 5, 9, 21, 22, 23, 24, 25, 28, 29] and references therein for models and techniques.”
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 [96], and the heat semigroup approach [36,19,37], which is used in Section 5, and elliptic extension approaches [33,37,36, 80,81] are also discussed for completeness.”