Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian

By Luis Caffarelli, Sandro Salsa, Luis Silvestre. 2007

589 citations recorded by the source corpus.

We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem. We write an equivalent characterization as a thin obstacle problem. In this way we are able to apply local type arguments to obtain sharp regularity estimates for the solution and study the regularity of the free boundary.

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

View Paper: https://www.semanticscholar.org/paper/0c7243dcdbb98c9487e99836ccaf503685130744

How researchers see this paper

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

In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

The Brezis–Nirenberg type problem involving the square root of the Laplacian (2011, background): “Moreover, Caffarelli et al [9],[8] studied a free boundary problem: the Signorini problem involving fractional Laplacians as well as random homogenization of fractional obstacle problems.”
Variational problems with free boundaries for the fractional Laplacian (2010, unknown): “This computation needs C1 regularity for u inside B1, which is provided in [10].”
Fractional Orlicz-Sobolev embeddings (2020, background): “A touch of recent contributions in this connection is furnished by [7, 8, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 41, 44, 45, 51, 52, 53, 58, 59, 61, 62, 63, 64, 65, 68, 70, 71, 77, 76, 78, 83, 86].”
Fractional powers of second order partial differential operators: extension problem and regularity theory (2010, methodology): “Silvestre y sus colaboradores [67, 68, 23, 22, 26] sobre estimaciones de regularidad para soluciones de problemas no lineales que involucran al Laplaciano fraccionario.”
Nonlinear Diffusion with Fractional Laplacian Operators (2012, background): “The linear or quasilinear elliptic theory has been actively studied recently in the works of Caffarelli and collaborators [6, 8, 24], Kassmann [44], Silvestre [60] and many others.”
Basic results of fractional Orlicz-Sobolev space and applications to non-local problems (2019, background): “[11, 12, 13], a large amount of papers were written on problems involving the fractional diffusion operator (−∆)s (0 < s < 1).”
On the regularity of the non-dynamic parabolic fractional obstacle problem (2016, methodology): “Observe, now, that, as it was done in [5], if we pass the term involving ut to the right hand side of the equation and ”freeze” time then we can apply the elliptic theory of [10] and [8].”
On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent (2017, background): “[6, 7, 8], a large amount of papers were written on problems involving the fractional diffusion operator (−∆) (0 < s < 1).”
Regularity of solutions to the parabolic fractional obstacle problem (2011, background): “First of all we recall that, for the stationary version of the obstacle problem, solutions belong to C1+s x (R n) (or equivalently, (−∆)su ∈ C1−s x (R n)), and such a regularity result is optimal [15, 6].”