Boundary regularity for fully nonlinear integro-differential equations

By Xavier Ros-Oton, J. Serra. 2014

159 citations recorded by the source corpus.

We study fine boundary regularity properties of solutions to fully nonlinear elliptic integro-differential equations of order 2s, with s ∈ (0,1). We consider the class of nonlocal operators L∗ ⊂L 0, which consists of all the infinitesimal generators of stable Levy processes belonging to the class L0 of Caffarelli-Silvestre. For fully nonlinear operators I elliptic with respec tt oL∗ ,w e prove that solutions to Iu = f in Ω, u = 0 in R n \ Ω, satisfy u/d s ∈ C s−ϵ (Ω) for all ϵ> 0, where d is the distance to ∂ Ωa ndf ∈ L ∞ . We expect the Holder exponent s − ϵ to be optimal (or almost optimal) for general right hand sides f ∈ L ∞ . Moreover, we also expect the class L∗ to be the largest scale invariant subclass of L0 for which this result is true. In this direction, we show that the class L0 is too large for all solutions to behave like d s . The constants in all the estimates in this paper remain bounded as th eo rder of the equation approaches 2.

Subfields: Applied Mathematics, Finance

View Paper: https://www.semanticscholar.org/paper/1ab2af28657cc77c61a837d49801c914617ebdd1

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Citations in the corpus

Obstacle problems for integro-differential operators: Higher regularity of free boundaries (2019, methodology): “, the higher order version of the blow-up technique from [31], and the proof of (7)) open the road to the study of the higher regularity of free boundaries in other obstacle problems that until now seemed out of reach, such as nonlocal operators with drift [15, 18, 27], or even the parabolic obstacl”
Regularity theory for general stable operators (2014, methodology): “An important difference with respect to the proofs [34, 31, 35] is that here we do not have any C estimate that we can iterate in order to prove a Liouville theorem, and hence the proofs of the present Liouville theorems must be completely different from the ones in [34, 31, 35].”
Stable solutions to semilinear elliptic equations are smooth up to dimension $9$ (2019, unknown): “We argue by contradiction, similarly to [34, 32].”
Local and nonlocal boundary conditions for μ-transmission and fractional elliptic pseudodifferential operators (2014, background): “Ros-Oton and Serra [2014b] showed, under weaker smoothness hypotheses on , the inclusion u ∈ daCα() for any α with 0< α <min{a, 1− a}, and improved it in [Ros-Oton and Serra 2014a] to α = a− ε.”
Nonlocal elliptic equations in bounded domains: a survey (2015, methodology): “[46] X. Ros-Oton, J. Serra, Boundary regularity for fully nonlinear integro-differential equations, preprint arXiv (2014).”
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].”
Global H\"older regularity for the fractional $p$-Laplacian (2014, background): “In the uniformly elliptic case (p = 2), the optimal regularity is C(Ω) due to the results of [19], but only for a subclass of rough symmetric kernels arising from stable Lévy processes, of the form”
On fractional elliptic equations in Lipschitz sets and epigraphs: regularity, monotonicity and rigidity results (2016, background): “The boundary regularity theory for fractional Laplacian, developed in [31,32,37,38], ensures that, for a solution u to (1.”
The obstacle problem for nonlinear integro-differential operators (2016, background): “As well known, in the contrary with respect to the interior case, boundary regularity for nonlocal equations driven by singular, possibly degenerate, operators as in (1) seems to be a di cult problem in a general nonlinear framework under natural assumptions on the involved quantities (while we refe”