Boundary regularity for fully nonlinear integro-differential equations

Abstract

We study fine boundary regularity properties of solutions to fully nonlinear elliptic integro-differential equations of order $2s$, with $s\in(0,1)$. We consider the class of nonlocal operators $\mathcal L_*\subset \mathcal L_0$, which consists of infinitesimal generators of stable L\'evy processes belonging to the class $\mathcal L_0$ of Caffarelli-Silvestre. For fully nonlinear operators $I$ elliptic with respect to $\mathcal L_*$, we prove that solutions to $I u=f$ in $\Omega$, $u=0$ in $\mathbb R^n\setminus\Omega$, satisfy $u/d^s\in C^{s+\gamma}(\overline\Omega)$, where $d$ is the distance to $\partial\Omega$ and $f\in C^\gamma$. We expect the class $\mathcal L_*$ to be the largest scale invariant subclass of $\mathcal L_0$ for which this result is true. In this direction, we show that the class $\mathcal L_0$ is too large for all solutions to behave like $d^s$. The constants in all the estimates in this paper remain bounded as the order of the equation approaches 2. Thus, in the limit $s\uparrow1$ we recover the celebrated boundary regularity result due to Krylov for fully nonlinear elliptic equations.Comment: To appear in Duke Math.

Obstacle problems for integro-differential operators: Higher regularity of free boundaries

“…This was one of the results that had to be proved in [31]; however the proof given therein only gave that L(d s ) ∈ C s (Ω) (and actually under a non-sharp assumption of the domain). To show that L(d s ) is more regular than C s (in C k,α domains) remained as an open problem after the results of [31]. We solve the first technical difficulty here by using some ideas by Dipierro, Savin, and Valdinoci [13]; notice however that our proofs are completely independent from those in [13], and we moreover show some new results concerning nonlocal operators for functions with polynomial growth.…”

“…This is a highly nontrivial task in the context of nonlocal operators, and even the sharp boundary Schauder-type estimates in C k,α domains was a completely open problem for operators of the type (2)- (3). The only known results in this direction are due to the second author and Serra [30][31][32][33] for k = 1, or to Grubb [21,22] for k = ∞, and are actually very delicate to establish.…”

“…On the proofs of Theorems 1.3 and 1.4. In order to establish our new fine estimates for nonlocal equations in C k,α domains, we develop a new, higher order version of the blow-up and compactness technique from [31]. This remained as an open problem after the results of [31] mainly because of two reasons.…”

Regularity of spectral fractional Dirichlet and Neumann problems

“…e.g. , and their references; in the case where K is outside 0, this defines an operator of the type P mentioned above.) More generally, K can be subject to estimates comparing with .…”

“…The properties of the restricted Dirichlet fractional Laplacian defined in the introduction were studied e.g. in Blumenthal and Getoor , Landkof , Hoh and Jacob , Kulczycki , Chen and Song , Jakubowski , Silvestre , Caffarelli and Silvestre , Frank and Geisinger , Ros‐Oton and Serra , , Felsinger, Kassmann and Voigt , Grubb , , Bonforte, Sire and Vazquez , Servadei and Valdinoci , Binlin, Molica Bisci and Servadei , and many more papers referred to in these works (see in particular the list in ).…”