Regularity Theory for Second Order Integro-PDEs

Abstract

This paper is concerned with higher Hölder regularity for viscosity solutions to nontranslation invariant second order integro-PDEs, compared to [24]. We first obtain C 1,α regularity estimates for fully nonlinear integro-PDEs. We then prove the Schauder estimates for solutions if the equation is convex.

Min–Max formulas for nonlocal elliptic operators on Euclidean Space

“…For translation invariant equations, these are the results that show solutions to translation invariant equations very often enjoy C 1,α regularity under mild assumptions; some examples are: [9], [17], [41], [44], [50], among others. Finally, for Schauder regularity, we mean results that show that for x-dependent operators, under certain regularity for the coefficients (such as Dini), solutions will have as much regularity as those equations with "constant coefficients"; some examples are: [20], [36], [43], among others. On top of questions of the type of Krylov-Safonov regularity mentioned above, there is another family of regularity results that accompanies existence and uniqueness techniques for viscosity solutions of elliptic partial-differential / integro-differential equations, and it is typically referred to as the Ishii-Lions method, going back to [34].…”

Boundary regularity of mixed local-nonlocal operators and its application

“…Thus, since u ∈ C 1 (Ω) by [28,Theorem 4.1] (as mentioned before, the proof of [28] works for inequations),…”

“…[21,Theorem 3.1.22]). Recently, inspired by [11,12], interior regularity of the solutions of (1.2) are studied in [27,28]. In this article, we are interested in up to the boundary regularity of the solutions.…”

“…Proof of Theorem 1.1 is based on two standard ingredients, interior estimates from [28] and barrier function. It can be easily shown that dist(•, Ω c ) gives a barrier function at the boundary (cf.…”

“…for all p, q ∈ R n and x ∈ Ω, then using the interior regularity of u, v from [28] and the coupling result [10, Theorem 5.1] it can be easily seen that w = u − v satisfies (1.1). Our result Theorem 1.3 then gives a C 1,γ estimate of w up to the boundary.…”