Aleksandrov–Bakelman–Pucci Type Estimates for Integro-Differential Equations

Abstract

In this work we provide an Aleksandrov-Bakelman-Pucci type estimate for a certain class of fully nonlinear elliptic integro-differential equations, the proof of which relies on an appropriate generalization of the convex envelope to a nonlocal, fractional-order setting and on the use of Riesz potentials to interpret second derivatives as fractional order operators. This result applies to a family of equations involving some nondegenerate kernels and as a consequence provides some new regularity results for previously untreated equations. Furthermore, this result also gives a new comparison theorem for viscosity solutions of such equations which only depends on the L ∞ and L n norms of the right hand side, in contrast to previous comparison results which utilize the continuity of the right hand side for their conclusions. These results appear to be new even for the linear case of the relevant equations.with the additional assumption that 0 ≤ f k ≤ 1. Then we pose the following question:Question 1.1. Under what conditions will it be true that |{x :(We will assume L is 1-homogeneous, and hence the constant 0 function is a supersolution, and so always u k ≤ 0.) In the case that L is a second order, uniformly elliptic operator, L(u, x) = a ij (x)u x i x j (x),

Uniqueness of viscosity solutions for a class of integro-differential equations

“…Theorem 5.1, combined with an Aleksandrov-Bakelman-Pucci estimate of [15], can be used as an alternative way to prove comparison theorem when γ = 0, at least for some class of equations which are independent of the u variable.…”

“…The only result in this direction in [12,Section 5], is for equations corresponding to the case when the measures μ x are independent of x. There is also a remark made in [15,Theorem 9.2], about comparison for a class of equations being a consequence of an Aleksandrov-Bakelman-Pucci estimate for nonlocal equations, however it is not supported by any proof and it is probably false without additional assumptions about the nonlocal operator. Our small contribution here in Sect.…”

“…We also show how to use these special sup/infconvolutions to prove that the difference of a viscosity subsolution and a viscosity supersolution of the same elliptic equation is a viscosity subsolution of a nonlocal Pucci extremal equation. Knowing this one can use an AleksandrovBakelman-Pucci estimate of [15] to prove a comparison principle but this part appears to be missing in [15].…”

The obstacle problem for a fractional Monge–Ampère equation

“…While the main tool in [8] is the solution of a purely nonlocal and degenerate elliptic obstacle problem, which stands as a replacement for the convex envelope in the fractional setting (with the usual convex envelope being such an obstacle solution in the second order setting), we stress that the problem (1.1) is different in nature than the one in [8]. Obstacle problems for the local Monge-Ampère equation (1.3) were considered by Lee [10] and Savin [12].…”

On a Fractional Monge–Ampère Operator

“…The interested reader should also check [7]. Remark 1.1 We can assume without loss of generality that the matrices A in the definition of D s u(x) are symmetric and positive definite.…”