Regularity Properties for a Class of Non-uniformly Elliptic Isaacs Operators

Abstract

Here we consider the elliptic differential operator defined as the sum of the minimum and the maximum eigenvalue of the Hessian matrix, which can be viewed as a degenerate elliptic Isaacs operator in dimension larger than two. Despite of nonlinearity, degeneracy, non-concavity and non-convexity, such operator generally enjoys the qualitative properties of the Laplace operator, as for instance maximum and comparison principle, Liouville theorem for subsolutions or supersolutions, ABP and Harnack inequalities. Existence and uniqueness for the Dirichlet problem are also proved as well as the local Hölder continuity of viscosity solutions.

A regularity result for a class of non-uniformly elliptic operators

“…We end this note by pointing out that the regularity issue for solutions of (1.1) in the case a 1 = 0 and a N > 0 (or a 1 > 0 and a N = 0) is an open problem and only partial results are known: Lipschitz regularity for P ± 1 , see [2], Hölder estimates for M a in the case of asymmetric distributions of weights concentrated on the smallest or on the largest eigenvalue, namely a 1 > a 2 + • • • + a N or a N > a 1 + • • • + a N −1 , see [15] and the references therein.…”

“…Let us point out that our result does not apply to P ± k when k < N since in these cases a 1 or a N are zero. Nevertheless, some regularity results concerning such operators, in particular for k = 1, can be found in [19] and [2, Propositions 3.1-3.2]- [15,Theorem 1.4]. Applying the Ishii-Lions approach to the problem (see [20]), we manage to prove that viscosity solutions of (1.1) are C 0,β loc (Ω), where…”

“…In particular, we recall the fundamental result [21]. Concerning the regularity issues of viscosity solutions of degenerate equations closely related to ours, we refer to [8,[13][14][15].…”

“…In [15], the authors studied qualitative properties of solutions of (1.1) under the further assumption that a 1 > 0 and a N > 0, having in mind as prototype the Isaacs operator…”

Game-theoretic approach to Hölder regularity for PDEs involving eigenvalues of the Hessian

“…In general, the PDE (1.2) is fully nonlinear and not uniformly elliptic, and we do not get Hölder regularity directly from the general theory. For this equation with lower order terms, Ferrari and Vitolo [FV20] used methods from the viscosity theory to study ABP, Harnack and Hölder estimates, and later Ferrari and Galise [FG21] showed C 0,δ -regularity for 0…”