The Bernstein Technique for Integro-Differential Equations

By X. Cabré, Serena Dipierro, Enrico Valdinoci. Archive for Rational Mechanics and Analysis, 2020

48 citations recorded by the source corpus.

We extend the classical Bernstein technique to the setting of integro-differential operators. As a consequence, we provide first and one-sided second derivative estimates for solutions to fractional equations, including some convex fully nonlinear equations of order smaller than two—for which we prove uniform estimates as their order approaches two. Our method is robust enough to be applied to some Pucci-type extremal equations and to obstacle problems for fractional operators, although several of the results are new even in the linear case. We also raise some intriguing open questions, one of them concerning the “pure” linear fractional Laplacian, another one being the validity of one-sided second derivative estimates for Pucci-type convex equations associated to linear operators with general kernels.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/02e1aaeb5790b1742585bc19b605742ab4f00a80

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

A Brezis-Nirenberg type result for mixed local and nonlocal operators (2022, unknown): “The contemporary investigation has speciﬁcally focused on several problems in the existence and regularity theory (see [1,4,5,8,11,12,16–18, The stirring motivation for the problems presented in this paper comes from the study of nonlinear problems with critical exponents, as the ones suggested by…”
Linear theory for a mixed operator with Neumann conditions (2020, unknown): “In terms of theory and applications, we recall that operators with mixed classical and fractional orders have been studied under different points of views, see for instance [1,2,4,6–10,12,13,18,19,21,25–27, 31–33,40,47,48] and the references therein.”
Regularity results for solutions of mixed local and nonlocal elliptic equations (2022, background): “Moreover, mixed operators have recently received a great attention from different points of view, including regularity theory [2, 3, 8, 10, 12, 14, 15, 17, 25], existence and non-existence results [1, 6, 28], eigenvalue problems [13, 16, 22], shape optimization and calculus of variations [7, 11], sy”
Mixed local and nonlocal elliptic operators: regularity and maximum principles (2020, background): “6 in [14]), regularization effects of Cahn-Hilliard equations (see [15]), numerics ([16]), probability and stochastics (see [17–19]), symmetry results for mixed range phase transitions (see [20]), porous medium equations (see [21]), decay estimates for parabolic equations (see [22]), specific Liouvi”