Hölder regularity for nonlocal double phase equations

By Cristiana De Filippis, Giampiero Palatucci. Journal of Differential Equations, 2019

145 citations recorded by the source corpus.

We prove some regularity estimates for viscosity solutions to a class of possible degenerate and singular integro-differential equations whose leading operator switches between two different types of fractional elliptic phases, according to the zero set of a modulating coefficient $a=a(\cdot,\cdot)$. The model case is driven by the following nonlocal double phase operator, $$ \int \!\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}}\, {\rm d}y + \int \!a(x,y)\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{n+tq}}\, {\rm d}y, $$ where $q\geq p$ and $a(\cdot,\cdot)\geqq 0$. Our results do also apply for inhomogeneous equations, for very general classes of measurable kernels. By simply assuming the boundedness of the modulating coefficient, we are able to prove that the solutions are H\"older continuous, whereas similar sharp results for the classical local case do require $a$ to be H\"older continuous. To our knowledge, this is the first (regularity) result for nonlocal double phase problems.

View Paper: https://www.semanticscholar.org/paper/81724e461801c4a33bbf6d01b32e013d02f75937

How researchers see this paper

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

Citations in the corpus

Existence of solutions for singular double phase problems via the Nehari manifold method (2021, methodology): “We mention the papers of Baroni-ColomboMingione [4, 5, 6], Baroni-Kuusi-Mingione [7], Byun-Oh [8], Colombo-Mingione [11, 12], De Filippis-Palatucci [13], Marcellini [27, 28, 29], Ok [34, 35], RagusaTachikawa [44] and the references therein.”
Hölder regularity for weak solutions to nonlocal double phase problems (2021, methodology): “[41] G. Palatucci, A. Pisante, and Y. Sire, Subcritical approximation of a Yamabe-type nonlocal equation: a gamma- convergence approach, Ann.”
Manifold Constrained Non-uniformly Elliptic Problems (2019, result): “We refer to the papers [2, 7, 9, 17, 18, 54, 65] for more results, different directions and related topics.”
Growth conditions and regularity for weak solutions to nonlinear elliptic pdes (2020, unknown): “For related recent references we also quote Duzgun-Marcellini-Vespri [39], [40], Rǎdulescu-Zhang [63], [64], Cencelja-Rădulescu-Repovš [19], Papageorgiou-Rădulescu-Repovš-Dušan [62], Ambrosio-Rădulescu [1], Ragusa-Tachikawa [65], Chlebicka [20], Chlebicka-DeFilippis [21], [22], CupiniGiannetti-Giova”
Fractional double-phase patterns: concentration and multiplicity of solutions (2020, background): “In [33] the authors establish a Hölder regularity result for nonlocal double phase equations.”