# Hölder regularity for nonlocal double phase equations

@article{DeFilippis2019HlderRF, title={H{\"o}lder regularity for nonlocal double phase equations}, author={Cristiana De Filippis and Giampiero Palatucci}, journal={Journal of Differential Equations}, year={2019} }

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…

## 51 Citations

Local boundedness and Hölder continuity for the parabolic fractional p-Laplace equations

- 2021

In this paper, we study the boundedness and Hölder continuity of local weak solutions to the following nonhomogeneous equation $$\begin{aligned} \partial _tu(x,t)+\mathrm{P.V.}\int _{{\mathbb…

H\"older regularity for weak solutions to nonlocal double phase problems

- Mathematics
- 2021

We prove local boundedness and Hölder continuity for weak solutions to nonlocal double phase problems concerning the following fractional energy functional ∫ Rn ∫ Rn |v(x) − v(y)| |x− y|n+sp + a(x,…

Global Lorentz estimates for nonuniformly nonlinear elliptic equations via fractional maximal operators

- Mathematics, Physics
- 2020

This paper is a contribution to the study of regularity theory for nonlinear elliptic equations. The aim of this paper is to establish some global estimates for non-uniformly elliptic in divergence…

Local H\"older continuity for fractional nonlocal equations with general growth

- Mathematics
- 2021

for some 0 < λ ≤ Λ. A main point is that the function G is an N-function satisfying the ∆2 and ∇2 conditions (see the next section) and that a simple example of the kernel K(x, y) is a(x, y)|x − y|−n…

Equivalence between distributional and viscosity solutions for the double-phase equation

- Mathematics, Physics
- 2020

We investigate the different notions of solutions to the double-phase equation $$ -\dive(|Du|^{p-2}Du+a(x)|Du|^{q-2}Du)=0, $$ which is characterized by the fact that both ellipticity and growth…

A priori estimates for solutions to a class of obstacle problems under p, q-growth conditions

- PhysicsJournal of Elliptic and Parabolic Equations
- 2019

In this paper we would like to complement the results contained in Gavioli (Forum Math, to appear) by dealing with the higher differentiability of integer order of solutions to a class of obstacle…

On weak and viscosity solutions of nonlocal double phase equations

- Mathematics
- 2021

We consider the nonlocal double phase equation P.V. ∫ Rn |u(x) − u(y)|(u(x) − u(y))Ksp(x, y) dy + P.V. ∫ Rn a(x, y)|u(x) − u(y)|(u(x) − u(y))Ktq(x, y) dy = 0, where 1 < p ≤ q and the modulating…

Regularity for multi-phase variational problems

- Mathematics, PhysicsJournal of Differential Equations
- 2019

We prove $C^{1,\nu}$ regularity for local minimizers of the \oh{multi-phase} energy: \begin{flalign*} w \mapsto \int_{\Omega}\snr{Dw}^{p}+a(x)\snr{Dw}^{q}+b(x)\snr{Dw}^{s} \ dx, \end{flalign*} under…

Multiple solutions for a class of double phase problem without the Ambrosetti–Rabinowitz conditions

- MathematicsNonlinear Analysis
- 2019

Abstract In the present paper, in view of the variational approach, we consider the existence and multiplicity of weak solutions for a class of the double phase problem − div ( | ∇ u | p − 2 ∇ u + a…

Regularity results for a class of obstacle problems with p, q−growth conditions

- MathematicsESAIM: Control, Optimisation and Calculus of Variations
- 2021

In this paper we prove the the local Lipschitz continuity for solutions to a class of obstacle problems of the type
min{ ∫ΩF(x, Dz) : z ∈ 𝛫ψ(Ω)}.
Here 𝛫ψ(Ω) is the set of admissible functions z ∈…

## References

SHOWING 1-10 OF 42 REFERENCES

Hölder estimates for viscosity solutions of equations of fractional p-Laplace type

- Mathematics
- 2014

AbstractWe prove Hölder estimates for viscosity solutions of a class of possibly degenerate and singular equations modelled by the fractional p-Laplace equation
$$PV…

Regularity for general functionals with double phase

- Mathematics
- 2017

We prove sharp regularity results for a general class of functionals of the type $$\begin{aligned} w \mapsto \int F(x, w, Dw) \, dx, \end{aligned}$$w↦∫F(x,w,Dw)dx,featuring non-standard growth…

Regularity for Double Phase Variational Problems

- Mathematics
- 2015

AbstractWe prove sharp regularity theorems for minimisers of a class of variational integrals whose integrand switches between two different types of degenerate elliptic phases, according to the zero…

Nonlinear Calderón–Zygmund Theory in the Limiting Case

- Mathematics
- 2018

AbstractWe prove a maximal differentiability and regularity result for solutions to nonlinear measure data problems. Specifically, we deal with the limiting case of the classical theory of Calderón…

Local and global minimizers for a variational energy involving a fractional norm

- Mathematics
- 2011

AbstractWe study existence, uniqueness, and other geometric properties of the minimizers of the energy functional
$$ \|u\|^2_{H^s(\Omega)}+\int\limits_\Omega W(u)\,{d}x, $$where…

Equivalence of solutions to fractional p-Laplace type equations

- Mathematics
- 2016

In this paper, we study different notions of solutions of nonlocal and nonlinear equations of fractional $p$-Laplace type $${\rm P.V.} \int_{\mathbb…

Bounded Minimisers of Double Phase Variational Integrals

- Mathematics
- 2015

AbstractBounded minimisers of the functional
$$w \mapsto \int (|Dw|^p+a(x)|Dw|^q)\,{\rm d}x,$$w↦∫(|Dw|p+a(x)|Dw|q)dx,where $${0 \leqq a(\cdot) \in C^{0, \alpha}}$$0≦a(·)∈C0,α and $${1 < p <…

Regularity for multi-phase variational problems

- Mathematics, PhysicsJournal of Differential Equations
- 2019

We prove $C^{1,\nu}$ regularity for local minimizers of the \oh{multi-phase} energy: \begin{flalign*} w \mapsto \int_{\Omega}\snr{Dw}^{p}+a(x)\snr{Dw}^{q}+b(x)\snr{Dw}^{s} \ dx, \end{flalign*} under…

Nonlocal self-improving properties

- Mathematics
- 2015

Solutions to nonlocal equations with measurable coefficients are higher differentiable.
Specifically, we consider nonlocal integrodifferential equations with measurable coefficients whose model is…

The Dirichlet problem for thep-fractional Laplace equation

- MathematicsNonlinear Analysis
- 2018

Abstract We deal with a class of equations driven by nonlocal, possibly degenerate, integro-differential operators of differentiability order s ∈ ( 0 , 1 ) and summability growth p ∈ ( 1 , ∞ ) ,…