$$C^{\sigma +\alpha }$$Cσ+α regularity for concave nonlocal fully nonlinear elliptic equations with rough kernels

  title={\$\$C^\{\sigma +\alpha \}\$\$C$\sigma$+$\alpha$ regularity for concave nonlocal fully nonlinear elliptic equations with rough kernels},
  author={Joaquim Serra},
  journal={Calculus of Variations and Partial Differential Equations},
  • J. Serra
  • Published 5 May 2014
  • Mathematics
  • Calculus of Variations and Partial Differential Equations
We establish $$C^{\sigma +\alpha }$$Cσ+α interior estimates for concave nonlocal fully nonlinear equations of order $$\sigma \in (0,2)$$σ∈(0,2) with rough kernels. Namely, we prove that if $$u\in C^{\alpha }(\mathbb {R}^n)$$u∈Cα(Rn) solves in $$B_1$$B1 a concave translation invariant equation with kernels in $$\mathcal L_0(\sigma )$$L0(σ), then u belongs to $$C^{\sigma +\alpha }(\overline{B_{1/2}})$$Cσ+α(B1/2¯), with an estimate. More generally, our results allow the equation to depend on x in… 

Existence-Uniqueness for nonlinear integro-differential equations with drift in $\mathbb{R}^d$

In this article we consider a class of nonlinear integro-differential equations of the form $$\inf_{\tau \in\mathcal{T}} \bigg\{\int_{\mathbb{R}^d}

Regularity results for nonlocal equations and applications

We introduce the concept of $$C^{m,\alpha }$$ C m , α -nonlocal operators, extending the notion of second order elliptic operator in divergence form with $$C^{m,\alpha }$$ C m , α -coefficients. We

Characterization of nonlocal diffusion operators satisfying the Liouville theorem. Irrational numbers and subgroups of $\mathbb{R}^d$

We investigate the characterization of generators $\mathcal{L}$ of Levy processes satisfying the Liouville theorem: Bounded functions $u$ solving $\mathcal{L}[u]=0$ are constant. These operators are

Regularity estimates for nonlocal Schrödinger equations

  • M. Fall
  • Mathematics
    Discrete & Continuous Dynamical Systems - A
  • 2019
We prove H\"older regularity estimates up to the boundary for weak solutions $u$ to nonlocal Schr\"odinger equations subject to exterior Dirichlet conditions in an open set $\Omega\subset

Regional fractional Laplacians: Boundary regularity

  • M. Fall
  • Mathematics
    Journal of Differential Equations
  • 2022

Obstacle problems for integro-differential operators: regularity of solutions and free boundaries

We study the obstacle problem for integro-differential operators of order 2s, with $$s\in (0,1)$$s∈(0,1). Our main result establish that the free boundary is $$C^{1,\gamma }$$C1,γ and $$u\in

On Schauder estimates for a class of nonlocal fully nonlinear parabolic equations

We obtain Schauder estimates for a class of concave fully nonlinear nonlocal parabolic equations of order σ∈(0,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym}



Regularity for fully nonlinear nonlocal parabolic equations with rough kernels

We prove space and time regularity for solutions of fully nonlinear parabolic integro-differential equations with rough kernels. We consider parabolic equations $$u_t = \mathrm{I}u$$ut=Iu, where

Elliptic Partial Differential Equations of Second Order

We study in this chapter a class of partial differential equations that generalize and are to a large extent represented by Laplace’s equation. These are the elliptic partial differential equations

Boundary regularity for fully nonlinear integro-differential equations

We study fine boundary regularity properties of solutions to fully nonlinear elliptic integro-differential equations of order 2s, with s ∈ (0,1). We consider the class of nonlocal operators L∗ ⊂L 0,

C 1, α Interior Regularity for Nonlinear Nonlocal Elliptic Equations with Rough Kernels

We prove a C 1, α interior regularity theorem for fully nonlinear uniformly elliptic integro-differential equations without assuming any regularity of the kernel. We then give some applications to

The Evans-Krylov theorem for nonlocal fully nonlinear equations

We prove a regularity result for solutions of a purely integro-differential Bellman equation. This regularity is enough for the solutions to be understood in the classical sense. If we let the order

Regularity theory for fully nonlinear integro‐differential equations

We consider nonlinear integro‐differential equations like the ones that arise from stochastic control problems with purely jump Lévy processes. We obtain a nonlocal version of the ABP estimate,

Fully Nonlinear Elliptic Equations

Introduction Preliminaries Viscosity solutions of elliptic equations Alexandroff estimate and maximum principle Harnack inequality Uniqueness of solutions Concave equations $W^{2,p}$ regularity

Regularity Results for Nonlocal Equations by Approximation

We obtain C1,α regularity estimates for nonlocal elliptic equations that are not necessarily translation-invariant using compactness and perturbative methods and our previous regularity results for

Interior a priori estimates for solutions of fully non-linear equations

On etend une theorie de perturbations aux solutions d'equations uniformement elliptiques d'ordre 2 totalement non lineaires