Related papers: Obstacle problems for nonlocal operators with sing…

We deal with a wide class of nonlinear integro-differential problems in the Heisenberg-Weyl group $\mathbb{H}^n$, whose prototype is the Dirichlet problem for the $p$-fractional subLaplace equation. These problems arise in many different…

In this article we consider the classical singular integral operator over a local field with rough kernels. We study the boundedness of such an operator on different function spaces by relaxing the smoothness condition on kernels.

We show how nonlocal boundary conditions of Robin type can be encoded in the pointwise expression of the fractional operator. Notably, the fractional Laplacian of functions satisfying homogeneous nonlocal Neumann conditions can be expressed…

We study the regularity for solutions of fully nonlinear integro differential equations with respect to nonsymmetric kernels. More precisely, we assume that our operator is elliptic with respect to a family of integro differential linear…

We study the regularity of weak solutions to nonlocal in time subdiffusion equations for a wide class of weakly singular kernels appearing in the generalised fractional derivative operator. We prove a weak Harnack inequality for nonnegative…

We provide sharp boundary regularity estimates for solutions to elliptic equations driven by an integro-differential operator obtained as the sum of a Laplacian with a nonlocal operator generalizing a fractional Laplacian. Our approach…

We analyze a nonlocal diffusion operator having as special cases the fractional Laplacian and fractional differential operators that arise in several applications. In our analysis, a nonlocal vector calculus is exploited to define a weak…

Anomalous behavior is ubiquitous in subsurface solute transport due to the presence of high degrees of heterogeneity at different scales in the media. Although fractional models have been extensively used to describe the anomalous transport…

We study a family of fractional integral operators whose kernels satisfying an non-isotropic dilation have singularity on a coordinate subspace. A characterization is given for these operators bounded from the classical, atom decomposable…

We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem in perforated domains.

We prove a higher regularity result for the free boundary in the obstacle problem for the fractional Laplacian via a higher order boundary Harnack inequality.

We analyse the obstacle problem for the nonlocal parabolic operator \[\partial_t u + (-\Delta)^{s} u - b \cdot \nabla u - \mathcal{I}u - ru,\] where $b\in\mathbb{R}^n$, $r\in\mathbb{R}$, and $\mathcal{I}$ is a nonlocal lower order diffusion…

The classical local Neumann problem is well studied and solutions of this problem lie, in general, in a Sobolev space. In this work, we focus on nonlocal Neumann problems with measurable, nonnegative kernels, whose solutions require less…

This paper investigates homogenization problems for the nonlocal operators with rapidly oscillating coefficients in the cases of periodic and random statistically homogeneous micro-structures. These operators involve the fractional…

We consider a generalization of the Bernoulli free boundary problem where the underlying differential operator is a nonlocal, non-translation-invariant elliptic operator of order $2s\in (0,2)$. Because of the lack of translation invariance,…

We study a nonlocal parabolic equation with an irregular kernel coefficient to establish higher H\"older regularity under an appropriate higher integrablilty on the nonhomogeneous terms and a minimal regularity assumption on the kernel…

We study the higher regularity in nonlocal free boundary problems posed for general integro-differential operators of order $2s$. Our main result is for the nonlocal one-phase (Bernoulli) problem, for which we establish that $C^{2,\alpha}$…

Recently, great attention has been focused on the study of fractional and non-local operators of elliptic type, both for pure mathematical research and in view of concrete real-world applications. Our problem is related to the fractional…

Kernel-based approach to operator approximation for partial differential equations has been shown to be unconditionally stable for linear PDEs and numerically exhibit unconditional stability for non-linear PDEs. These methods have the same…

We show that the solutions to the nonlocal obstacle problems for the nonlocal $-\Delta_p^s$ operator, when the fractional parameter $s\to\sigma$ for $0<\sigma\leq1$, converge to the solution of the corresponding obstacle problem for…