Related papers: Nonlocal problems in perforated domains

We investigate nonlocal field theories, a subject that has attracted some renewed interest in connection with nonlocal gravity models. We study, in particular, scalar theories of interacting delocalized fields, the delocalization being…

We prove a local Faber-Krahn inequality for solutions $u$ to the Dirichlet problem for $\Delta + V$ on an arbitrary domain $\Omega$ in $\mathbb{R}^n$. Suppose a solution $u$ assumes a global maximum at some point $x_0 \in \Omega$ and…

In this paper we introduce and study the concept of nonlocal ordered curvature. In the classical (differential) setting, the problem was introduced by Nirenberg and Li, where they conjectured that if a bounded, smooth surface has its mean…

We study large deviations for some non-local parabolic type equations. We show that, under some assumptions on the non-local term, problems defined in a bounded domain converge with an exponential rate to the solution of the problem defined…

In this paper we study the dependence of the H\"older estimates on the geometry of a domain with holes for the Neumann problem. For this, we study the H\"older regularity of the solutions to the Dirichlet and Neumann problems in the disk…

We study the asymptotic behavior for nonlocal diffusion equations $\partial_tu=\mathcal{J}u-\chi_0u$ in $\mathbb{R}^n\times(0,\infty)$ and obtain a sufficient condition so that solutions of the Cauchy problem decay in time at the rate of a…

We study Dirichlet problems for fractional Laplace equations of the form $(-\Delta)^{\frac{\alpha}{2}} u = f(x,u)$ in $\mathbb{R}^{n}$ for $0<\alpha<n$ where the nonlinearity $f(x,u) = \sum_{i=1}^{M} \sigma_{i} u^{q_i} + \omega$ involves…

This work showcases level set estimates for weak solutions to the $p$-Poisson equation on a bounded domain, which we use to establish Lebesgue space inclusions for weak solutions. In particular we show that if $\Omega\subset\mathbb{R}^n$ is…

We consider a minimization problem that combines the Dirichlet energy with the nonlocal perimeter of a level set, namely $$ \int_\Om |\nabla u(x)|^2\,dx+\Per\Big(\{u > 0\},\Om \Big),$$ with $\sigma\in(0,1)$. We obtain regularity results for…

We deal with Dirichlet problems of the form $$ \Delta u+f(u)=0 \mbox{ in }\Omega,\qquad u=0\ \mbox{ on }\partial \Omega $$ where $\Omega$ is a bounded domain of $\mathbb{R}^n$, $n\ge 3$, and $f$ has supercritical growth from the viewpoint…

The paper deals with second order parabolic equations on bounded domains with Dirichlet conditions in arbitrary Euclidean spaces. Their interest comes from being models for describing reaction-diffusion processes in several frameworks. A…

In this paper we study the Dirichlet problem for Laplace's equation in a domain $\omega_{\epsilon, \eta}$ perforated periodically with small holes in $\mathbb{R}^d$, where $\epsilon$ represents the scale of the minimal distances between…

We consider the inviscid Leray-$\alpha$ equations - an inviscid nonlocal regularisation of the Euler equations. In the first part, we prove the convergence of strong solutions of the Leray-$\alpha$ equations to strong solutions of the Euler…

This paper considers the initial-boundary value problem for the nonlocal Cahn--Hilliard equation $$ \partial_t\varphi + (-\Delta+1)(a(\cdot)\varphi -J\ast\varphi + G'(\varphi)) = 0 \quad \mbox{in}\ \Omega\times(0, T) $$ in an unbounded…

Solutions to nonlinear integro-differential systems are regular outside a negligible closed subset whose Hausdorff dimension can be explicitly bounded from above. This subset can be characterized using quantitative, universal energy…

Let $D$ be a bounded $C^2$-domain. Consider the following Dirichlet initial-boundary problem of nonlocal operators with a drift: $$ \partial_t u={\mathscr L}^{(\alpha)}_\kappa u+b\cdot \nabla u+f\ \mathrm{in}\ \mathbb R_+\times D,\ \…

Let $\Omega$ be a bounded $C^2$ domain in $\mathbb{R}^n$ and $u\in C(\mathbb{R}^n)$ solves \begin{equation*} \begin{aligned} \Delta u + a Iu + C_0|Du| \geq -K\quad \text{in}\; \Omega, \quad \Delta u + a Iu - C_0|Du|\leq K \quad \text{in}\;…

In this paper we establish optimal regularity estimates and smoothness of free boundaries for nonlocal obstacle problems governed by a very general class of integro-differential operators with possibly singular kernels. More precisely, in…

We investigate the space of non-local Sobolev functions associated with an integral kernel. We prove an extension result, Sobolev and Poincar\'e inequalities and an isoperimetric inequality for the non-local perimeter restricted to a set.…

We consider elliptic equations of order $2m$ in a domain $G\subset\mathbb R^n$ with nonlocal conditions that connect the values of the unknown function and its derivatives on $(n-1)$-dimensional submanifolds $\Upsilon_i$ (where…