Nonlocal criteria for compactness in the space of LpL^{p} vector fields

This work presents a set of sufficient conditions that guarantee a compact inclusion in the function space of $L^p$ vector fields defined on a domain that is either a bounded domain in $\mathbb{R}^{d}$ or $\mathbb{R}^{d}$ itself. The criteria are nonlocal and are given with respect to nonlocal interaction kernels that may not be necessarily radially symmetric. Moreover, these criteria for vector fields are also different from those given for scalar fields in that the conditions are based on nonlocal interactions involving only parts of the components of the vector fields

Characterizations of fractional Sobolev--Poincar\'e and (localized) Hardy inequalities

In this paper, we prove capacitary versions of the fractional Sobolev--Poincar\'e inequalities. We characterize localized variant of the boundary fractional Sobolev--Poincar\'e inequalities through uniform fatness condition of the domain in $\mathbb{R}^n$. Existence type results on the fractional Hardy inequality are established in the supercritical case $sp>n$ for $s\in(0,1)$, $p>1$. Characterization of the fractional Hardy inequality through weak supersolution of the associate problem is also addressed.Comment: 15 pages, there is a slight mistake in the proof of Theorem 1.12 in the version 1, so we remove this resul

Optimal design problems governed by the nonlocal p -Laplacian equation

In the present work, a nonlocal optimal design model has been considered as an approximation of the corresponding classical or local optimal design problem. The new model is driven by the nonlocal p-Laplacian equation, the design is the diffusion coefficient and the cost functional belongs to a broad class of nonlocal functional integrals. The purpose of this paper is to prove the existence of an optimal design for the new model. This work is complemented by showing that the limit of the nonlocal p-Laplacian state equation converges towards the corresponding local problem. Also, as in the paper by F. Andrés and J. Muñoz [J. Math. Anal. Appl. 429:288– 310], the convergence of the nonlocal optimal design problem toward the local version is studied. This task is successfully performed in two different cases: when the cost to minimize is the compliance functional, and when an additional nonlocal constraint on the design is assumed

On the Optimal Control of a Linear Peridynamics Model

We study a non-local optimal control problem involving a linear, bond-based peridynamics model. In addition to existence and uniqueness of solutions to our problem, we investigate their behavior as the horizon parameter $\delta$, which controls the degree of nonlocality, approaches zero. We then study a finite element-based discretization of this problem, its convergence, and the so-called asymptotic compatibility as the discretization parameter $h$ and the horizon parameter $\delta$ tend to zero simultaneously.Comment: 35 page