相关论文: A variational theory for integral functionals invo…

Based on recent developments in the theory of fractional Sobolev spaces, an interesting new class of nonlocal variational problems has emerged in the literature. These problems, which are the focus of this work, involve integral functionals…

This work revolves around the rigorous asymptotic analysis of models in nonlocal hyperelasticity. The corresponding variational problems involve integral functionals depending on nonlocal gradients with a finite interaction range $\delta$,…

In this paper we consider the minimization of a novel class of fractional linear growth functionals involving the Riesz fractional gradient. These functionals lack the coercivity properties in the fractional Sobolev spaces needed to apply…

In this paper, we consider a class of variational problems with integral functionals involving nonlocal gradients. These models have been recently proposed as refinements of classical hyperelasticity, aiming for an effective framework to…

We extend classical results on variational inequalities with convex sets with gradient constraint to a new class of fractional partial differential equations in a bounded domain with constraint on the distributional Riesz fractional…

In this paper we develop a new set of results based on a nonlocal gradient jointly inspired by the Riesz s-fractional gradient and Peridynamics, in the sense that its integration domain depends on a ball of radius delta > 0 (horizon of…

This paper addresses the asymptotics of functionals with linear growth depending on the Riesz $s$-fractional gradient on piecewise constant functions. We consider a general class of varying energy densities and, as $s\to 1$, we characterize…

This work revolves around properties and applications of functions whose nonlocal gradient, or more precisely, finite-horizon fractional gradient, vanishes. Surprisingly, in contrast to the classical local theory, we show that this class…

Singular perturbations have been used to select solutions of (non-convex) variational problems with a multiplicity of minimizers. The prototype of such an approach is the gradient theory of phase transitions by L. Modica, who specialized…

We present a systematic study on a class of nonlocal integral functionals for functions defined on a bounded domain and the naturally induced function spaces. The function spaces are equipped with a seminorm depending on finite differences…

We introduce a variational theory for processes adapted to the multi-dimensional Brownian motion filtration. The theory provides a differential structure which describes the infinitesimal evolution of Wiener functionals at very small…

The paper introduces a general strategy for identifying strong local minimizers of variational functionals. It is based on the idea that any variation of the integral functional can be evaluated directly in terms of the appropriate…

Fractional derivatives are a well-studied generalization of integer order derivatives. Naturally, for optimization, it is of interest to understand the convergence properties of gradient descent using fractional derivatives. Convergence…

In this paper we provide a different approach for existence of the variational solutions of the gradient flows associated to functionals on Sobolev spaces studied in \cite{BDDMS20}. The crucial condition is the convexity of the functional…

We provide a general treatment of perturbations of a class of functionals modeled on convolution energies with integrable kernel which approximate the $p$-th norm of the gradient as the kernel is scaled by letting a small parameter…

Nonlocal and fractional-order models capture effects that classical partial differential equations cannot describe; for this reason, they are suitable for a broad class of engineering and scientific applications that feature multiscale or…

Motivated by problems where jumps across lower dimensional subsets and sharp transitions across interfaces are of interest, this paper studies the properties of fractional bounded variation ($BV$)-type spaces. Two different natural…

In this paper, the aim of our work is to establish global weighted gradient estimates via fractional maximal functions and the point-wise regularity estimates of Dirichlet problem for divergence elliptic equations of the type \begin{align*}…

In this paper, a class of nonlocal fractional Dirichlet problems is studied. By using a variational principle due to Ricceri (whose original version was given in J. Comput. Appl. Math. 113 (2000), 401-410), the existence of infinitely many…

Local solutions for variational and quasi-variational inequalities are usually the best type of solutions that could practically be obtained when in case of lack of convexity or else when available numerical techniques are too limited for…