Related papers: Extending linear growth functionals to functions o…
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…
The center of interest in this work are variational problems with integral functionals depending on special nonlocal gradients. The latter correspond to truncated versions of the Riesz fractional gradient, as introduced in [Bellido, Cueto &…
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…
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…
We prove a relaxation result for a quasi-convex bulk integral functional with variable exponent growth in a suitable space of bounded variation type. A key tool is a decomposition under mild assumptions of the energy into absolutely…
We study the relaxation of multiple integrals of the calculus of variations, where the integrands are nonconvex with convex effective domain and can take the value \infty. We use local techniques based on measure arguments to prove integral…
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 article studies an integral representation of functionals of linear growth on metric measure spaces with a doubling measure and a Poincar\'e inequality. Such a functional is defined through relaxation, and it defines a Radon measure on…
We study the minimization of convex, variational integrals of linear growth among all functions in the Sobolev space $W^{1,1}$ with prescribed boundary values (or its equivalent formulation as a boundary value problem for a degenerately…
We establish partial regularity results for minimizers of a class of functionals depending on differential expressions based on elliptic operators. Specifically, we focus on functionals of Orlicz growth with a natural strong quasiconvexity…
We establish the first partial regularity results for (strongly) symmetric quasiconvex functionals of linear growth on BD, the space of functions of bounded deformation. By Rindler's foundational work (Lower semicontinuity for integral…
It is well-known that convex variational problems with linear growth and Dirichlet boundary conditions might not have minimizers if the boundary condition is not suitably relaxed. We show that for a wide range of integrands, including the…
We study the existence of positive solutions for nonlocal systems in gradient form and set in the whole $\mathbb R^N$. A quasilinear fractional Schr\"odinger equation, where the leading operator is the $\frac Ns$-fractional Laplacian, is…
We give an extension of the theory of relaxation of variational integrals in classical Sobolev spaces to the setting of metric Sobolev spaces. More precisely, we establish a general framework to deal with the problem of finding an integral…
The classic Riesz representation theorem characterizes all linear and increasing functionals on the space $C_{c}(X)$ of continuous compactly supported functions. A geometric version of this result, which characterizes all linear increasing…
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…
Motivated by the direct method in the calculus of variations in $L^{\infty}$, our main result identifies the notion of convexity characterizing the weakly$^*$ lower semicontinuity of nonlocal supremal functionals: Cartesian level convexity.…
We establish $\mathrm{C}^{\infty}$-partial regularity results for relaxed minimizers of strongly quasiconvex functionals \begin{align*} \mathscr{F}[u;\Omega]:=\int_{\Omega}F(\nabla u)\,\mathrm{d} x,\qquad u\colon\Omega\to\mathbb{R}^{N},…
An extension of gradient elasticity through the inclusion of spatial derivatives of fractional order to describe power-law type of non-locality is discussed. Two phenomenological possibilities are explored. The first is based on the Caputo…
We characterize lower semicontinuity of integral functionals with respect to weak$^*$ convergence in $\mathrm{BV}$, including integrands whose negative part has linear growth. In addition, we allow for sequences without a fixed trace at the…