Related papers: Non-local approximation of the Griffith functional

The approximation in the sense of $\Gamma$-convergence of nonisotropic Griffith-type functionals, with $p-$growth ($p>1$) in the symmetrized gradient, by means of a suitable sequence of non-local convolution type functionals defined on…

We analyze a finite-difference approximation of a functional of Ambrosio-Tortorelli type in brittle fracture, in the discrete-to-continuum limit. In a suitable regime between the competing scales, namely if the discretization step $\delta$…

We prove a compactness result with respect to $\Gamma$-convergence for a class of integral functionals which are expressed as a sum of a local and a non-local term. The main feature is that, under our hypotheses, the local part of the…

We analyse the $\Gamma$-convergence of general non-local convolution type functionals with varying densities depending on the space variable and on the symmetrized gradient. The limit is a local free-discontinuity functional, where the bulk…

We study the $\Gamma$-convergence of a family of non-local, non-convex functionals in $L^p(I)$ for $p \ge 1$, where $I$ is an open interval. We show that the limit is a multiple of the $W^{1, p}(I)$ semi-norm to the power $p$ when $p>1$…

Classical results show that gradient descent converges linearly to minimizers of smooth strongly convex functions. A natural question is whether there exists a locally nearly linearly convergent method for nonsmooth functions with quadratic…

We propose a possible nonlocal approximation of the Willmore functional, in the sense of Gamma-convergence, based on the first variation ot the fractional Allen-Cahn energies, and we prove the corresponding $\Gamma$-limsup estimate. Our…

We prove that E. De Giorgi's conjecture for the nonlocal approximation of free-discontinuity problems extends to the case of functionals defined in terms of the symmetric gradient of the admissible field. After introducing a suitable class…

We approximate functionals depending on the gradient of $u$ and on the behaviour of $u$ near the discontinuity points, by families of non-local functionals where the gradient is replaced by finite differences. We prove pointwise…

We consider solving nonconvex composite optimization problems in which the sum of a smooth function and a nonsmooth function is minimized. Many of convergence analyses of proximal gradient-type methods rely on global descent property…

For a locally Lipschitz continuous function $f:X\to\mathbb{R}$ the generalized gradient $\partial f(x)$ of Clarke is used to develop some (set-valued) gradient on a set $A\subset X$. Existence, uniqueness and some approximation are…

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 prove the $\Gamma$-convergence of sequences of differentially constrained, random integral functionals of the form \begin{equation*} \int_{U} f\Big(\omega, x/\varepsilon, \mathbb{A} u\Big) \mathrm{d} x \end{equation*} for the class of…

We show that gradient descent can converge to any local minimum of a smooth semi-algebraic function. This holds if the step sizes are nonsummable and sufficiently small. The same results hold for the subgradient method on locally Lipschitz…

We prove that certain nonlocal functionals defined on partitions made of measurable sets Gamma-converge to a local functional modeled on the perimeter in the sense of De Giorgi. Those nonlocal functionals involve generalized surface tension…

We prove a partial regularity result for local minimizers of quasiconvex variational integrals with general growth. The main tool is an improved A-harmonic approximation, which should be interesting also for classical growth.

Given a family of locally Lipschitz vector fields $X(x)=(X_1(x),\dots,X_m(x))$ on $\mathbb{R}^n$, $m\leq n$, we study functionals depending on $X$. We prove an integral representation for local functionals with respect to $X$ and a result…

We show that the minimization problem of any non-convex and non-lower semi-continuous function on a compact convex subset of a locally convex real topological vector space can be studied via an associated convex and lower semi-continuous…

We investigate the $\Gamma$-convergence of Ambrosio-Tortorelli type-functionals for circle valued functions, in the case of volume terms with linear growth. We show the emergence of a non-local $\Gamma$-limit, which is due to the…

We study the $\Gamma$-convergence of sequences of free discontinuity functionals with linear growth defined in the space ${\rm BD}$ of functions with bounded deformation. We prove a compactness result with respect to $\Gamma$-convergence…