Related papers: $\Gamma$-Convergence of Free-discontinuity Problem…
We prove results on the relaxation and weak* lower semicontinuity of integral functionals of the form \[ \mathcal{F}[u] := \int_{\Omega} f \bigg( \frac{1}{2} \bigl( \nabla u(x) + \nabla u(x)^T \bigr) \bigg)\,\mathrm{d} x, \qquad u : \Omega…
We study homogenization by Gamma-convergence of periodic multiple integrals of the calculus of variations when the integrand can take infinite values outside of a convex set of matrices.
We discuss the $\Gamma$-convergence, under the appropriate scaling, of the energy functional $$ \|u\|_{H^s(\Omega)}^2+\int_\Omega W(u)dx,$$ with $s \in (0,1)$, where $\|u\|_{H^s(\Omega)}$ denotes the total contribution from $\Omega$ in the…
We derive, by means of variational techniques, a limiting description for a class of integral functionals under linear differential constraints. The functionals are designed to encode the energy of a high-contrast composite, that is, a…
Starting from 3D elasticity equations we derive the model of the homogenized von K\'arm\'an plate by means of $\Gamma$-convergence. This generalizes the recent results, where the material oscillations were assumed to be periodic.
A periodic homogenization result of nonconvex integral functionals in the vectorial case with convex bounded constraints on gradients is proved. The class of integrands considered have singular behavior near the boundary of the convex set…
We analyze the asymptotic behavior of a multiscale problem given by a sequence of integral functionals subject to differential constraints conveyed by a constant-rank operator with two characteristic length scales, namely the film thickness…
We present new results concerning the approximation of the total variation, $\int_{\Omega} |\nabla u|$, of a function $u$ by non-local, non-convex functionals of the form $$ \Lambda_\delta u = \int_{\Omega} \int_{\Omega} \frac{\delta…
Given a $C^1$ function $\mathcal{H}$ defined in the unit sphere $\mathbb{S}^2$, an $\mathcal{H}$-surface $M$ is a surface in the Euclidean space $\mathbb{R}^3$ whose mean curvature $H_M$ satisfies $H_M(p)=\mathcal{H}(N_p)$, $p\in M$, where…
We establish a framework that allows to prove Gamma-converge of functionals of Lagrangian form on spaces of trajectories based on convergence of viscosity solutions of associated Hamilton-Jacobi equations. Gamma convergence follows from a:…
We prove that certain nonlocal functionals defined on partitions made of measurable sets Gamma-converge to a local functional modeled on the perimeter in the sense of De Giorgi. Those nonlocal functionals involve generalized surface tension…
Let $\Gamma$ be the fundamental group of a closed orientable surface of genus at least two. Consider the composition of a uniformly random element of $\mathrm{Hom}(\Gamma,S_n)$ with the $(n-1)$-dimensional irreducible representation of…
We consider in 2D the following special case of the Mumford-Shah functional $$ J(u, \Gamma)=\int_{B_1\backslash\Gamma} |\nabla u|^2 dx + \lambda^2 \frac{\pi}{2} \mathcal{H}^1(\Gamma). $$ It is known that if the minimizer has a crack-tip in…
The end compactification |\Gamma| of the locally finite graph \Gamma is the union of the graph and its ends, endowed with a suitable topology. We show that \pi_1(|\Gamma|) embeds into a nonstandard free group with hyperfinitely many…
We study the effective behavior of random, heterogeneous, anisotropic, second order phase transitions energies that arise in the study of pattern formations in physical-chemical systems. Specifically, we study the asymptotic behavior, as…
Given a bounded open set $\Omega\subset \mathbb{R}^n$, we study sequences of quadratic functionals on the Sobolev space $H^1_0(\Omega)$, perturbed by sequences of bounded linear functionals. We prove that their $\Gamma$-limits, in the weak…
Let $\Omega$ be a domain in $\mathbb R^3$ with $\partial\Omega = \partial\left(\mathbb R^3\setminus \overline{\Omega}\right)$, where $\partial\Omega$ is unbounded and connected, and let $u$ be the solution of the Cauchy problem for the heat…
This work develops a dynamic homogenization approach for metamaterials. It finds an approximate macroscopic homogenized equation with constant coefficients posed in space and time; however, the resulting homogenized equation is higher order…
By introducing height dependency in the surface energy density, we propose a novel regularized variational model to simulate wetting/dewetting problems. The regularized model leads to the appearance of a precursor layer which covers the…
We study the asymptotic behaviour of sequences of integral functionals depending on moving anisotropies. We introduce and describe the relevant functional setting, establishing uniform Meyers-Serrin type approximations, Poincar\'e…