Related papers: $\Gamma$-convergence for nonlocal phase transition…

We analyze a family of non-local integral functionals of convolution-type depending on two small positive parameters $\varepsilon,\delta$: the first rules the length-scale of the non-local interactions and produces a `localization' effect…

We develop the free boundary regularity for nonnegative minimizers of the Alt-Phillips functional for negative power potentials $$\int_\Omega \left(\frac 1 2 |\nabla u|^2 + u^{\gamma} \chi_{\{u>0\}}\right) \, dx, \quad \quad \gamma \in…

We propose an abstract framework for the homogenization of random functionals which may contain non-convex terms, based on a two-scale $\Gamma$-convergence approach and a definition of Young measures on micropatterns which encodes the…

The $\Gamma $-limit of a family of functionals $u\mapsto \int_{\Omega }f\left( \frac{x}{\varepsilon },\frac{x}{\varepsilon ^{2}},D^{s}u\right) dx$ is obtained for $s=1,2$ and when the integrand $f=f\left( y,z,v\right) $ is a continous…

We discuss a model for phase transitions in which a double-well potential is singularly perturbed by possibly several terms involving different, arbitrarily high orders of derivation. We study by $\Gamma$-convergence the asymptotic…

$\Gamma$-convergence techniques are used to give a characterization of the behavior of a family of heterogeneous multiple scale integral functionals. Periodicity, standard growth conditions and nonconvexity are assumed whereas a stronger…

We study the limit behaviour of singularly-perturbed elliptic functionals of the form \[ \mathcal F_k(u,v)=\int_A v^2\,f_k(x,\nabla u)\.dx+\frac{1}{\varepsilon_k}\int_A g_k(x,v,\varepsilon_k\nabla v)\.dx\,, \] where $u$ is a vector-valued…

We prove a $\Gamma$-convergence result for space dependent weak membrane energies, that is for 'truncated quadratic potentials', that are quadratic below some threshold (depending on the pair of points that we are considering) and constant…

Given $p\in[1,\infty)$ and a bounded open set $\Omega\subset\mathbb R^d$ with Lipschitz boundary, we study the $\Gamma$-convergence of the weighted fractional seminorm \[ [u]_{s,p,f}^p = \int_{\mathbb R^d} \int_{\mathbb R^d}…

We consider a Dirichlet to Neumann operator $\mathcal{L}_a$ arising in a model for water waves, with a nonlocal parameter $a\in(-1,1)$. We deduce the expression of the operator in terms of the Fourier transform, highlighting a local…

This survey offers an overview of recent advances in nonlocal phase transition problems, modeled by Ginzburg--Landau type energies of the form \[ \frac{1}{4}\iint_{\R^{2n}\setminus (\R^n \setminus \Omega)^2}…

We consider degenerate nonautonomous energies $$ \int_\Omega f(x, Dv)\, dx, $$ for vector-valued functions $v \in W^{1,1}(\Omega, \mathbb{R}^N)$, where the integrand $f(x,P)$ satisfies growth and weak uniform quasiconvexity assumption…

For a fixed smooth map $u_0$ between two Riemann surfaces $\Sigma$ and $S$ with non-zero degree, we consider the energy function on Teichm\"uller space $\mc{T}$ of $\Sigma$ that assigns to a complex structure $t\in \mc{T}$ on $\Sigma$ the…

We investigate a homogenization problem related to a non-local interface energy with a periodic forcing term. We show the existence of planelike minimizers for such energy. Moreover, we prove that, under suitable assumptions on the…

We discuss some results related to a phase transition model in which the potential energy induced by a double-well function is balanced by a fractional elastic energy. In particular, we present asymptotic results (such as…

We establish some new results about the $\Gamma$-limit, with respect to the $L^1$-topology, of two different (but related) phase-field approximations of the so-called Euler's Elastica Bending Energy for curves in the plane.

We extend the duality principle for the $\Gamma$-convergence of convex lower semicontinuous functions, which was previously established only in separable reflexive Banach spaces, to the broader class of weakly compactly generated (WCG)…

In this paper we explore the potential of the double phase functional in an image processing context. To this end, we study minimizers of the double phase energy for functions with bounded variation and show that this energy can be obtained…

For $s\in (0,1)$ we introduce a notion of fractional $s$-mass on $(n-2)$-dimensional closed, orientable surfaces in $\R^n$. Moreover, we prove its $\Gamma$-convergence, with respect to the flat topology, and pointwise convergence to the…

In this paper, using the theory developed in [8], we obtain some results of a totally new type about a class of non-local problems. Here is a sample: Let $\Omega\subset {\bf R}^n$ be a smooth bounded domain, with $n\geq 4$, let $a, b,…