Related papers: A rigidity result for global Mumford-Shah minimize…

We show that if $(u,K)$ is a global minimizer for the Mumford-Shah functional in $R^N$, and if K is a smooth enough cone, then (modulo constants) u is a homogenous function of degree 1/2. We deduce some applications in $R^3$ as for instance…

We establish a new monotonicity formula for minimizers of the Mumford-Shah functional in planar domains. Our formula follows the spirit of Bucur-Luckhaus, but works with the David-L\'eger entropy instead of the energy. Interestingly, this…

We show that if (u;K) is a minimizer of the Mumford-Shah functional in an open set of R^3, and if x, K and r > 0 are such that K is close enough to a minimal cone of type P (a plane), Y (three half planes meeting with 120 degrees angles) or…

Let P be an hyperplane in R^N, and denote by dH the Hausdorff distance. We show that for all positive radius r < 1 there is an epsilon > 0, such that if K is a Reifenberg-flat set in B(0; 1), a ball in R^N, that contains the origin, with…

In [8], G. David suggested a new type of global minimizer for the Mumford-Shah functional in $\R^3$, for which the singular sets belong to a three parameters family of sets ($0<\delta\_1,\delta\_2,\delta\_3<\pi$). We first derive necessary…

We give a more elementary proof of a result by Ambrosio, Fusco and Hutchinson to estimate the Hausdorff dimension of the singular set of minimizers of the Mumford-Shah energy (see [2, Theorem 5.6]). On the one hand, we follow the strategy…

In this paper we exhibit a family of stationary solutions of the Mumford-Shah functional in $\mathbb{R}^3$, arbitrary close to a crack-front. Unlike other examples, known in the literature, those are topologically non-minimizing in the…

The goal of this paper is to establish a monotonicity formula for perimeter minimizing sets in RCD(0,N) metric measure cones, together with the associated rigidity statement. The applications include sharp Hausdorff dimension estimates for…

We consider $\mathbb{S}^2$-valued maps on a domain $\Omega\subset\mathbb{R}^N$ minimizing a perturbation of the Dirichlet energy with vertical penalization in $\Omega$ and horizontal penalization on $\partial\Omega$. We first show the…

The study of singular perturbations of the Dirichlet energy is at the core of the phenomenological-description paradigm in soft condensed matter. Being able to pass to the limit plays a crucial role in the understanding of the…

We review some classical results and more recent insights about the regularity theory for local minimizers of the Mumford and Shah energy and their connections with the Mumford and Shah conjecture. We discuss in details the links among the…

We consider regularity of the crack set associated to a minimizer of the Griffith fracture energy, often used in modeling brittle materials. We show that the crack is uniformly rectifiable which in conjunction with our previous…

Given a compact Riemannian manifold $(M^n,g)$ and a fixed cohomology class, $[\alpha^*] \in H^k(M)$, we consider the existence of a minimizer $\alpha \in [\alpha^*]$ of the generalized minimal surface energy $\int_M \sqrt{1+|\alpha|^2}…

In any dimension $N \geq 1$, for given mass $m > 0$ and for the $C^1$ energy functional \begin{equation*} I(u):=\frac{1}{2}\int_{\mathbb{R}^N}|\nabla u|^2dx-\int_{\mathbb{R}^N}F(u)dx, \end{equation*} we revisit the classical problem of…

In this paper, we study semilinear fractional equations $$(-\Delta)^s u(x) = f(u(x))$$ in a half-space and prove that all positive solutions are strictly increasing in the $x_n$-direction. Previous results typically require the solution $u$…

We are interested in the study of local and global minimizers for an energy functional of the type $$ \frac{1}{4} \iint_{\mathbb{R}^{2 N} \setminus \left( \mathbb{R}^N \setminus \Omega \right)^2} |u(x) - u(y)|^2 K(x - y) \, dx dy +…

We study existence, unicity and other geometric properties of the minimizers of the energy functional $$ \|u\|^2_{H^s(\Omega)}+\int_\Omega W(u)\,dx, $$ where $\|u\|_{H^s(\Omega)}$ denotes the total contribution from $\Omega$ in the $H^s$…

This note establishes, first of all, the monotonic increase with $N$ of the average $K$-body energy of classical $N$-body ground state configurations with $N\geq K$ monomers that interact solely through a permutation-symmetric $K$-body…

The aim of these notes is to give a complete self-contained account of the current state of the art in the regularity for planar minimizers and critical points of the Mumford-Shah functional.

In this paper we deal with the existence, regularity and Beltrami field property of magnetic energy minimisers under a helicity constraint. We in particular tackle the problem of characterising local as well as global minimisers of the…