Related papers: Numerical study of a new global minimizer for the …

Recently, the distribution-dependent Mumford-Shah model for hyperspectral image segmentation was introduced. It approximates an image based on first and second order statistics using a data term, that is built of a Mahalanobis distance plus…

We prove that, if u is a function satisfying all Euler conditions for the Mumford-Shah functional and the discontinuity set of u is given by three line segments meeting at the origin with equal angles, then there exists a neighbourhood U of…

We study global Mumford-Shah minimizers in $\R^N$, introduced by Bonnet as blow-up limits of Mumford-Shah minimizers. We prove a new monotonicity formula for the energy of $u$ when the singular set $K$ is contained in a smooth enough cone.…

This paper deals with the numerical optimization of the first three eigenvalues of the Laplace-Beltrami operator of domain in the Euclidean sphere in $\mathbb{R}^3$ with Neumann boundary conditions. We address two approaches : the first one…

This paper is the first part of an ongoing project aimed at providing a local minimality criterion, based on a second variation approach, for the triple point configurations of the Mumford-Shah functional.

A new necessary minimality condition for the Mumford-Shah functional is derived by means of second order variations. It is expressed in terms of a sign condition for a nonlocal quadratic form on $H^1_0(\Gamma)$, $\Gamma$ being a submanifold…

Using a calibration method, we prove that, if w is a function which satisfies all Euler conditions for the Mumford-Shah functional on a two-dimensional domain, and the discontinuity set S of w is a regular curve connecting two boundary…

Let \phi\ be a Dirichlet or Neumann eigenfunction of the Laplace-Beltrami operator on a compact Riemannian manifold with boundary. We prove lower bounds for the size of the nodal set {\phi=0}.

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…

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…

Let $M$ be a compact Riemannian manifold with or without boundary, and let $-\Delta $ be its Laplace-Beltrami operator. For any bounded scalar potential $q$, we denote by $\lambda\_i(q)$ the $i$-th eigenvalue of the Schr\"{o}dinger type…

Using a calibration method, we prove that, if $w$ is a function which satisfies all Euler conditions for the Mumford-Shah functional on a two-dimensional open set $\Omega$, and the discontinuity set of $w$ is a segment connecting two…

We prove a formula for the spectrum of the Laplace-Beltrami operator on functions for compact naturally reductive homogeneous spaces in terms of eigenvalues of a generalized Casimir operator and spherical representations. We apply this…

In this paper we consider minimizers of the Mumford-Shah functional with Dirichlet boundary conditions. We study blow-ups at the boundary and prove an epsilon-regularity theorem.

A new algorithm to determine the position of the crack (discontinuity set) of certain minimizers of Mumford-Shah functional in situations when a crack-tip occurs is introduced. The conformal mapping $w=\sqrt{z}$ in the complex plane is used…

We establish the existence and find some qualitative properties of open sets that minimize functionals of the form $ F(\lambda_1(\Omega;\beta),\dots,\lambda_k(\Omega;\beta))$ under measure constraint on $\Omega$, where…

We analyze the topological structure of the Nehari set for a class of functionals depending on a real parameter $\lambda$, and having two degrees of homogeneity. A special attention is paid to the extremal parameter $\lambda^*$, which is…

We describe a highly efficient numerical scheme for finding two-sided bounds for the eigenvalues of the fractional Laplace operator (-Delta)^{alpha/2} in the unit ball D in R^d, with a Dirichlet condition in the complement of D. The…

It is shown that eigenvalues of Laplace-Beltrami operators on compact Riemannian manifolds can be determined as limits of eigenvalues of certain finite-dimensional operators in spaces of polyharmonic functions with singularities. In…

In this short note, we answer a question raised by E. De Giorgi, showing that a Mumford-Shah minimizer in dimension 2 can only admit three maximum limit values as approaching the singular set. This result stems from tools developed in the…