Related papers: Manifold-constrained free discontinuity problems a…

We study the regularity of minimizers to the composite membrane problem in the plane (ie given a domain omega and a positive number A, smaller than the measure of omega, minimize the first Dirichlet eigenvalue for the Schrodinger operator…

We prove reversed Hardy-Littlewood-Sobolev inequalities by carefully studying the natural associated free energies with direct methods of calculus of variations. Tightness is obtained by a dyadic argument, which quantifies the relative…

We mainly discuss superquadratic minimization problems for splitting-type variational integrals on a bounded Lipschitz domain $\Omega \subset \mathbb{R}^2$ and prove higher integrability of the gradient up to the boundary by incorporating…

We consider the task of recovering a Sobolev function on a connected compact Riemannian manifold $M$ when given a sample on a finite point set. We prove that the quality of the sample is given by the $L_\gamma(M)$-average of the geodesic…

We study the validity of the so-called Smoluchowski-Kramers approximation for a two dimensional system of stochastic partial differential equations, subject to a constant magnetic field. As the small mass limit does not yield to the…

We prove a lower bound on the sharp Poincar\'e-Sobolev embedding constants for general open sets, in terms of their inradius. We consider the following two situations: planar sets with given topology; open sets in any dimension, under the…

We establish an epsilon-regularity theorem at points in the free boundary of almost-minimizers of the energy $\mathrm{Per}_{w}(E)=\int_{\partial^*E}w\,\mathrm{d} {\mathscr{H}}^{n-1}$, where $w$ is a weight asymptotic to…

This paper deals with fractional Sobolev spaces on a compact Riemannian manifold. We prove a Sobolev inequality in the critical range with an optimal constant for these fractional Sobolev spaces. We use this result to study the existence of…

We study a variant of the Alt, Caffarelli, and Friedman free boundary problem with many phases and a slightly different volume term, which we originally designed to guess the localization of eigenfunctions of a Schr\"odinger operator in a…

We study almost-minimizers of anisotropic surface energies defined by a H\"older continuous matrix of coefficients acting on the unit normal direction to the surface. In this generalization of the Plateau problem, we prove almost-minimizers…

For a given domain $\Omega \subset \Bbb{R}^n$, we consider the variational problem of minimizing the $L^1$-norm of the gradient on $\Omega$ of a function $u$ with prescribed continuous boundary values and satisfying a continuous lower…

The present work constitutes a first step towards establishing a systematic framework for treating variational problems that depend on a given input function through a mixture of its derivatives of different orders in different directions.…

In [David-Toro 15] and [David-Engelstein-Toro 19], (some of) the authors studied almost minimizers for functionals of the type first studied by Alt and Caffarelli in [Alt-Caffarelli 81] and Alt, Caffarelli and Friedman in…

We prove a monotone Sobolev extension theorem for maps to Jordan domains with rectifiable boundary in metric surfaces of locally finite Hausdorff 2-measure. This is then used to prove a uniformization result for compact metric surfaces by…

We study boundary regularity of maps from two-dimensional domains into manifolds which are critical with respect to a generic conformally invariant variational functional and which, at the boundary, enter perpendicularly into a support…

We consider the simplest optimal control problem with one nonregular mixed inequality constraint, i.e. when its gradient in the control can vanish on the zero surface. Using the Dubovitskii--Milyutin theorem on the approximate separation of…

We prove quasi-monotonicity formulae for classical obstacle-type problems with quadratic energies with coefficients in fractional Sobolev spaces, and a linear term with a Dini-type continuity property. These formulae are used to obtain the…

We study the regularity properties of H\"older continuous minimizers to non-autonomous functionals satisfying $(p,q)$-growth conditions, under Besov assumptions on the coefficients. In particular, we are able to prove higher integrability…

We investigate the properties of minimizers of one-dimensional variational problems when the Lagrangian has no higher smoothness than continuity. An elementary approximation result is proved, but it is shown that this cannot be in general…

The solvability in Sobolev spaces is proved for divergence form second order elliptic equations in the whole space, a half space, and a bounded Lipschitz domain. For equations in the whole space or a half space, the leading coefficients…