Related papers: A nonlocal isoperimetric problem with dipolar repu…
We consider a class of variational problems for densities that repel each other at distance. Typical examples are given by the Dirichlet functional and the Rayleigh functional \[ D(\mathbf{u}) = \sum_{i=1}^k \int_{\Omega} |\nabla u_i|^2…
In 2001 Wolansky \cite{Wol} introduced a particle number-Casimir functional for the Einstein-Vlasov system. Two open questions are associated with this functional. First, a meaningful variational problem should be formulated and the…
The discrete isoperimetric inequality in Euclidean geometry states that among all $n$-gons having a fixed perimeter $p$, the one with the largest area is the regular $n$-gon. The statement is true in spherical geometry and hyperbolic…
In this paper we prove the uniqueness and radial symmetry of minimizers for variational problems that model several phenomena. The uniqueness is a consequence of the convexity of the functional. The main technique is Fourier transform of…
We investigate the static critical behaviour of a uniaxial magnetic layer, with finite thickness L in one direction, yet infinitely extended in the remaining d dimensions. The magnetic dipole-dipole interaction is taken into account. We…
The main thrust of our current work is to exploit very specific characteristics of a given problem in order to acquire improved compactness for supercritical problems and to prove existence of new types of solutions. To this end, we shall…
For a complete noncompact connected Riemannian manifold with bounded geometry, we prove a compactness result for sequences of finite perimeter sets with uniformly bounded volume and perimeter in a larger space obtained by adding limit…
The ancient Gamow liquid drop model of nuclear energies has had a renewed life as an interesting problem in the calculus of variations: Find a set $\Omega \subset \mathbb R^3$ with given volume A that minimizes the sum of its surface area…
This contribution considers optimal control problems subject to nonlocal conservation laws -- those in which the velocity depends nonlocally (i.e., via a convolution) on the solution -- and the so-called singular limit. First, the existence…
Given a positive definite kernel in a locally compact space, we study a minimal energy problem in the presence of an external field over the class of all nonnegative Radon measures that are supported by a given closed noncompact set,…
We consider the inverse conductivity problem with discontinuous conductivities. We show in a rigorous way, by a convergence analysis, that one can construct a completely discrete minimization problem whose solution is a good approximation…
Consider a convex function that is invariant under an group of transformations. If it has a minimizer, does it also have an invariant minimizer? Variants of this problem appear in nonparametric statistics and in a number of adjacent fields.…
We formulate a variational principle for a collection of projectors in an indefinite inner product space. The existence of minimizers is proved in various situations.
We investigate the class of geodesic metric discs satisfying a uniform quadratic isoperimetric inequality and uniform bounds on the length of the boundary circle. We show that the closure of this class as a subset of Gromov-Hausdorff space…
We study the standing waves for a fourth-order Schr\"odinger equation with mixed dispersion that minimize the associated energy when the $L^2-$norm (the \textit{mass}) } is kept fixed. We need some non-homogeneous Gagliardo-Nirenberg-type…
Existence and regularity of minimizers in elliptic free boundary problems have been extensively studied in the literature. The corresponding study of higher critical points was recently initiated in Jerison and Perera [30, 31]. In…
In this note, we propose a variational framework in which the minimization of the acceleration on the group of diffeomorphisms endowed with a right-invariant metric is well-posed. It relies on constraining the acceleration to belong to a…
We derive upper and lower bounds on the Casimir--Polder force between an anisotropic dipolar body and a macroscopic body separated by vacuum via algebraic properties of Maxwell's equations. These bounds require only a coarse…
A class of causal variational principles on a compact manifold is introduced and analyzed both numerically and analytically. It is proved under general assumptions that the support of a minimizing measure is either completely timelike, or…
We introduce the nonlocal analogue of the classical free boundary minimal hypersurfaces in an open domain $\Omega$ of $\mathbb{R}^n$ as the (boundaries of) critical points of the fractional perimeter $\operatorname{Per}_s(\cdot,\,\Omega )$…