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The study deals with a minimal energy problem in the presence of an external field over noncompact classes of vector measures of infinite dimension in a locally compact space. The components are positive measures (charges) satisfying…
We discuss a variational model, given by a weighted sum of perimeter, bending and Riesz interaction energies, that could be considered as a toy model for charged elastic drops. The different contributions have competing preferences for…
We study compressible and incompressible nonlinear elasticity variational problems in a general context. Our main result gives a sufficient condition for an equilibrium to be a global energy minimizer, in terms of convexity properties of…
We consider the volume-constrained minimization of the sum of the perimeter and the Riesz potential. We add an external potential of the form $\|x\|^{\beta}$ that provides the existence of a minimizer for any volume constraint, and we study…
We study a variational problem for the perimeter associated with the Grushin plane, called minimal partition problem with trace constraint. This consists in studying how to enclose three prescribed areas in the Grushin plane, using the…
This work presents an Iterative Constraint Energy Minimizing Generalized Multiscale Finite Element Method (ICEM-GMsFEM) for solving the contact problem with high contrast coefficients. The model problem can be characterized by a variational…
We study a large family of axisymmetric Riesz-type singular interaction potentials with anisotropy in three dimensions. We generalize some of the results of our recent work in two dimensions to the present setting. For potentials with…
We compute the equilibrium measure in dimension d=s+4 associated to a Riesz s-kernel interaction with an external field given by a power of the Euclidean norm. Our study reveals that the equilibrium measure can be a mixture of a continuous…
In view of a recent example of a positive Radon measure $\mu$ on a domain $D\subset\mathbb R^n$, $n\geqslant3$, such that $\mu$ is of finite energy $E_g(\mu)$ relative to the $\alpha$-Green kernel $g$ on $D$, though the energy of…
The purpose of this paper is to present a new mathematical model for the deformation of thin Cosserat elastic plates. Our approach, which is based on a generalization of the classical Reissner plate theory, takes into account the transverse…
Derived from the concentration-compactness principle, the concept of generalized minimizer can be used to define generalized solutions of variational problems which may have components ``infinitely'' distant from each other. In this article…
We consider a weakly-interacting, harmonically-trapped Bose-Einstein condensed gas under rotation and investigate the connection between the energies obtained from mean-field calculations and from exact diagonalizations in a subspace of…
We establish lower semi-continuity and strict convexity of the energy functionals for a large class of vector equilibrium problems in logarithmic potential theory. This in particular implies the existence and uniqueness of a minimizer for…
In this paper, we investigate Riesz energy problems on unbounded conductors in $\R^d$ in the presence of general external fields $Q$, not necessarily satisfying the growth condition $Q(x)\to\infty$ as $x\to\infty$ assumed in several…
The main goal of this paper is to design multiscale basis functions within GMsFEM framework such that the convergence of method is independent of the contrast and linearly decreases with respect to mesh size if oversampling size is…
We investigate the problem of dimension reduction for plates in nonlinear magnetoelasticity. The model features a mixed Eulerian-Lagrangian formulation, as magnetizations are defined on the deformed set in the actual space. We consider…
We consider control constrained optimal control problems governed by parameterized stationary Maxwell's system with the Gauss's law. The parameters enter through dielectric, magnetic permeability, and charge density. Moreover, the parameter…
We establish an energy quantization for constrained Willmore surfaces, where the constraints are given by area, volume, and total mean curvature, assuming that the underlying conformal structures remain bounded. Furthermore, we show strong…
We show that in the setting of proper metric spaces one obtains a solution of the classical two-dimensional Plateau problem by minimizing the energy, as in the classical case, once a definition of area (in the sense of convex geometry) has…
We consider thin plates whose energy density is a quadratic function of the difference between the second fundamental form of the deformed configuration and a "natural" curvature tensor. This tensor either denotes the second fundamental…