Asymptotics of eigenstates of elliptic problems with mixed boundary data on domains tending to infinity

We analyze the asymptotic behavior of eigenvalues and eigenfunctions of an elliptic operator with mixed boundary conditions on cylindrical domains when the length of the cylinder goes to infinity. We identify the correct limiting problem and show in particular, that in general the limiting behavior is very different from the one for the Dirichlet boundary conditions.Comment: Asymptotic Analysis, 201

On global minimizers of repulsive-attractive power-law interaction energies

We consider the minimisation of power-law repulsive-attractive interaction energies which occur in many biological and physical situations. We show existence of global minimizers in the discrete setting and get bounds for their supports independently of the number of Dirac Deltas in certain range of exponents. These global discrete minimizers correspond to the stable spatial profiles of flock patterns in swarming models. Global minimizers of the continuum problem are obtained by compactness. We also illustrate our results through numerical simulations.Comment: 14 pages, 2 figure

Correctors for some asymptotic problems

In the theory of anisotropic singular perturbation boundary value problems, the solution u ɛ does not converge, in the H 1-norm on the whole domain, towards some u 0. In this paper we construct correctors to have good approximations of u ɛ in the H 1-norm on the whole domain. Since the anisotropic singular perturbation problems can be connected to the study of the asymptotic behaviour of problems defined in cylindrical domains becoming unbounded in some directions, we transpose our results for such problem

Boundary layer solutions to functional elliptic equations

The goal of this paper is to study a class of nonlinear functional elliptic equations using very simple comparison principles. We first construct a nontrivial solution and then study its asymptotic behaviour when the diffusion coefficient goes to

Correctors for some asymptotic problems

In the theory of anisotropic singular perturbation boundary value problems, the solution u ɛ does not converge, in the H 1-norm on the whole domain, towards some u 0. In this paper we construct correctors to have good approximations of u ɛ in the H 1-norm on the whole domain. Since the anisotropic singular perturbation problems can be connected to the study of the asymptotic behaviour of problems defined in cylindrical domains becoming unbounded in some directions, we transpose our results for such problems

A magneto-viscoelasticity problem with a singular memory kernel

The existence of solutions to a one-dimensional problem arising in magneto-viscoelasticity is here considered. Specifically, a non-linear system of integro-differential equations is analyzed, it is obtained coupling an integro-differential equation modeling the viscoelastic behaviour, in which the kernel represents the relaxation function, with the non-linear partial differential equations modeling the presence of a magnetic field. The case under investigation generalizes a previous study since the relaxation function is allowed to be unbounded at the origin, provided it belongs to $L^1$; the magnetic model equation adopted, as in the previous results [21,22, 24, 25] is the penalized Ginzburg-Landau magnetic evolution equation.Comment: original research articl