A remark on the radial minimizer of the Ginzburg-Landau functional

Denote by $E_\epsilon$ the Ginzburg-Landau functional in the plane and let $\tilde u_\varepsilon$ be the radial solution to the Euler equation associated to the problem $\min \left\{E_\varepsilon(u,B_1): \>\left. u\right\vert _{\partial B_{1}}=(\cos \vartheta,\sin \vartheta)\right\}$. Let $\Omega\subset \R^2$ be a smooth, bounded domain with the same area as $B_1$. Denoted by $$\mathcal{K}=\left\{v=(v_1,v_2) \in H^1(\Omega;\R^2):\> \int_\Omega v_1\,dx=\int_\Omega v_2\,dx=0,\> \int_\Omega |v|^2\,dx\ge \int_{B_1} |\tilde u_\varepsilon|^2\,dx\right\},$$ we prove $$ \min_{v \in \mathcal{K}} E_\varepsilon (v,\Omega)\le E_\varepsilon (\tilde u_\varepsilon,B_1). $

Some sharp Hardy inequalities on spherically symmetric domains

We prove some sharp Hardy inequalities for domains with a spherical symmetry. In particular, we prove an inequality for domains of the unit $n$-dimensional sphere with a point singularity, and an inequality for functions defined on the half-space $\R_+^{n+1}$} vanishing on the hyperplane $\{x_{n+1}=0\}$, with singularity along the $x_{n+1}$-axis. The proofs rely on a one-dimensional Hardy inequality involving a weight function related to the volume element on the sphere, as well as on symmetrization arguments. The one-dimensional inequality is derived in a general form.Comment: 15 page

Weighted isoperimetric inequalities in cones and applications

This paper deals with weighted isoperimetric inequalities relative to cones of $\mathbb{R}^{N}$. We study the structure of measures that admit as isoperimetric sets the intersection of a cone with balls centered at the vertex of the cone. For instance, in case that the cone is the half-space $\mathbb{R}_{+}^{N}={x \in \mathbb{R}^{N} : x_{N}>0}$ and the measure is factorized, we prove that this phenomenon occurs if and only if the measure has the form $d\mu=ax_{N}^{k}\exp(c|x|^{2})dx$, for some $a>0$, $k,c\geq 0$. Our results are then used to obtain isoperimetric estimates for Neumann eigenvalues of a weighted Laplace-Beltrami operator on the sphere, sharp Hardy-type inequalities for functions defined in a quarter space and, finally, via symmetrization arguments, a comparison result for a class of degenerate PDE's

Existence of minimizers for eigenvalues of the Dirichlet-Laplacian with a drift

This paper deals with the eigenvalue problem for the operator $L=-\Delta -x\cdot \nabla$ with Dirichlet boundary conditions. We are interested in proving the existence of a set minimizing any eigenvalue $\lambda_k$ of $L$ under a suitable measure constraint suggested by the structure of the operator. More precisely we prove that for any $c>0$ and $k\in \mathbb{N}$ the following minimization problem \min\left\{\lambda_k(\Omega): \> \Omega \>\mbox{quasi-open} \>\mbox{set}, \> \int_\Omega e^{|x|^2/2}dx\le c\right\} has a solution

Tuning the relaxation dynamics of ultracold atoms with an optical cavity

We investigate the out-of-equilibrium dynamics of ultracold atoms trapped in an optical lattice and loaded into an optical resonator that is driven transversely. We derive an effective quantum master equation for weak atom-light coupling that can be brought into Lindblad form both in the bad and good cavity limits. In the so-called bad cavity regime, we find that the steady state is always that of infinite temperature, but that the relaxation dynamics can be highly non-trivial. For small hopping, the interplay between dissipation and strong interactions generally leads to anomalous diffusion in the space of atomic configurations. However, for a fine-tuned ratio of cavity-mediated and on-site interactions, we discover a limit featuring normal diffusion. In contrast, for large hopping and vanishing on-site interactions, the system can be described by a linear rate equation leading to an exponential approach of the infinite-temperature steady state. Finally, in the good cavity regime, we show that for vanishing on-site interactions, the system allows for optical pumping between momentum mode pairs enabling cavity cooling