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

Magnetic-Field-Induced Topological Reorganization of a P-wave Superconductor

In this work we illustrate the detrimental impact of the Cooper pair's spin-structure on the thermodynamic and topological properties of a spin-triplet superconductor in an applied Zeeman field. We particularly focus on the paradigmatic one-dimensional case (Kitaev chain) for which we self-consistently retrieve the energetically preferred Cooper pair spin-state in terms of the corresponding spin d-vector. The latter undergoes a substantial angular and amplitude reorganization upon the variation of the strength and the orientation of the field and results to a modification of the bulk topological phase diagram. Markedly, when addressing the open chain we find that the orientation of the d-vector varies spatially near the boundary, affecting in this manner the appearance of Majorana fermions at the edge or even altering the properties of the bulk region. Our analysis reveals the limitations and breakdown of the bulk-boundary correspondence in interacting topological systems.Comment: 5 pages, 3 panels of figures; Minor corrections in the new version, which will appear in Phys. Rev. B as a Rapid Communicatio

On isoperimetric inequalities with respect to infinite measures

We study isoperimetric problems with respect to infinite measures on $R ^n$. In the case of the measure $\mu$ defined by $d\mu = e^{c|x|^2} dx$, $c\geq 0$, we prove that, among all sets with given $\mu-$measure, the ball centered at the origin has the smallest (weighted) $\mu-$perimeter. Our results are then applied to obtain Polya-Szego-type inequalities, Sobolev embeddings theorems and a comparison result for elliptic boundary value problems.Comment: 25 page

Microwave Electrodynamics of the Antiferromagnetic Superconductor GdBa_2Cu_3O_{7-\delta}

The temperature dependence of the microwave surface impedance and conductivity are used to study the pairing symmetry and properties of cuprate superconductors. However, the superconducting properties can be hidden by the effects of paramagnetism and antiferromagnetic long-range order in the cuprates. To address this issue we have investigated the microwave electrodynamics of GdBa_2Cu_3O_{7-\delta}, a rare-earth cuprate superconductor which shows long-range ordered antiferromagnetism below T_N=2.2 K, the Neel temperature of the Gd ion subsystem. We measured the temperature dependence of the surface resistance and surface reactance of c-axis oriented epitaxial thin films at 10.4, 14.7 and 17.9 GHz with the parallel plate resonator technique down to 1.4 K. Both the resistance and the reactance data show an unusual upturn at low temperature and the resistance presents a strong peak around T_N mainly due to change in magnetic permeability.Comment: M2S-HTCS-VI Conference Paper, 2 pages, 2 eps figures, using Elsevier style espcrc2.st

Neumann problems for nonlinear elliptic equations with L1L^1 data

In the present paper we prove existence results for solutions to nonlinear elliptic Neumann problems whose prototype is \begin{equation*} \begin{cases} -\Delta_{p} u -\text{div} (c(x)|u|^{p-2}u)) =f & \text{in}\ \Omega, \\ \left( |\nabla u|^{p-2}\nabla u+ c(x)|u|^{p-2}u \right)\cdot\underline n=0 & \text{on}\ \partial \Omega \,, \end{cases} \end{equation*} when $f$ is just a summable function. Our approach allows also to deduce a stability result for renormalized solutions and an existence result for operator with a zero order term