Optimal boundary holes for the Sobolev trace constant

In this paper we study the problem of minimizing the Sobolev trace Rayleigh quotient $\|u\|_{W^{1,p}(\Omega)}^p / \|u\|_{L^q(\partial\Omega)}^p$ among functions that vanish in a set contained on the boundary $\partial\Omega$ of given boundary measure. We prove existence of extremals for this problem, and analyze some particular cases where information about the location of the optimal boundary set can be given. Moreover, we further study the shape derivative of the Sobolev trace constant under regular perturbations of the boundary set.Comment: 22 page

The Elusive Part of the Standard-Model Extension Gravitational Sector

In the minimal gravitational sector of the Standard-Model Extension, there is a coefficient whose physical consequences are unknown, and the reason behind this lack of effects is still puzzling. This contribution summarizes several studies where the goal was to find a fundamental explanation of this puzzle. So far, no evidence of such a fundamental explanation has been found, suggesting that this coefficient could actually produce physical effects. Nevertheless, while looking for this fundamental reason, several relevant lessons have been revealed.Comment: Presented at the Seventh Meeting on CPT and Lorentz Symmetry, Bloomington, Indiana, June 20-24, 201

Estimates for the Sobolev trace constant with critical exponent and applications

In this paper we find estimates for the optimal constant in the critical Sobolev trace inequality S\|u\|^p_{L^{p_*}(\partial\Omega) \hookrightarrow \|u\|^p_{W^{1,p}(\Omega)} that are independent of $\Omega$. This estimates generalized those of [3] for general $p$. Here $p_* := p(N-1)/(N-p)$ is the critical exponent for the immersion and $N$ is the space dimension. Then we apply our results first to prove existence of positive solutions to a nonlinear elliptic problem with a nonlinear boundary condition with critical growth on the boundary, generalizing the results of [16]. Finally, we study an optimal design problem with critical exponent.Comment: 22 pages, submitte

Quantum particles and an effective spacetime geometry

Spacetime geometry is supposed to be measured by identifying the trajectories of free test particles with geodesics. In practice, this cannot be done because, being described by Quantum Mechanics, particles do not follow trajectories. As a first step to study how it is possible to read spacetime geometry with quantum particles, we model these particles with classical extended objects. We propose to represent such extended objects by its covariant center of mass, which generically does not follow a geodesic of the background metric. We present a scheme that allows to extract some of components of an "effective" connection, namely, the connection that would be obtained if the locus of the center of mass is regarded as a geodesic. We discuss some issues that arise when trying to obtain all the components of the effective connection and its possible implications.Comment: Presented at the IX Workshop of the Mexican Gravity Divisio