1,772 research outputs found
Optimal boundary holes for the Sobolev trace constant
In this paper we study the problem of minimizing the Sobolev trace Rayleigh
quotient among
functions that vanish in a set contained on the boundary 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 . This estimates
generalized those of [3] for general . Here is the
critical exponent for the immersion and 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
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