717 research outputs found
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
An optimization problem for the first weighted eigenvalue problem plus a potential
In this paper, we study the problem of minimizing the first eigenvalue of the
Laplacian plus a potential with weights, when the potential and the weight
are allowed to vary in the class of rearrangements of a given fixed potential
and weight . Our results generalized those obtained in [9] and [5].Comment: 15 page
convergence result for nonlocal elliptic-type problems via Tartar's method
In this work we obtain a compactness result for the convergence of a
family of nonlocal and nonlinear monotone elliptic-type problems by means of
Tartar's method of oscillating test functions.Comment: In this revision we added a new section that shows the
Gamma-convergence of the associated energy functional
Quasilinear eigenvalues
In this work, we review and extend some well known results for the
eigenvalues of the Dirichlet Laplace operator to a more general class of
monotone quasilinear elliptic operators. As an application we obtain some
homogenization results for nonlinear eigenvalues.Comment: 23 pages, Rev. UMA, to appea
Energy nonconservation and relativistic trajectories: Unimodular gravity and beyond
Energy conservation has the status of a fundamental physical principle.
However, measurements in quantum mechanics do not comply with energy
conservation. Therefore, it is expected that a more fundamental theory of
gravity -- one that is less incompatible with quantum mechanics -- should admit
energy nonconservations. This paper begins by identifying the conditions for a
theory to have an energy-momentum tensor that is not conserved. Then, the
trajectory equation for pointlike particles that lose energy is derived,
showing that energy nonconservation produces a particular acceleration. As an
example, the unimodular theory of gravity is studied. Interestingly, in
spherical symmetry, given that there is a generalized Birkhoff theorem and that
the energy-momentum tensor divergence is a closed form, the trajectories of
test particles that lose energy can be found using well known methods. Finally,
limits on the energy nonconservation parameters are set using Solar system
observations.Comment: 13 pages. Accepted in Phys. Rev.
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