215,506 research outputs found
Regularity of stable solutions up to dimension 7 in domains of double revolution
We consider the class of semi-stable positive solutions to semilinear
equations in a bounded domain of
double revolution, that is, a domain invariant under rotations of the first
variables and of the last variables. We assume . When
the domain is convex, we establish a priori and bounds for each
dimension , with when . These estimates lead to the
boundedness of the extremal solution of in every
convex domain of double revolution when . The boundedness of extremal
solutions is known when for any domain , in dimension
when the domain is convex, and in dimensions in the radial case.
Except for the radial case, our result is the first partial answer valid for
all nonlinearities in dimensions
Sobolev and isoperimetric inequalities with monomial weights
We consider the monomial weight in ,
where is a real number for each , and establish Sobolev,
isoperimetric, Morrey, and Trudinger inequalities involving this weight. They
are the analogue of the classical ones with the Lebesgue measure replaced
by , and they contain the best or critical
exponent (which depends on , ..., ). More importantly, for the
Sobolev and isoperimetric inequalities, we obtain the best constant and
extremal functions.
When are nonnegative \textit{integers}, these inequalities are exactly
the classical ones in the Euclidean space (with no weight) when
written for axially symmetric functions and domains in .Comment: The proof of Theorem 1.6 in the previous version of this paper was
not correct. Indeed, Lemma 5.1 in that version was not true as stated
therein. We thank Georgios Psaradakis for pointing this to u
Experimental Validation of Simplified Free Jet Turbulence Models Applied to the Vocal Tract
Sound production due to turbulence is widely shown to be an important
phenomenon involved in a.o. fricatives, singing, whispering and speech
pathologies. In spite of its relevance turbulent flow is not considered in
classical physical speech production models mostly dealing with voiced sound
production. The current study presents preliminary results of an experimental
validation of simplified turbulence models in order to estimate the time-mean
velocity distribution in a free jet downstream of a tube outlet. Aiming a
future application in speech production the influence of typical vocal tract
shape parameters on the velocity distribution is experimentally and
theoretically explored: the tube shape, length and the degree and geometry of
the constriction. Simplified theoretical predictions are obtained by applying
similarity solutions of the bidimensional boundary layer theory to a plane and
circular free jet in still air. The orifice velocity and shape are the main
model input quantities. Results are discussed with respect to the upper airways
and human sound production.Comment: 6 pages; 19th International Congress on Acoustics, Madrid : Espagne
(2007
Angular momentum-induced circular dichroism in non-chiral nanostructures
Circular dichroism (CD), i.e. the differential response of a system to left
and right circularly polarized light, is one of the only techniques capable of
providing morphological information of certain samples. In biology, for
instance, CD spectroscopy is widely used to study the structure of proteins.
More recently, it has also been used to characterize metamaterials and
plasmonic structures. Typically, CD can only be observed in chiral objects.
Here, we present experimental results showing that a non-chiral sample such as
a sub-wavelength circular nano-aperture can produce giant CD when a vortex beam
is used to excite it. These measurements can be understood by studying the
symmetries of the sample and the total angular momentum that vortex beams
carry. Our results show that CD can provide a wealth of information about the
sample when combined with the control of the total angular momentum of the
input field
Sharp isoperimetric inequalities via the ABP
Given an arbitrary convex cone of Rn, we find a geometric class of homogeneous weights for which balls centered at the origin and intersected with the cone are minimizers of the weighted isoperimetric problem in the convex cone. This leads to isoperimetric inequalities with the optimal constant that were unknown even for a sector of the plane. Our result applies to all nonnegative homogeneous weights in Rn satisfying a concavity condition in the cone. The condition is equivalent to a natural curvature-dimension bound and also to the nonnegativeness of a Bakry-Emery Ricci tensor. Even that our weights are nonradial, still balls are minimizers of the weighted isoperimetric problem. A particular important case is that of monomial weights. Our proof uses the ABP method applied to an appropriate linear Neumann problem. We also study the anisotropic isoperimetric problem in convex cones for the same class of weights. We prove that the Wulff shape (intersected with the cone) minimizes the anisotropic weighted perimeter under the weighted volume constraint. As a particular case of our results, we give new proofs of two classical results: the Wulff inequality and the isoperimetric inequality in convex cones of Lions and PacellaPeer ReviewedPostprint (published version
Equivalence principle and the gauge hierarchy problem
We show that the gauge hierarchy problem can be solved in the framework of scalar-tensor theories of gravity very much in the same way as it is solved in the Randall-Sundrum scenario. Our solution involves a fine-tuning of the gravitational sector which can, however, be avoided if a supergravity extension of the dilaton sector is considered. However our mechanism does not require the introduction of extra dimensions or new physics strongly coupled to the standard model in the low energy regime. We do introduce a new scalar field which is, however, coupled only gravitationally to regular matter. The physical reason for the splitting between the weak scale and the Planck scale is a violation of Einstein's equivalence principle
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