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 $-\Delta u=f(u)$ in a bounded domain $\Omega\subset\mathbb R^n$ of double revolution, that is, a domain invariant under rotations of the first $m$ variables and of the last $n-m$ variables. We assume $2\leq m\leq n-2$. When the domain is convex, we establish a priori $L^p$ and $H^1_0$ bounds for each dimension $n$, with $p=\infty$ when $n\leq7$. These estimates lead to the boundedness of the extremal solution of $-\Delta u=\lambda f(u)$ in every convex domain of double revolution when $n\leq7$. The boundedness of extremal solutions is known when $n\leq3$ for any domain $\Omega$, in dimension $n=4$ when the domain is convex, and in dimensions $5\leq n\leq9$ in the radial case. Except for the radial case, our result is the first partial answer valid for all nonlinearities $f$ in dimensions $5\leq n\leq 9$

Sobolev and isoperimetric inequalities with monomial weights

We consider the monomial weight $|x_1|^{A_1}...|x_n|^{A_n}$ in $\mathbb R^n$, where $A_i\geq0$ is a real number for each $i=1,...,n$, and establish Sobolev, isoperimetric, Morrey, and Trudinger inequalities involving this weight. They are the analogue of the classical ones with the Lebesgue measure $dx$ replaced by $|x_1|^{A_1}...|x_n|^{A_n}dx$, and they contain the best or critical exponent (which depends on $A_1$, ..., $A_n$). More importantly, for the Sobolev and isoperimetric inequalities, we obtain the best constant and extremal functions. When $A_i$ are nonnegative \textit{integers}, these inequalities are exactly the classical ones in the Euclidean space $\mathbb R^D$ (with no weight) when written for axially symmetric functions and domains in $\mathbb R^D=\mathbb R^{A_1+1}\times...\times\mathbb R^{A_n+1}$.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