On Consistence of Material Coupling in a GL(3,R) Gauge Formulation of Gravity

A covariant scheme for material coupling with $GL(N,R)$ gauge formulation of gravity is studied. We revisit a known idea of a Yang-Mills type construction, where quadratical power of cosmological constant have to be considered in consistence with vacuum Einstein's gravity. Then, matter coupling with gravity is introduced and some constraints on fields and background appear. Finally, exploring the N=3 case we elucidate that introduction of auxiliary fields decreases the number of these constraints.Comment: 9 pages, LaTeX, title, and abstract changed. To appear in Mod.Phys.Lett.A, Vol. 18, No. 25 (2003) pp. 1753-176

Multiple nonradial solutions for a nonlinear elliptic problem with singular and decaying radial potential

Many existence and nonexistence results are known for nonnegative radial solutions $u\in D^{1,2}(\mathbb{R}^{N})\cap L^{2}(\mathbb{R}^{N},\left|x\right| ^{-\alpha }dx)$ to the equation $$-\triangle u+\dfrac{A}{\left| x\right| ^{\alpha }}u=f\left( u\right) \quad \textrm{in }\mathbb{R}^{N},\quad N\geq 3,\quad A,\alpha >0,$$ with nonlinearites satisfying $\left| f\left( u\right) \right| \leq \left(\mathrm{const.}\right) u^{p-1}$ for some $p>2$. Existence of nonradial solutions, by contrast, is known only for $N\geq 4$, $\alpha =2$, $f\left( u\right) =u^{(N+2)/(N-2)}$ and $A$ large enough. Here we show that the equation has multiple nonradial solutions as $A\rightarrow +\infty$ for $N\geq 4$, $2/(N-1)<\alpha <2N-2$, $\alpha\neq 2$, and nonlinearities satisfying suitable assumptions. Our argument essentially relies on the compact embeddings between some suitable functional spaces of symmetric functions, which yields the existence of nonnegative solutions of mountain-pass type, and the separation of the corresponding mountain-pass levels from the energy levels associated to radial solutions

An introduction to the study of critical points of solutions of elliptic and parabolic equations

We give a survey at an introductory level of old and recent results in the study of critical points of solutions of elliptic and parabolic partial differential equations. To keep the presentation simple, we mainly consider four exemplary boundary value problems: the Dirichlet problem for the Laplace's equation; the torsional creep problem; the case of Dirichlet eigenfunctions for the Laplace's equation; the initial-boundary value problem for the heat equation. We shall mostly address three issues: the estimation of the local size of the critical set; the dependence of the number of critical points on the boundary values and the geometry of the domain; the location of critical points in the domain.Comment: 34 pages, 13 figures; a few slight changes and some references added; to appear in the special issue, in honor of G. Alessandrini's 60th birthday, of the Rendiconti dell'Istituto Matematico dell'Universit\`a di Triest

Consistence of a GL(3,R) gauge formulation for topological massive gravity

We include a Chern-Simons term in a GL(3,R) gauge formulation of gravity with a cosmological contribution in 2+1 dimension and we explore consistence showing that excitations must be causal and standard topological massive gravity is recovered from this type of construction at the torsionless limit.Comment: Talk given at the Spanish Relativity Meeting 2007, Relativistic Astrophysics and Cosmology, Puerto de la Cruz, Tenerife, Spai