On the existence of extremals for the critical Sobolev immersion with variable exponents

In this work we review some recent results conserning the existence problem of an extremal for the immersion W 1,p(x) 0 (Ω),→ L q(x) (Ω) in the critical range, i.e. A = {x ∈ Ω: q(x) = p ∗ (x)} 6= /0, where p ∗ (x) = N p(x)/(N − p(x)) is the critical Sobolev exponent.Fil: Fernandez Bonder, Julian. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "luis A. Santaló"; Argentin

On the Sobolev trace Theorem for variable exponent spaces in the critical range

In this paper we study the Sobolev Trace Theorem for variable exponent spaces with critical exponents. We find conditions on the best constant in order to guaranty the existence of extremals. Then we give local conditions on the exponents and on the domain (in the spirit of Adimurthy and Yadava) in order to satisfy such conditions, and therefore to ensure the existence of extremals.Comment: 21 pages, submitte

On the first eigenvalue of the generalized laplacian

In this work we investigate the energy of minimizers of Rayleigh-type quotients of the form $$\frac{\int_\Omega A(|\nabla u|)\, dx}{\int_\Omega A(|u|)\, dx}.$$ These minimizers are eigenfunctions of the generalized laplacian defined as $\Delta_a u = \text{div}\left(a(|\nabla u|)\frac{\nabla u}{|\nabla u|}\right)$ where $a(t)=A'(t)$ and the Rayleigh quotient is comparable to the associated eigenvalue. On the function $A$ we only assume that it is a Young function but no $\Delta_2$ condition is imposed. Since the problem is not homogeneous, the energy of minimizers is known to strongly depend on the normalization parameter $\alpha =\int_\Omega A(|u|)\, dx$. In this work we precisely analyze this dependence and show differentiability of the energy with respect to $\alpha$ and, moreover, the limits as $\alpha\to 0$ and $\alpha\to \infty$ of the Rayleigh quotient. The nonlocal version of this problem is also analyzed.Comment: 22 pages. Submitte

Quasilinear eigenvalues

In this work, we review and extend some well known results for the eigenvalues of the Dirichlet $p-$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