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

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