Tres nuevas clases de espacios localmente convexos

Local and nonlocal weighted pLaplacian evolution equations with Neumann boundary conditions

In this paper we study existence and uniqueness of solutions to the local diffusion equation with Neumann boundary conditions and a bounded nonhomogeneous diffusion coefficient g ≥ 0, {ut = div (g|∇u|p-2∇u) in ]0; T[×Ωg|∇u|p-2u·n = 0 on ]0; T[×∂Ω; for 1 ≤ p < ∞. We show that a nonlocal counterpart of this diffusion problem is ut(t; x)= ∫ω J(x-y)g(x+y/2)|u(t; y)-u(t; x)| p-2 (u(t; y)-u(t; x)) dy in ]0; T[× Ω,where the diffusion coefficient has been reinterpreted by means of the values of g at the point x+y/2 in the integral operator. The fact that g ≥ 0 is allowed to vanish in a set of positive measure involves subtle difficulties, specially in the case p = 1.

Existence and uniqueness of a solution for a parabolic quasilinear problem for linear growth functionals with $L^1$ data

We introduce a new concept of solution for the Dirichlet problem for quasilinear parabolic equations in divergent form for which the energy functional has linear growth. Using Kruzhkov's method of doubling variables both in space and time we prove uniqueness and a comparison principle in $L^1$ for these solutions. To prove the existence we use the nonlinear semigroup theory.

Ecuaciones en derivadas parciales gobernadas por operadores acretivos

Una teoria que ha resultado ser de gran utilidad en el estudio de muchas ecuaciones en derivadas parciales no lineales es la teoria de semigrupos no lineales generados por operadores acretivos en espacios de Banach. Dicha teoria se basa fundamentalmente en el Teorema de Crandall-Ligget y en las aportaciones de Ph. Benilan. En este articulo, despues de hacer una exposicion esquematica de esta teoria general, veremos como la hemos aplicado a algunas ecuaciones en derivadas parciales no lineales que aparecen en diversos campos de la Ciencia.

Partial differential equations governed by accretive operators

The theory of nonlinear semigroups in Banach spaces generated by accretive operators has been very useful in the study of many nonlinear partial differential equations Such a theory is fundamentally based in the Crandall-Liggett Theorem and in the contributions of Ph. Benilan. In this paper, after outlining some of the main points of this theory, we present some of the applications to some nonlinear partial differential equations that appear in different fields of Science.