Resumen del Proyecto

Liouville-type theorems for Pucci's operator. One remarkable result in elliptic partial differential equations is the Liouville-type theorem for the Laplace operator proved by Gidas and Spruck. This theorem stands at the heart of the theory of elliptic partial differential equations, with deep consequences in the existence theory, connecting critical Sobolev embeddings, a priori estimates and non-existence results. Since its proof in 1981, generalizations in various directions have been obtained, both for more general operators and systems. During the last five years, a series of advances on fully non-linear operators, Pucci's extremal operators, conformally invariant operators, degenerate p-laplacian like operators, etc have re-invigorated again the investigations in this area of research which is now very active. We plan to consider various questions related to Liouville type theorems and related existence results. One major problem has to do with the existence and characterization of the critical exponent for nonlinear autonomous extremal operators, in the general sense of Pucci, in the case of radial functions.

Fourth Order problems. During the past two years, we have started the study of variational principles associated to nonlinear fourth order equations. These equations arise in many problems in Mathematical Physics. It appears for instance in Lieb-Thirring inequalities for fourth order linear operators in the line and we have already produced a paper on this question. We would like to focus first on simple problems like Gagliardo-Nirenberg type inequalities involving second derivatives and understand some qualitative properties like the dependence of the best constants showing up in the inequalities in terms of the parameters of the problem, and the properties of the minimizers or of the minimizing sequences. We have for instance noted in some special cases that the minimizers are achieved only when periodic boundary conditions are imposed. On the whole line, minimizing sequences are oscillating and compactness is lost due to the oscillations, even after taking into account the symmetry properties of the variational problem (translation and scaling invariance). The goal is therefore to understand if these new properties are a generic feature of fourth order operators and to extend the examples we have obtained to more generic situations and, for instance, in higher dimensions.

Quantum drift-diffusion equations. This topic deserves a long justification which is partially given in the "scientific project" annexed to this proposal. The goal is to understand mathematically what are the corrections to standard models of drift-diffusion and fluid mechanics due to quantum mechanics, with applications to semi-conductor devices. It is located at the intersection of mathematics (mathematical physics) and modelization. We plan to obtain a systematic derivation of quantum drift-diffusion equations based on physical quantities (free energy) and elaborate mathematical tools (gradient flows with respect to Wasserstein's distance, new Lieb-Thirring inequalities which have recently been obtained). Next, we would like to justify diffusion limits which have been written by Degond and Ringhofer at a formal level. A last goal  would be to justify and understand some quantum corrections to classical equations which have been introduced on an empirical basis. 

Proyecto Fondecyt 1040794 NONLINEAR ELLIPTIC AND PARABOLIC EQUATIONS:     NON-DIVERGENCE FORM OPERATORS AND QUALITATIVE PROPERTIES.

Proyecto Fondecyt 1030929 NONLINEAR ELLIPTIC EQUATIONS: SINGULAR PERTURBATIONS AND OTHER TOPICS.

Proyecto Fondecyt 1020844 2002 A STUDY OF SOME NOLINEAR PARTIAL     

§DIFFERENTIAL EQUATIONS ARISING FROM MATHEMATICAL PHYSICS.