Seminario di Analisi - a.a. 2007/2008

Abstract: Taking as a model the p(x)-Laplace equation, we extend the theory of entropy solutions to elliptic equations with nonlinearities involving variable exponents. We obtain existence and uniqueness for L^1 data, as well as integrability results for the solution and its gradient. The obstacle problem is also studied: we prove some convergence and stability properties of the coincidence set, extending the Lewy-Stampacchia inequalities to the general framework of L^1 data. This is a topic where PDEs again meet Functional Analysis in a truly two-way street and we make a brief tour of the properties of functional spaces involving variable exponents. For Marcinkiewicz spaces, we present new inclusions of independent interest.

Concentrazione di positività e disuguaglianza di Harnack

Un modello variazionale per mezzi porosi in Meccanica delle Fratture

Sunto. In alcuni recenti modelli matematici della visione umana, compaiono equazioni alle derivate parziali di tipo Kolomogorov-Fokker-Planck. I principali prototipi degli operatori coinvolti rientrano nella generale classe di operatori di tipo Hormander. Mostriamo che essi risultano invarianti per traslazioni a sinistra su opportuni gruppi di Lie non nilpotenti, e che dispongono di una soluzione fondamentale globale, non negativa e col supporto contenuto in un semispazio. Queste proprieta' consentono lo studio degli operatori con metodi di Analisi Reale e di Analisi Armonica sui gruppi di Lie.

A short introduction to geometric tomography will describe how X-rays relate to support functions. After this, the talk focuses on algorithms for reconstructing an approximation to a convex body from finitely many noisy measurements of its support function. The first algorithm for this purpose was proposed by Prince and Willsky in 1990, but only implemented in the plane. A new algorithm developed in joint work with Markus Kiderlen will be described. This algorithm, based on a least squares procedure, is very easy to program in standard software such as Matlab and allows, for the first time, good three-dimensional reconstructions to be performed on an ordinary PC. Under mild conditions, theory guarantees that outputs of the algorithm will converge to the input shape as the number of measurements increases. Reconstructions may be obtained without any pre- or post-processing steps and with no restriction on the sets of measurement directions except their number, a limitation dictated only by computing time. In addition we describe a linear program version of the new algorithm that is much faster and better, or at least comparable, in performance at low levels of noise and reasonably small numbers of measurements.

Mathematical discovery often involves the interplay - back and forth - between rough intuition and more precise mathematics. Unfortunately, only the smoothed-out (and perhaps over-generalized) mathematical version gets published, and the original rough intuitive model which led to the discovery goes forever unexplained and unpublished. In 1972, while on sabbatical in Firenze, I constructed an example of backwards nonuniqueness for a uniformly parabolic equation in self-adjoint divergence form with space and time varying conductivity coefficients. The solution u(x,y,t) begins proportional to cos(x)exp(-1t) and then, after an intricate 7-phase dance of varying the conductivity matrix, becomes later proportional to cos(2x)exp(-4t). In later steps one makes the solution oscillate faster and faster in space and thus decay faster and faster in time until it vanishes in finite time and continues as zero thereafter. But the crucial discovery, that one can vary the conductivity so as to make a solution go from a cos(x) form to a cos(2y) form, did not first occur to me in its published mathematically precise 7-phase form. Instead it occurred as a rough physically motivated construction, never published, which I would like to now explain.