Seminario di Analisi 2011/12

Abstract

Abstract

Abstract

Recently, several investigators have been studying, especially in ODE field, some nonlinear inverse and direct bifurcation problems in several frameworks. In this talk, we introduce several typical inverse and direct bifurcation problems from a view point of $L^q$-approach, variational approach, and recent results with respect to the inverse and direct bifurcation problems.

We consider a class W of Weingarten hypersurfaces in R^{N+1} with N>=2, which contains an example related to stationary isothermic hypersurfaces. Denote by C the class of continuous entire graphs x_{N+1} = f(x), x in R^N over R^N such that the oscillation of f restricted on each unit ball in R^N is bounded. Then, our main theorem is stated that, if S in C belongs to W in the viscosity sense, then S must be a hyperplane. This theorem gives a considerable improvement of the previous results of S. Sakaguchi "A Liouville-type theorem for some Weingarten hypersurfaces", Discrete and Continuous Dynamical Systems - Series S 4 (2011), 887--895.

L'esistenza di soluzioni per equazione di Monge-Amp&egravere omogenea complessa gioca un ruolo importante nello studio molti problemi della geometria complessa. In particolare l'esistenza di una soluzione di rango massimo dell'equazione di Monge-Amp&egravere omogenea complessa definisce sull'insieme di definizione una foliazione in superficie di Riemann caratterizzata dal fatto che i fogli sono esattamente le curve complesse lungo le quali la soluzione è una funzione armonica. Spiegato questo fenomeno e dato conto di alcune applicazioni geometriche, illustreremo come si trovano soluzioni regolari per mezzo della costruzione di tali foliazioni. Il seminario cercher&agrave di offrire una panoramica delle idee che sono coinvolte e eviter&agrave deliberatamente gli aspetti tecnici.

Recenti sviluppi sui metamateriali hanno permesso di creare dispositivi elettromagnetici in grado di nascondere oggetti da un'interrogazione elettromagnetica ad una data frequenza. In questo seminario faremo una panoramica dello stato dell'arte di questo problema e presenteremo alcuni risultati matematici ad esso collegati.

&Egrave noto che un insieme mean-convex rappresenta una barriera locale per superfici minime e inoltre, come segue dai lavori di Meeks e Yau, pu&ograve essere utilizzato come barriera locale per la costruzione delle superfici minime. Ci&ograve non di meno, in generale un insieme mean-convex pu&ograve non essere una barriere globale. In questo seminario mostriamo alcuni risultati sulla relazione tra insiemi mean-convex e la nozione di soprasoluzione per superfici minime, caratterizzando la pi&ugrave piccola barriera globale in termini di un flusso per curvature media con ostacolo.

I will recall the definition of the combinatorial Laplacian associated with a connected locally finite simplicial graph and prove that it is surjective whenever the graph is infinite.This is joint work with Tullio Ceccherini-Silberstein and Jozef Dodziuk, arXiv:1103.4901.

Abstract

Abstract

We consider the problem of minimizing the first non trivial Stekloff eigenvalue of the Laplacian, among sets with given measure. We prove that the Brock--Weinstock inequality, asserting that optimal shapes for this spectral optimization problem are balls, can be improved by means of a (sharp) quantitative stability estimate. This result is based on the analysis of a certain class of weighted isoperimetric inequalities already proved in Betta et al. (J. of Inequal. & Appl. 4: 215--240, 1999): we provide some new (sharp) quantitative versions of these, achieved by means of a suitable calibration technique.

A set K is said to be lattice convex if K is the intersection of the integer lattice with a convex set. With a finite, lattice convex set K one can associate the so called covariogram information, that is, the cardinaliy of K interesected with K+u, for each integer vector u. The problem of determination of K by the covariogram information of K (up to translations and reflections) was raised independently by Gardner, Gronchi, Zong and Daurat, Gerard, Nivat. In my talk I am going to report on partial results to this problem, which were obtained in a joint work with Barbara Langfeld (University of Kiel, Germany). On the one hand, in certain general situations, K is determined by the covariogram information. On the other hand, we were able to find an infinite family of examples for which the determination is not unique.

Abstract

Abstract: In this talk we will discuss about the maximum principles and their applications to the study of partial differential equations. More exactly, we will show how one may employ the maximum principles to get a priori estimates, isoperimetric inequalities, symmetry results, convexity results, the shape of some free boundaries and Liouville type theorems in the elliptic case or spatial and temporal asymptotic behaviour of solutions in the parabolic case.

Abstract

Abstract

Abstract

Abstract