1,935 research outputs found
A direct proof of the Sharp Gårding inequality for symbols with limited smoothness
No full textWe give a proof of the (possibly optimal) Sharp Gårding inequality for system oper- ators with symbol of limited smoothness directly from the original symmetrization arguments by Friedrichs and Kumano-Go. The fact that only a few derivatives of the regularized symbol are really important was already there
Analisi e Geometria nello spazio delle fasi ed Equazioni alle Derivate Parziali
No full textLinee guida portanti ed unificanti della ricerca che si
intende portare avanti sono l’Analisi e la Geometria dello
Spazio delle Fasi in vari problemi e modelli riconducibili
alle Equazioni alle Derivate Parziali.
All'interno del progetto si intendono continuare varie
collaborazioni multilaterali tra i partecipanti e con
studiosi stranieri, con l'ulteriore apporto e
coinvolgimento di alcuni giovani ricercatori. Le tematiche
sono descritte qui di seguito in maggior dettaglio assieme
alle principali collaborazioni, ferme restando le sinergie
tra tutti i membri del progetto su tutti gli argomenti che
si intendono affrontare nelle linee comuni di indirizzo.
A) CONTINUAZIONE UNICA
F. Colombini (Pisa), C. Grammatico (Bologna), D. Tataru
(Berkeley).
Si intendono studiare problemi di "continuazione unica
forte" per equazioni e disequazioni scalari nonché per
sistemi ellittici, cercando anche controesempi che mostrino
l’ottimalità dei risultati.
B) OPERATORI IPERBOLICI E DI EVOLUZIONE
A. Ascanelli (Ferrara), M. Cicognani (Bologna), F. Colombini
(Pisa), D. Del Santo (Trieste), G. Métivier (Bordeaux).
In riferimento a operatori iperbolici, o piu’ in generale a
operatori di evoluzione con caratteristiche reali, ci si
propone di studiare le relazioni intercorrenti tra la
regolarità dei coefficienti, ovvero il comportamento
oscillante di questi, con l'esistenza, l'unicità e a
dipendenza continua dai dati delle soluzioni del problema di
Cauchy. Nel caso non kovalevskiano, ad esempio per operatori
di tipo Schrodinger, si vuole determinare anche la
decrescita ottimale dei coefficienti all’infinto rispetto
alle variabili spaziali.
Si intende inoltre proseguire una ricerca gia’ avviata sullo
Scattering per equazioni delle onde in cui sono
principalmente coinvolti F. Colombini (Pisa), V. Petkov
(Bordeaux) e J. Rauch (Michigan).
C) PROBLEMI DI RISOLUBILTA’ E IPOELLITTICITA’
F. Colombini (Pisa), P. Cordaro (San Paulo), M. Mughetti
(Bologna), A. Parmeggiani (Bologna), L. Pernazza (Pavia).
Si intendono affrontare i problemi di risolubilita' locale
per operatori non di tipo principale e studiare condizioni
geometriche necessarie e/o sufficienti per la risolubilita`
di sistemi NxN.
Si vuole continuare lo studio, in termini di geometria dello
spazio delle fasi, della validita` della stima di
Fefferman-Phong per sistemi e continuare lo studio dello
spettro di sistemi globalmente ellittici diequazioni
differenziali a coefficienti polinomiali e delle relative
funzioni zeta-spettrali e formule di Poisson in termini
della geometria del simbolo del sistema.
Si intende poi studiare l'ipoellitticità C^infty per una
classe di operatori (pseudo)differenziali la cui perdita di
regolarità sia maggiore di quella classicamente
caratterizzata; in tal senso, e’ necessario individuare
invarianti di ordine superiore da associare all'operatore
esaminato, in quanto quelli usuali non sono intrinsecamente
sufficienti;
D) PROPAGAZIONE DELLA REGOLARITA’/SINGOLARITA’
V. Murthy (Pisa).
Si intende studiare la regolarita’ delle soluzioni di
equazioni subellittiche associate a campi vettoriali che
soddisfano la condizione di Hormander al di fuori di
superfici non caratteristiche e che su queste superfici
possono degenerare di ordine infinito. Si tratta di imporre
condizioni di tipo microlocale su commutatori, sufficienti
per assicurare stime a priori di tipo logaritmico - Sobolev.
Si vuole poi studiare una classe di equazioni quasilineari
subellittiche associate a campi vettoriali che soddisfano la
condizione di Hormander e la propagazione della singolarita’
per sistemi di Dirac su campi spinoriali sulle
varieta’ globalmente iperboliche,
Infine, si vogliono analizzare i punti critici stabili per
il funzionale di Ginzburg - Landau con un vettore potenziale
magnetico su domini limitati in spazi di due e tre
dimensioni e la connessione con le proprieta’ topologiche e
le proprieta’ geomet..
On the nonlinear Cauchy problem
No full textThe paper deals with the Cauchy problem for quasilinear higher-order hyperbolic equations. For different cases, using a unified procedure, local existence and uniqueness of solutions as well as propagation-of-regularity results are obtained. In particular, in the case when the characteristic equation has only real roots, each having constant multiplicity (call this case CM), the well-posedness of the Cauchy problem in and in Sobolev-Gevrey spaces is proved. The authors also consider the case when the equation is strictly hyperbolic and the coefficients of the principal part are Log-Lipschitz in the variable or only in , obey the estimate , , and are smooth in the other variables, as well as the case (CM) under the assumption that the coefficients , are only Hölder continuous with exponent less than 1 in the variable and smooth in the other variables
Strictly hyperbolic equations with coefficients low-regular in time and smooth in space
No full textWe consider the Cauchy problem for strictly hyperbolic m-th order partial
differential equations with coefficients low-regular in time and smooth in space. It
is well-known that the problem is L2 well-posed in the case of Lipschitz continuous
coefficients in time, Hs well-posed in the case of Log-Lipschitz continuous coefficients
in time (with an, in general, finite loss of derivatives) and Gevrey well-posed
in the case of Hölder continuous coefficients in time (with an, in general, infinite loss
of derivatives). Here, we use moduli of continuity to describe the regularity of the
coefficients with respect to time, weight sequences for the characterization of their
regularity with respect to space and weight functions to define the solution spaces.We
establish sufficient conditions for the well-posedness of the Cauchy problem, that link
the modulus of continuity and the weight sequence of the coefficients to the weight
function of the solution space. The well-known results for Lipschitz, Log-Lipschitz
and Hölder coefficients are recovered
Operators of p-evolution with nonregular coefficients in the time variable
No full textAbstractWe study the Cauchy problem for a class of p-evolution operators P(t,x,Dt,Dx) in [0,T]×Rn,p>1, with less than C1 coefficients with respect to the time variable.According to Lipschitz, log-lipschitz or Hölder regularity we find well-posedness in Sobolev spaces or in Gevrey classes
Energy estimate and fundamental solution for degenerate hyperbolic Cauchy problems
No full textAbstractThe aim of this paper is to give an uniform approach to different kinds of degenerate hyperbolic Cauchy problems. We prove that a weakly hyperbolic equation, satisfying an intermediate condition between effective hyperbolicity and the C∞ Levi condition, and a strictly hyperbolic equation with non-regular coefficients with respect to the time variable can be reduced to first-order systems of the same type. For such a kind of systems, we prove an energy estimate in Sobolev spaces (with a loss of derivatives) which gives the well-posedness of the Cauchy problem in C∞. In the strictly hyperbolic case, we also construct the fundamental solution and we describe the propagation of the space singularities of the solution which is influenced by the non-regularity of the coefficients with respect to the time variable
Coefficients with unbounded derivatives in hyperbolic equations
No full textWe are concerned with the problem of determining the sharp regularity of the coefficients with respect to the time variable in order to have a well posed Cauchy problem in or in Gevrey classes for linear or quasilinear hyperbolic operators of higher order.
We use and mix two different scales of regularity of global and local type: the modulus of H"older continuity and/or the behaviour with respect to of the first derivative as tends to a point . Both are ways to weaken the Lipschitz regularity
Modulus of continuity of the coefficients and loss of derivatives in the strictly hyperbolic Cauchy problem
No full textAbstractWe deal with the Cauchy problem for a strictly hyperbolic second-order operator with non-regular coefficients in the time variable. It is well-known that the problem is well-posed in L2 in case of Lipschitz continuous coefficients and that the log-Lipschitz continuity is the natural threshold for the well-posedness in Sobolev spaces which, in this case, holds with a loss of derivatives. Here, we prove that any intermediate modulus of continuity between the Lipschitz and the log-Lipschitz one leads to an energy estimate with arbitrary small loss of derivatives. We also provide counterexamples to show that the following classification:modulusofcontinuity→lossofderivativesis sharpLipschitz→noloss,intermediate→arbitrarysmallloss,log-Lipschitz→finiteloss
The cauchy problem for nonlinear hyperbolic equations with Levi condition
No full textAbstractThe Cauchy problem for weakly hyperbolic equations is generally not C∞ well posed without assuming conditions on lower order terms: this is well known since the famous E.E. Levi paper [7], generalized many years later by several authors.Here we want to study the same problem in nonlinear framework, hence it is natural to impose “Levi conditions” on the linearized operator. We shall confine ourselves to consider equations with constant multiplicity for which Levi conditions are plain (see for example J. Chazarian [2], H. Flascka and G. Strang [3], S. Mizohata and Y. Ohya [9], J. Vaillant [12]) and several applications to Mathematical Physics are possible.As far as we know, the only result of this type is proved by D. Gourdin [4], where he treats, with different methods, a class of equations having small intersection with the one we consider here
Schrödinger equations in modulation spaces
No full textWe consider the linear propagator for Schrödinger equations with variable coefficients in R^d. We show that it is bounded on some spaces arising in Time- frequency Analysis, known as modulation spaces. This generalizes recent results where the case of constant coefficients was considered
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