148 research outputs found
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Symmetry and rigidity results for composite membranes and plates
Full text linkThe composite membrane problem is an eigenvalue optimization problem deeply studied from the beginning of the '00's. In this note we survey most of the results proved by several authors over the last twenty years, up to the recent paper [14] written in collaboration with Giovanni Cupini.We finally introduce an eigenvalue optimization problem for a fourth order operator, called composite plate problem and we present the symmetry and rigidity results obtained in this framework. These last mentioned results are part of the papers [12,13], written in collaboration with Francesca Colasuonno.Il problema della membrana composita è un problema di ottimizzazione di autovalori i cui primi contributi risalgono agli inizi degli anni '00. In questa nota presentiamo una sintesi dei principali risultati ottenuti negli ultimi venti anni, fino al recente contributo [14] scritto in collaborazione con Giovanni Cupini.Introdurremo poi un problema di ottimizzazione di autovalori per un operatore del quart'ordine noto come problema della piastra composita, e presenteremo alcuni risultati di simmetria e rigidità in questo ambito. Questi ultimi risultati sono contenuti nei lavori [12,13] scritti in collaborazione con Francesca Colasuonno
Enhanced boundary regularity of planar nonlocal minimal graphs and a butterfly effect
Full text linkIn this note, we showcase some recent results obtained in [DSV19] concerning the stickiness properties of nonlocal minimal graphs in the plane. To start with, the nonlocal minimal graphs in the planeenjoy an enhanced boundary regularity, since boundary continuity with respect to the external datum is sufficient to ensure differentiability across the boundary of the domain. As a matter of fact, the Hoelder exponent of the derivative is in this situation sufficiently high to provide the validity of the Euler-Lagrange equation at boundary points as well. From this, using a sliding method, one also deduces that the stickiness phenomenon is generic for nonlocal minimal graphs in the plane, since an arbitrarily small perturbation of continuous nonlocal minimal graphs can produce boundary discontinuities (making the continuous case somehow ``exceptional'' in this framework.In questa nota, presentiamo alcuni risultati recenti ottenuti in [DSV19] relativi alla proprietà di ``appiccicosità'' dei grafici minimi nonlocali nel piano. I grafici minimi non locali nel piano godono di una regolarità ``accresciuta'' al bordo, in quanto la continuità al bordo rispetto al dato esterno è sufficiente a garantire la differenziabilità attraverso il bordo del dominio. Inoltre, l'esponente di Hoelder della derivata è sufficientemente grande da garantire la validità dell'equazione di Eulero-Lagrange anche ai punti di bordo del dominio. Da ciò, usando un metodo di scivolamento, si ottiene anche cheil fenomeno di appiccicosità è generico per grafici minimi non locali nel piano, nel senso che una perturbazione arbitrariamente piccola di i grafici minimi nonlocali continui produce discontinuità al bordo (rendendo quindi il caso continuo in qualche modo ``eccezionale'')
Some global Sobolev inequalities related to Kolmogorov-type operators
Full text linkIn this note we review a recent result in [17] in collaboration with N. Garofalo, where we establish global versions of Hardy-Littlewood-Sobolev inequalities attached to hypoelliptic equations of Kolmogorov type. The relevant Sobolev spaces are defined through the fractional powers of the operator under consideration. We outline the main steps of the semigroup approach we adopt.Viene qui presentato un recente risultato ottenuto in [17] in collaborazione con N. Garofalo, in cui si dimostrano disuguaglianze globali di tipo Hardy-Littlewood-Sobolev relative ad una classe di operatori ipoellittici di tipo Kolmogorov. Nell'approccio adottato gli spazi di Sobolev sono definiti attraverso le potenze frazionarie dell'operatore in questione
The fractional mean curvature flow
Full text linkIn this note, we present some recent results in the study of the fractional mean curvature flow, that is a geometric evolution of the boundary of a set whose speed is given by the fractional mean curvature. The flow under consideration is of nonlocal type and presents several interesting difference with respect to the classical mean curvature flow. We will describe the main contributions in this field, with particular emphasis on some tipically nonlocal behaviors which are in contrast with the classical local case.In questa nota, presentiamo alcuni risultati recenti riguardanti lo studio del moto per curvatura media frazionaria, che descrive l'evoluzione del bordo di un insieme la cui velocita è data dalla curvatura media frazionaria. Tale flusso ha natura nonlocale e presenta alcune interessanti differenze rispetto al flusso per curvatura media classica. Descriviamo i principali contributi in questo ambito, con particolare enfasi ai comportamente tipicamente nonlocali che sono in contrasto col caso classico
Regularity for quasilinear PDEs in Carnot groups via Riemannian approximation
Full text link We study the interior regularity of weak solutions to subelliptic quasilinear PDEs in Carnot groups of the formΣi=1m1Xi (Φ(|∇Hu|2)Xiu) = 0.Here ∇Hu = (X1u,...,Xmiu) is the horizontal gradient, δ > 0 and the exponent p ∈ [2, p*), where p* depends on the step ν and the homogeneous dimension Q of the group, and it is given byp* = min {2ν ∕ ν-1 , 2Q+8 ∕ Q-2}.Sunto. Studiamo la regolarità interna delle soluzioni deboli di EDP, quasilineari subellittiche in gruppi di Carnot, della formaΣi=1m1Xi (Φ(|∇Hu|2)Xiu) = 0.Qui ∇Hu = (X1u,...,Xmiu) è il gradiente orizzontale, δ > 0 e l'esponente p ∈ [2, p*), dove p* dipende dal passo ν e dalla dimensione omogenea Q del gruppo ed è dato dap* = min {2ν ∕ ν-1 , 2Q+8 ∕ Q-2}
Radial solutions of Lane-Emden-Fowler equations with Pucci's extremal operators
Full text linkWe report on some recent results obtained for positive radial solutions of Lane-Emden-Fowler type equations with Pucci's operators as principal parts. The presented results include the asymptotic analysis of almost critical solutions in the unit ball, existence results in annular domains and sharp Liouville-type results for exterior Dirichlet problems.
Large sets at infinity and Maximum Principle on unbounded domains for a class of sub-elliptic operators
Full text linkMaximum Principles on unbounded domains play a crucial role in several problems related to linear second-order PDEs of elliptic and parabolic type. In the present notes, based on a joint work with prof. E. Lanconelli, we consider a class of sub-elliptic operators L in R^N and we establish some criteria for an unbounded open set to be a Maximum Principle set for L. We extend some classical results related to the Laplacian(proved by Deny, Hayman and Kennedy) and to the sub-Laplacians on homogeneous Carnot groups (proved by Bonfiglioli and Lanconelli)
The nodal set of solutions to anomalous equations
Full text linkThis note focuses on the geometric-theoretic analysis of the nodal set of solutions to specific degenerate or singular equations. As they belong to the Muckenhoupt class A_2, these operators appear in the seminal works of Fabes, Kenig, Jerison and Serapioni. In particular, they have recently attracted a lot of attention in the last decade due to their link to the local realization of the fractional Laplacian. The goal is to get a glimpse of the complete theory of the nodal set of solutions of such equations in the spirit of the seminal works of Hardt, Simon, Han and Lin
Sobolev-Poincaré inequalities for differential forms and currents
Full text linkIn this note we collect some results in R^n about (p,q) Poincaré and Sobolev inequalities for differential forms obtained in a joint research with Franchi and Pansu. In particular, we focus to the case p=1. From the geometric point of view, Poincaré and Sobolev inequalities for differential forms provide a quantitative formulation of the vanishing of the cohomology. As an application of the results obtained in the case p=1 we obtain Poincaré and Sobolev inequalities for Euclidean currents
The stickiness phenomena of nonlocal minimal surfaces: new results and a comparison with the classical case
Full text linkWe discuss in this note the stickiness phenomena for nonlocal minimal surfaces. Classical minimal surfaces in convex domains do not stick to the boundary of the domain, hence examples of stickiness can be obtained only by removing the assumption of convexity. On the other hand, in the nonlocal framework, stickiness is ''generic''. We provide various examples from the literature, and focus on the case of complete stickiness in highly nonlocal regimes