1,720,982 research outputs found
Characterizing spheres in C^2 by their Levi curvature: a result à la Jellett
Full text linkWe investigate rigidity problems for a class of real hypersurfaces in C^2 with constant Levi curvature. We present a recent result obtained in [18] in collaboration with V. Martino for the boundaries of starshaped circular domains
Double Ball Property for non-divergence horizontally elliptic operators on step two Carnot groups
No full textLet L be a linear second order horizontally elliptic operator on a Carnot group of step two. We assume L in non-divergence form and with measurable coefficients. Then, we prove the Double Ball Property for the nonnegative sub-solutions of L. With our result, in order to solve the Harnack inequality problem for this kind of operators, it becomes sufficient to prove the so called ε-Critical Density
A Universal Heat Semigroup Characterisation of Sobolev and BV Spaces in Carnot Groups
No full textIn sub-Riemannian geometry there exist, in general, no known explicit representations of the heat kernels, and these functions fail to have any symmetry whatsoever. In particular, they are not a function of the control distance, nor they are for instance spherically symmetric in any of the layers of the Lie algebra. Despite these unfavourable aspects, in this paper we establish a new heat semigroup characterisation of the Sobolev and
spaces in a Carnot group by means of an integral decoupling property of the heat kernel
A Bourgain–Brezis–Mironescu–Dávila theorem in Carnot groups of step two
Full text linkIn this note we prove the following theorem in any Carnot group of step two G: lim s↗1/2 (1 − 2s)PH,s(E) = 4 √_ PH(E). Here, PH(E) represents the horizontal perimeter of a measurable set E ⊂ G, whereas the nonlocal horizontal perimeter PH,s(E) is a heat based Besov seminorm. This result represents a dimensionless sub-Riemannian counterpart of a famous characterisation of Bourgain-Brezis-Mironescu and Davila
On the Hopf–Oleinik lemma for degenerate-elliptic equations at characteristic points
No full textIn this paper we discuss the validity of the Hopf lemma at boundary points which are characteristic with respect to certain degenerate-elliptic equations. In the literature there are some positive results under the assumption that the boundary of the domain reflects the underlying geometry of the specific operator. We focus here on conditions on the boundary which are suitable for some families of degenerate operators, also in presence of first order terms
A certain critical density property for invariant Harnack inequalities in H-type groups
No full textWe consider second order linear degenerate-elliptic operators which are elliptic with respect to horizontal directions generating a stratified algebra of H-type. Extending a result by Gutiérrez and Tournier (2011) for the Heisenberg group, we prove a critical density estimate by assuming a condition of Cordes-Landis type. We then deduce an invariant Harnack inequality for the non-negative solutions from a result by Di Fazio, Gutiérrez, and Lanconelli (2008)
Double ball property: an overview and the case of step two Carnot groups
Full text linkWe investigate the notion of the so-called Double Ball Property, which concerns the nonnegative sub-solutions of some differential operators. Thanks to the axiomatic approach developed in [6], this is an important tool in order to solve the Krylov-Safonov's Harnack inequality problem for this kind of operators. In particular, we are interested in linear second order horizontally-elliptic operators in non-divergence formand with measurable coefficients. In the setting of homogeneous Carnot groups, we would like to stress the relation between the Double Ball Property and a kind of solvability of the Dirichlet problem for the operator in the exterior of some homogeneous balls. We present a recent result obtained in [15], where the double ball property has been proved in a generic Carnot group of step two
Levi curvature with radial symmetry : a sphere theorem for bounded Reinhardt domains of
No full textWe study bounded Reinhardt domains of C^2, whose Levi curvature depends on the distance from the origin. We prove that a radial decrease of the Levi curvature forces the domain to be a ball
Some Zaremba-Hopf-Oleinik Boundary Comparison Principles at Characteristic Points
Full text linkWe investigate the so-called Hopf lemma for certain degenerate-elliptic equations at characteristic boundary points of bounded open sets. For such equations, the validity of the Hopf lemma is related to the fact that the boundary of the open set reflects the underlying geometry of the specific operator. We present here some recent results obtained in [21] in collaboration with V. Martino. Our main focus is on conditions on the boundary which are stable by changing our operators in some particular classes, for example in the class of horizontally elliptic operators in non-divergence form. We also study what happens to these conditions for degenerate operators with first order terms
Complex group actions on the sphere and sign changing solutions for the CR-Yamabe equation
No full textIn this paper we prove that the CR-Yamabe equation on the sphere has infinitely many sign changing solutions. The problem is variational but the related functional does not satisfy the Palais-Smale condition, therefore the standard topological methods fail to apply directly. To overcome this lack of compactness, we will exploit different group actions on the sphere in order to find suitable closed subspaces, on which the restricted functional is Palais-Smale: this will allow us to use the minimax argument of Ambrosetti-Rabinowitz to find critical points. By a classification of the positive solutions and by considerations on the energy blow-up, we will get the desired result
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