Maria Manfredini

IMPLICIT FUNCTION THEOREM IN CARNOT–CARATHÉODORY SPACES

In this paper, we prove an implicit function theorem and we study the regularity of the function implicitly defined. The implicit function theorem had already been proved in homogeneous Lie groups by Franchi, Serapioni and Serra Cassano, while the regularity problem of the function implicitly defined was still open even in the simplest Lie group.

From neural oscillations to variational problems in the visual cortex.

Aim of this study is to provide a formal link between connectionist neural models and variational psycophysical ones. We show that the solution of phase difference equation of weakly connected neural oscillators gamma-converges as the dimension of the grid tends to 0, to the gradient flow relative to the Mumford-Shah functional in a Riemannian space. The Riemannian metric is directly induced by the pattern of neural connections. Next, we embed the energy functional in the specific geometry of the functional space of the primary visual cortex, that is described in terms of a subRiemannian Heisenberg space. Namely, we introduce the Mumford-Shah functional with the Heisenberg metric and discuss the applicability of our main gamma-convergence result to subRiemannian spaces.

NEURONAL OSCILLATIONS IN THE VISUAL CORTEX: Γ-CONVERGENCE TO THE RIEMANNIAN MUMFORD-SHAH FUNCTIONAL ∗

The aim of this paper is to provide a formal link between an oscillatory neural model, whose phase is represented by a difference equation, and the Mumford and Shah functional. A Riemannian metric is induced by the pattern of neural connections, and in this setting the difference equation is studied. Its Euler--Lagrange operator

Uniform Gaussian bounds for subelliptic heat kernels and an application to the total variation flow of graphs over Carnot groups

\Gamma

A note on the Poincaré inequality for Lipschitz vector fields of step two

-converges as the dimension of the grid tends to 0 to the Mumford and Shah functional in the same Riemannian space. Correspondingly, the solutions of the phase equation converge to a BV function, which is interpreted as the flow associated with the Mumford and Shah functional. In this way we provide a biological motivation to this celebrated functional.

Long time behavior of Riemannian mean curvature flow of graphs

Abstract In this paper we study heat kernels associated with a Carnot group G, endowed with a family of collapsing left-invariant Riemannian metrics σε which converge in the Gromov- Hausdorff sense to a sub-Riemannian structure on G as ε→ 0. The main new contribution are Gaussian-type bounds on the heat kernel for the σε metrics which are stable as ε→0 and extend the previous time-independent estimates in [16]. As an application we study well posedness of the total variation flow of graph surfaces over a bounded domain in a step two Carnot group (G; σε ). We establish interior and boundary gradient estimates, and develop a Schauder theory which are stable as ε → 0. As a consequence we obtain long time existence of smooth solutions of the sub-Riemannian flow (ε = 0), which in turn yield sub-Riemannian minimal surfaces as t → ∞.

A priori estimates for quasilinear degenerate parabolic equations

We provide a Poincare inequality for families of Lipschitz continuous vector fields satisfying a Hormander-type condition of step two.

The Role of Fundamental Solution in Potential and Regularity Theory for Subelliptic PDE

Abstract In this paper we consider long time behavior of a mean curvature flow of nonparametric surface in R n , with respect to a conformal Riemannian metric. We impose zero boundary value, and we prove that the solution tends to 0 exponentially fast as t→∞ . Its normalization u/ sup u tends to the first eigenfunction of the associated linearized problem.

LpEstimates for Some Ultraparabolic Operators with Discontinuous Coefficients

We prove some maximum and gradient estimates for classical solutions to a wide class of quasilinear degenerate parabolic equations, including first order ones. The proof is elementary and exploits the smallness of the domain in the time direction.

Regularity of non-characteristic minimal graphs in the Heisenberg group mathbb{H}^{1}

In this survey we consider a general Hormander type operator, represented as a sum of squares of vector fields plus a drift and we outline the central role of the fundamental solution in developing Potential and Regularity Theory for solutions of related PDEs. After recalling the Gaussian behavior at infinity of the kernel, we show some mean value formula on the level set of the fundamental solution, which allow to obtain a comprehensive parallel of the classical Potential Theory. Then we show that a precise knowledge of the fundamental solution leads to global regularity results: estimates at the boundary or on the whole space. Finally in the problem of regularity of non linear differential equations we need an ad hoc modification of the parametrix method, based on the properties of the fundamental solution of an approximating problem.