Bruno Franchi

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We describe intrinsically regular submanifolds in Heisen-berg groups H n. Low dimensional and low codimensional submanifolds turn out to be of a very different nature. The first ones are Legendrian surfaces, while low codimensional ones are more general objects, possibly non Euclidean rectifiable. Nevertheless we prove that they are graphs in a natural(More)
The purpose of this note is to study the relationship between the validity of L 1 versions of Poincaré's inequality and the existence of representation formulas for functions as (fractional) integral transforms of first-order vector fields. The simplest example of a representation formula of the type we have in mind is the following familiar inequality for(More)
In this paper we propose a mathematical model for the onset and progression of Alzheimer's disease based on transport and diffusion equations. We regard brain neurons as a continuous medium, and structure them by their degree of malfunctioning. Two different mechanisms are assumed to be relevant for the temporal evolution of the disease: i) diffusion and(More)
In this paper we present a mathematical model for the aggregation and diffusion of Aβ amyloid in the brain affected by Alzheimer's disease, at the early stage of the disease. The model is based on a classical discrete Smoluchowski aggregation equation modified to take diffusion into account. We also describe a numerical scheme and discuss the results of the(More)
This paper completes a previous work on a Black and Scholes equation with stochastic volatility. This is a degenerate parabolic equation, which gives the price of a European option as a function of the time, of the price of the underlying asset, and of the volatility, when the volatility is a function of a mean reverting Orstein–Uhlenbeck process, possibly(More)
In this paper we prove a compensated compactness theorem for differential forms of the intrinsic complex of a Carnot group. The proof relies on a L s –Hodge decomposition for these forms. Because of the lack of homogeneity of the intrinsic exterior differential, Hodge decomposition is proved using the parametrix of a suitable 0-order Laplacian on forms.