Gagliardo-Nirenberg Inequalities for Differential Forms in Heisenberg Groups

The L 1-Sobolev inequality states that the L n/(n--1)-norm of a compactly supported function on Euclidean n-space is controlled by the L 1-norm of its gradient. The generalization to differential forms (due to Lanzani & Stein and Bourgain & Brezis) is recent, and states that a the L n/(n--1)-norm of a compactly supported differential h-form is controlled by the L 1-norm of its exterior differential du and its exterior codifferential $\delta$u (in special cases the L 1-norm must be replaced by the H 1-Hardy norm). We shall extend this result to Heisenberg groups in the framework of an appropriate complex of differential forms

L1L^1-Poincar\'e and Sobolev inequalities for differential forms in Euclidean spaces

In this paper, we prove Poincar\'e and Sobolev inequalities for differential forms in $L^1(\mathbb R^n)$. The singular integral estimates that it is possible to use for $L^p$, $p>1$, are replaced here with inequalities which go back to Bourgain-Brezis.Comment: Accepted for publication in Science China Mathematics. arXiv admin note: text overlap with arXiv:1902.0481

Well-posedness of a mathematical model for Alzheimer's disease

We consider the existence and uniqueness of solutions of an initial-boundary value problem for a coupled system of PDE's arising in a model for Alzheimer's disease. Apart from reaction diffusion equations, the system contains a transport equation in a bounded interval for a probability measure which is related to the malfunctioning of neurons. The main ingredients to prove existence are: the method of characteristics for the transport equation, a priori estimates for solutions of the reaction diffusion equations, a variant of the classical contraction theorem, and the Wasserstein metric for the part concerning the probability measure. We stress that all hypotheses on the data are not suggested by mathematical artefacts, but are naturally imposed by modelling considerations. In particular the use of a probability measure is natural from a modelling point of view. The nontrivial part of the analysis is the suitable combination of the various mathematical tools, which is not quite routine and requires various technical adjustments

Exceptional families of measures on Carnot groups

We study the families of measures on Carnot groups that have vanishing $p$-module, which we call $p$-exceptional families. We found necessary and sufficient condition for the family of intrinsic Lipschitz surfaces passing through a common point to be $p$-exceptional for $p\geq 1$. We described a wide class of $p$-exceptional intrinsic Lipschitz surfaces for $p\in(0,\infty)$.Comment: 46 page

Intrinsic Lipschitz Graphs Within Carnot Groups

A Carnot group is a connected, simply connected, nilpotent Lie group with stratified Lie algebra. We study the notions of intrinsic graphs and of intrinsic Lipschitz graphs within Carnot groups. Intrinsic Lipschitz graphs are the natural local analogue inside Carnot groups of Lipschitz submanifolds in Euclidean spaces, where \u201cnatural\u201d emphasizes that the notion depends only on the structure of the algebra. Intrinsic Lipschitz graphs unify different alternative approaches through Lipschitz parameterizations or level sets. We provide both geometric and analytic characterizations and a clarifying relation between these graphs and Rumin\u2019s complex of differential forms

Steiner's formula in the Heisenberg group

Steiner's tube formula states that the volume of an ∈-neighborhood of a smooth regular domain in ℝn is a polynomial of degree n in the variable ∈ whose coefficients are curvature integrals (also called quermassintegrals). We prove a similar result in the sub-Riemannian setting of the first Heisenberg group. In contrast to the Euclidean setting, we find that the volume of an ∈-neighborhood with respect to the Heisenberg metric is an analytic function of ∈ that is generally not a polynomial. The coefficients of the series expansion can be explicitly written in terms of integrals of iteratively defined canonical polynomials of just five curvature terms

A mathematical model for Alzheimer's disease: An approach via stochastic homogenization of the Smoluchowski equation

In this note, we apply the theory of stochastic homogenization to find the asymptotic behavior of the solution of a set of Smoluchowski's coagulation-diffusion equations with non-homogeneous Neumann boundary conditions. This system is meant to model the aggregation and diffusion of β-amyloid peptide (Aβ) in the cerebral tissue, a process associated with the development of Alzheimer's disease. In contrast to the approach used in our previous works, in the present paper we account for the non-periodicity of the cellular structure of the brain by assuming a stochastic model for the spatial distribution of neurons. Further, we consider non-periodic random diffusion coefficients for the amyloid aggregates and a random production of Aβ in the monomeric form at the level of neuronal membranes

A mathematical model for Alzheimer's disease: An approach via stochastic homogenization of the Smoluchowski equation

In this note, we apply the theory of stochastic homogenization to find the asymptotic behavior of the solution of a set of Smoluchowski's coagulation-diffusion equations with non-homogeneous Neumann boundary conditions. This system is meant to model the aggregation and diffusion of β-amyloid peptide (Aβ) in the cerebral tissue, a process associated with the development of Alzheimer's disease. In contrast to the approach used in our previous works, in the present paper we account for the non-periodicity of the cellular structure of the brain by assuming a stochastic model for the spatial distribution of neurons. Further, we consider non-periodic random diffusion coefficients for the amyloid aggregates and a random production of Aβ in the monomeric form at the level of neuronal membranes