We derive several properties of the heat equation with the Hodge operator associated with the Rum... more We derive several properties of the heat equation with the Hodge operator associated with the Rumin complex on Heisenberg groups and prove several properties of the fundamental solution. As an application, we use the heat kernel for Rumin's differential forms to construct a Calderón reproducing formula on Rumin forms. Contents 0 12 3.3. The Calderón reproducing formula 23 4. Appendix A: Rumin's complex on Heisenberg groups 27 5. Appendix B: Kernels in Carnot groups and Folland-Stein Spaces 34 5.1. Convolution in Carnot groups 34 5.2. Folland-Stein-Sobolev spaces and homogeneous kernels 35 References 39
In this paper we prove existence and interior regularity of convex solutions of the Dirichlet pro... more In this paper we prove existence and interior regularity of convex solutions of the Dirichlet problem for a class of Monge-Ampre type equations with right hand side vanishing on the boundary, by means of the fixed point index theory for Bakelman's weak solutions. The regularity in the interior follows from recent results of Caffarelli.
In this note, we apply the theory of stochastic homogenization to find the asymptotic behavior of... more 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.
We study the families of measures on Carnot groups that have vanishing p-module, which we call p-... more 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 ≥ 1. We described a wide class of p-exceptional intrinsic Lipschitz surfaces for p ∈ (0, ∞).
In this Note we study a mathematical model for the progression of Alzheimer's Disease in the huma... more In this Note we study a mathematical model for the progression of Alzheimer's Disease in the human brain. The novelty of our approach consists in the representation of the brain as two superposed graphs where toxic proteins diffuse, the connectivity graph which represents the neural network, and the proximity graph which takes into account the extracellular space. Toxic proteins such as β-amyloid and τ play in fact a crucial role in the development of Alzheimer's disease and, separately, have been targets of medical treatments. Recent biomedical literature stresses the potential impact of the synergetic action of these proteins. We numerically test various modelling hypotheses which confirm the relevance of this synergy.
We propose a mathematical model for the onset and progression of Alzheimer's disease based on tra... more We propose a mathematical model for the onset and progression of Alzheimer's disease based on transport and diffusion equations. We treat brain neurons as a continuous medium and structure them by their degree of malfunctioning. Three different mechanisms are assumed to be relevant for the temporal evolution of the disease: i) diffusion and agglomeration of soluble Amyloid beta, ii) effects of phosphorylated tau protein and iii) neuron-to-neuron prion-like transmission of the disease. We model these processes by a system of Smoluchowski equations for the Amyloid beta concentration, an evolution equation for the dynamics of tau protein and a kinetic-type transport equation for the distribution function of the degree of malfunctioning of neurons. The latter equation contains an integral term describing the random onset of the disease as a jump process localized in particularly sensitive areas of the brain. We are particularly interested in investigating the effects of the synergistic interplay of Amyloid beta and tau on the dynamics of Alzheimer's disease. The output of our numerical simulations, although in 2D with an oversimplified geometry, is in good qualitative agreement with clinical findings concerning both the disease distribution in the brain, which varies from early to advanced stages, and the effects of tau on the dynamics of the disease.
In this Note we study a mathematical model for the progression of Alzheimer’s Disease in the huma... more In this Note we study a mathematical model for the progression of Alzheimer’s Disease in the human brain. The novelty of our approach consists in the representation of the brain as two superposed graphs where toxic proteins diffuse, the connectivity graph which represents the neural network, and the proximity graph which takes into account the extracellular space. Toxic proteins such as $$\beta $$ β amyloid and Tau play in fact a crucial role in the development of Alzheimer’s disease and, separately, have been targets of medical treatments. Recent biomedical literature stresses the potential impact of the synergetic action of these proteins. We numerically test various modelling hypotheses which confirm the relevance of this synergy.
In the last few years the authors proved Poincaré and Sobolev type inequalities in Heisenberg gro... more In the last few years the authors proved Poincaré and Sobolev type inequalities in Heisenberg groups ℍ^n for differential forms in the Rumin's complex. The need to substitute the usual de Rham complex of differential forms for Euclidean spaces with the Rumin's complex is due to the different stratification of the Lie algebra of Heisenberg groups. The crucial feature of Rumin's complex is that d_c is a differential operator of order 1 or 2 according to the degree of the form. Roughly speaking, Poincaré and Sobolev type inequalities are quantitative formulations of the well known topological problem whether a closed form is exact. More precisely, for suitable p and q, we mean that every exact differential form ω in L^p admits a primitive ϕ in L^q such thatϕ_L^q≤ Cω_L^p. The cases of the norm L^p, p≥ 1 and q<∞ have been already studied in a series of papers by the authors. In the present paper we deal with the limiting case where q=∞: it is remarkable that, unlike in the scalar case, when the degree of the forms ω is at least 2, we can take q=∞ in the left-hand side of the inequality. The corresponding inequality in the Euclidean setting ℝ^N (p=N and q=∞) was proven by Bourgain and Brezis.
In order to enhance the academic communication and stimulate international exchange and research ... more In order to enhance the academic communication and stimulate international exchange and research collaboration, an International Conference on PDEs and Harmonic Analysis will be held in Beijing, June 25-29, 2007. The purpose of this conference is to provide a chance for mathematicians to exchange their new ideas and recent research developments in PDEs and Harmonic Analysis with applications
We derive several properties of the heat equation with the Hodge operator associated with the Rum... more We derive several properties of the heat equation with the Hodge operator associated with the Rumin complex on Heisenberg groups and prove several properties of the fundamental solution. As an application, we use the heat kernel for Rumin's differential forms to construct a Calderón reproducing formula on Rumin forms. Contents 0 12 3.3. The Calderón reproducing formula 23 4. Appendix A: Rumin's complex on Heisenberg groups 27 5. Appendix B: Kernels in Carnot groups and Folland-Stein Spaces 34 5.1. Convolution in Carnot groups 34 5.2. Folland-Stein-Sobolev spaces and homogeneous kernels 35 References 39
In this paper we prove existence and interior regularity of convex solutions of the Dirichlet pro... more In this paper we prove existence and interior regularity of convex solutions of the Dirichlet problem for a class of Monge-Ampre type equations with right hand side vanishing on the boundary, by means of the fixed point index theory for Bakelman&amp;#39;s weak solutions. The regularity in the interior follows from recent results of Caffarelli.
In this note, we apply the theory of stochastic homogenization to find the asymptotic behavior of... more 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.
We study the families of measures on Carnot groups that have vanishing p-module, which we call p-... more 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 ≥ 1. We described a wide class of p-exceptional intrinsic Lipschitz surfaces for p ∈ (0, ∞).
In this Note we study a mathematical model for the progression of Alzheimer's Disease in the huma... more In this Note we study a mathematical model for the progression of Alzheimer's Disease in the human brain. The novelty of our approach consists in the representation of the brain as two superposed graphs where toxic proteins diffuse, the connectivity graph which represents the neural network, and the proximity graph which takes into account the extracellular space. Toxic proteins such as β-amyloid and τ play in fact a crucial role in the development of Alzheimer's disease and, separately, have been targets of medical treatments. Recent biomedical literature stresses the potential impact of the synergetic action of these proteins. We numerically test various modelling hypotheses which confirm the relevance of this synergy.
We propose a mathematical model for the onset and progression of Alzheimer's disease based on tra... more We propose a mathematical model for the onset and progression of Alzheimer's disease based on transport and diffusion equations. We treat brain neurons as a continuous medium and structure them by their degree of malfunctioning. Three different mechanisms are assumed to be relevant for the temporal evolution of the disease: i) diffusion and agglomeration of soluble Amyloid beta, ii) effects of phosphorylated tau protein and iii) neuron-to-neuron prion-like transmission of the disease. We model these processes by a system of Smoluchowski equations for the Amyloid beta concentration, an evolution equation for the dynamics of tau protein and a kinetic-type transport equation for the distribution function of the degree of malfunctioning of neurons. The latter equation contains an integral term describing the random onset of the disease as a jump process localized in particularly sensitive areas of the brain. We are particularly interested in investigating the effects of the synergistic interplay of Amyloid beta and tau on the dynamics of Alzheimer's disease. The output of our numerical simulations, although in 2D with an oversimplified geometry, is in good qualitative agreement with clinical findings concerning both the disease distribution in the brain, which varies from early to advanced stages, and the effects of tau on the dynamics of the disease.
In this Note we study a mathematical model for the progression of Alzheimer’s Disease in the huma... more In this Note we study a mathematical model for the progression of Alzheimer’s Disease in the human brain. The novelty of our approach consists in the representation of the brain as two superposed graphs where toxic proteins diffuse, the connectivity graph which represents the neural network, and the proximity graph which takes into account the extracellular space. Toxic proteins such as $$\beta $$ β amyloid and Tau play in fact a crucial role in the development of Alzheimer’s disease and, separately, have been targets of medical treatments. Recent biomedical literature stresses the potential impact of the synergetic action of these proteins. We numerically test various modelling hypotheses which confirm the relevance of this synergy.
In the last few years the authors proved Poincaré and Sobolev type inequalities in Heisenberg gro... more In the last few years the authors proved Poincaré and Sobolev type inequalities in Heisenberg groups ℍ^n for differential forms in the Rumin's complex. The need to substitute the usual de Rham complex of differential forms for Euclidean spaces with the Rumin's complex is due to the different stratification of the Lie algebra of Heisenberg groups. The crucial feature of Rumin's complex is that d_c is a differential operator of order 1 or 2 according to the degree of the form. Roughly speaking, Poincaré and Sobolev type inequalities are quantitative formulations of the well known topological problem whether a closed form is exact. More precisely, for suitable p and q, we mean that every exact differential form ω in L^p admits a primitive ϕ in L^q such thatϕ_L^q≤ Cω_L^p. The cases of the norm L^p, p≥ 1 and q<∞ have been already studied in a series of papers by the authors. In the present paper we deal with the limiting case where q=∞: it is remarkable that, unlike in the scalar case, when the degree of the forms ω is at least 2, we can take q=∞ in the left-hand side of the inequality. The corresponding inequality in the Euclidean setting ℝ^N (p=N and q=∞) was proven by Bourgain and Brezis.
In order to enhance the academic communication and stimulate international exchange and research ... more In order to enhance the academic communication and stimulate international exchange and research collaboration, an International Conference on PDEs and Harmonic Analysis will be held in Beijing, June 25-29, 2007. The purpose of this conference is to provide a chance for mathematicians to exchange their new ideas and recent research developments in PDEs and Harmonic Analysis with applications
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Papers by Bruno Franchi