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MARIA MANFREDINI
Professore Associato
Dipartimento di Scienze Fisiche, Informatiche e Matematiche sede ex-Matematica
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Pubblicazioni
2023
- Hölder Continuity and Boundedness Estimates for Nonlinear Fractional Equations in the Heisenberg Group
[Articolo su rivista]
Manfredini, Maria; Palatucci, Giampiero; Piccinini, Mirco; Polidoro, Sergio
abstract
We extend the celebrate De Giorgi-Nash-Moser theory to a wide class of nonlinear equations driven by nonlocal, possibly degenerate, integro-differential operators, whose model is the fractional p-Laplacian operator on the Heisenberg-Weyl group Hn. Among other results, we prove that the weak solutions to such a class of problems are bounded and Hölder continuous, by also establishing general estimates as fractional Caccioppoli-type estimates with tail and logarithmic-type estimates.
2021
- Intrinsic fractional Taylor formula
[Articolo su rivista]
Manfredini, Maria
abstract
2017
- Fundamental solutions and local solvability for nonsmooth Hormander's operators
[Articolo su rivista]
Bramanti, Marco; Brandolini, Luca; Manfredini, Maria; Pedroni, Marco
abstract
We consider operators sum of squareo vector fields in a bounded domain where the vector fields are nonsmooth Hormander's vector fields of step r such that the highest order commutators are only holder continuous. Applying Levi's parametrix method we construct a local fundamental solution Gamma for L and provide growth estimates for Gamma and its first derivatives with respect to the vector fields. Requiring the existence of one more derivative of the coefficients we prove that Gamma also possesses second derivatives, and we deduce the local solvability of L, constructing, by means of Gamma, a solution to Lu = f with holder continuous f. We also prove C^2,alpha_loc estimates on this solution.
2016
- Poincaré-type inequality for Lipschitz continuous vector fields
[Articolo su rivista]
Citti, Giovanna; Manfredini, Maria; Pinamonti, Andrea; Serra Cassano, Francesco
abstract
The scope of this paper is to prove a Poincaré type inequality for a family
of non linear vector fields, whose coefficients are only Lipschitz continuous with respect to the distance induced by the vector fields themselves.
2015
- Regularity of mean curvature flow of graphs on Lie groups free up to step 2
[Articolo su rivista]
Capogna, Luca; Citti, Giovanna; Manfredini, Maria
abstract
We consider (smooth) solutions of the mean curvature flow of graphs over bounded domains in a Lie group free up to step two (and not necessarily nilpotent), endowed with a one parameter family of Riemannian metrics σ_ε collapsing to a subRiemannian metric σ as ε → 0. We establish C^(k,α) estimates for this flow, that are uniform as ε → 0 and as a consequence prove long time existence for the sub Riemannian mean curvature flow of the graph.
2015
- The role of fundamental solution in Potential and Regularity Theory for subelliptic PDE
[Capitolo/Saggio]
Bonfiglioli, Andrea; Citti, Giovanna; Cupini, Giovanni; Manfredini, Maria; Montanari, Annamaria; Morbidelli, Daniele; Pascucci, Andrea; Polidoro, Sergio; Uguzzoni, Francesco
abstract
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 formulas on the level sets of the fundamental solution, which are the starting point 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, namely 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.
2014
- Smooth approximation for intrinsic Lipschitz functions in the Heisenberg group
[Articolo su rivista]
Citti, Giovanna; Manfredini, Maria; Pinamonti, Andrea; Serra Cassano, Francesco
abstract
We characterize intrinsic Lipschitz functions as maps which can be approximated by a sequence of smooth maps, with pointwise convergent intrinsic gradient. We also provide an estimate of the Lipschitz constant of an intrinsic Lipschitz function in terms of the TeX -norm of its intrinsic gradient
2013
- Geometric Methods in PDE ́s: I.N.d.A.M. Meeting on the occasion of the 70 th birthday of Ermanno Lanconelli, Cortona, 27-31 maggio 2013,
[Altro]
Bonfiglioli, Andrea; Citti, Giovanna; Cupini, Giovanni; Manfredini, Maria; Montanari, Annamaria; Morbidelli, Daniele; Pascucci, Andrea; Polidoro, Sergio; Uguzzoni, Francesco
abstract
The scope of this conference is to celebrate the 70th birthday of Ermanno Lanconelli and to bring together Italian and foreign Mathematicians to favour the discussion in the areas of research where Ermanno Lanconelli has been particularly active:
- Second order linear and nonlinear partial differential equations with non-negative characteristic form;
- Geometric problems related to the underlying algebraic, geometrical or topological structure;
- Application to complex geometry and CR manifolds.
These fields are the objects of active research and development, and possess a remarkable degree of interrelation in their pure and applied aspects.
2013
- Uniform Gaussian Bounds for Subelliptic Heat Kernels and an Application to the Total Variation Flow of Graphs over Carnot Groups
[Articolo su rivista]
Capogna, Luca; Citti, Giovanna; Manfredini, Maria
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 → ∞.
2012
- Fundamental solutions for sum of squares of vector fields operators with C^1,α coefficients
[Articolo su rivista]
Manfredini, M.
abstract
We adapt Levi's parametrix method to construct a local fundamental solutions for operators of the form sum_i=1^m X_i^2,
where X_1,ldots, X_m are H"ormander
vector fields of step 2 having non-smooth coefficients. We
also provide estimates of the fundamental solution and of its derivatives.
2010
- A note on the Poincaré inequality for Lipschitz vector fields of step two
[Articolo su rivista]
Manfredini, M.
abstract
We provide a Poincarè inequality for families
of Lipschitz continuous vector fields satisfying a Hormander-type condition of step two.
2010
- Finite difference approximation of the Mumford and Shah functional in a contact manifold of the Heisenberg space
[Articolo su rivista]
Citti, G.; Manfredini, M.; Sarti, A.
abstract
The functionality of the visual cortex has been described by Peititot as a contact manifold of dimension three and in a previous paper of the authors the Mumford and Shah functional has been proposed to segment lifting of an image in the three dimensional cortical space. Hence, we study here this functional and we provide a constructive approach to the problem, extending to the sub- Riemannian setting an approximation technique proposed by De Giorgi in the Euclidean case.
2010
- Smoothness of Lipschitz minimal intrinsic graphs in Heisenberg groups H^n
[Articolo su rivista]
Citti, G.; Manfredini, M.; Capogna, L.
abstract
We prove that Lipschitz intrinsic graphs in the Heisenberg groups , with n > 1, which are vanishing viscosity solutions of the minimal surface equation, are smooth and satisfy the PDE in a strong sense.
2009
- Regularity of non-characteristic minimal graphs in the Heisenberg group $mathbbH^1$
[Articolo su rivista]
Capogna, L.; Citti, G.; Manfredini, M.
abstract
Minimal surfaces in the sub-Riemannian Heisenberg group can be constructed by means of a Riemannian approximation scheme, as limit of Riemannian minimal surfaces. We study the regularity of Lipschitz, non-characteristic minimal surfaces which arise as such limits.
Our main results are a-priori estimates on the solutions of the approximating Riemannian PDE and the ensuing C∞ regularity of the sub-Riemannian minimal surface along its Legendrian foliation.
2009
- Uniform Schauder estimates for regularized hypoelliptic equations
[Articolo su rivista]
Manfredini, Maria
abstract
In this paper we are concerned with a family of
elliptic operators represented as sum of square vector fields and the limit operator is hypoelliptic. Here we establish
Schauder's estimates, uniform with respect to the parameter ,
of solution of the approximated equation, using a modification of
the lifting technique of Rothschild and Stein.
These estimates can be
used in particular while studying regularity of viscosity
solutions of nonlinear equations represented in terms of vector
fields.
2006
- Implicit function theorem in Carnot-Caratheodory spaces
[Articolo su rivista]
Citti, G.; Manfredini, M.
abstract
In this paper we study the notion of regular surface in non homogeneous Lie groups. In
particular we prove an implicitly function theorem and prove the
regularity of the function implicitly defined. Indeed the
implicit function theorem had already been proved in homogeneous
Lie groups by [FSS], while the regularity problem of the function
implicitly defined was still open even in the simplest Lie group.
2006
- Uniform Estimates of the fundamental solution for a family of hypoelliptic operators
[Articolo su rivista]
Citti, G.; Manfredini, M.
abstract
In this paper we are concerned with a family of elliptic operators L-epsilon represented as sum of square vector fields, whose limit is an Hormander type operator L. It is well known that each operator L_epsilon
admits a fundamental solution. Here we establish a priori estimates uniform in epsilon of the fundamental solution, using a generalization
of the freezing and lifting technique of Rothschild and Stein. As
a consequence we deduce some a priori estimates uniform in epsilon,
for solutions of the approximated equation. These
estimates can be used in particular while studying regularity of
viscosity solutions of nonlinear equations represented in terms of
vector fields.
2005
- Blow-up in non homogeneous Lie groups and rectifiability,
[Articolo su rivista]
Citti, G.; Manfredini, M.
abstract
In this paper we extend the De Giorgi notion of rectifiability of surfaces in non homogeneous Lie groups. This notion and the principal properties of Cacciopoli sets had already been proved in homogeneous Lie group, using a blow-up method, with respect to the natural dilations. In non homogeneous Lie groups no dilations are defined, so that we need to apply a freezing method, locally approximating the non homogeneous structure, with an homogeneous one.
2004
- A degenerate parabolic equation arising in image processing
[Articolo su rivista]
Citti, G.; Manfredini, M.
abstract
2004
- Neuronal oscillation in the visual cortex: Gamma-convergence to the Riemannian Mumford-Shah functional
[Articolo su rivista]
Citti, G.; Manfredini, M.; Sarti, A.
abstract
2003
- A priori estimates for quasilinear degenarate parabolic equations
[Articolo su rivista]
Manfredini, M; Pascucci, A
abstract
2003
- From neural oscillations to variational problems in the visual cortex
[Articolo su rivista]
Sarti, A.; Citti, G.; Manfredini, M.
abstract
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 Γ-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 Γ-convergence result to subRiemannian spaces. © 2003 Elsevier Ltd. All rights reserved.
2002
- Long time behavior of Riemannian mean curvature flow of graphs
[Articolo su rivista]
Citti, Giovanna; Manfredini, M
abstract
In this paper we consider long time behavior of a mean curvature flow of nonparametric surface in ℝn, 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
1997
- The Dirichlet problem for a class of ultraparabolic equation
[Articolo su rivista]
Manfredini, M
abstract
1996
- L^p estimates for some ultraparabolic operators with discontinuous coefficients,
[Articolo su rivista]
Manfredini, M; M. C., Cerutti; M., Bramanti
abstract
1993
- Compact embedding theorems for generalized Sobolev spaces
[Articolo su rivista]
Manfredini, M
abstract
1992
- Asymptotic behavior of solutions of variational equations
[Articolo su rivista]
Manfredini, M
abstract
1992
- Stability properties for solutions of general Euler-Lagrange systems
[Articolo su rivista]
G., Leoni; Manfredini, M; Pucci, ; P,
abstract