Cecilia De Zan - Academia.edu (original) (raw)

Papers by Cecilia De Zan

Research paper thumbnail of Singular limits of reaction diffusion equations and geometric flows with discontinuous velocity

arXiv (Cornell University), May 23, 2019

We consider the singular limit of a bistable reaction diffusion equation in the case when the vel... more We consider the singular limit of a bistable reaction diffusion equation in the case when the velocity of the traveling wave solution depends on the space variable and converges to a discontinuous function. We show that the family of solutions converges to the stable equilibria off a front propagating with a discontinuous velocity. The convergence is global in time by applying the weak geometric flow uniquely defined through the theory of viscosity solutions and the level-set equation.

Research paper thumbnail of Singular limits of reaction diffusion equations and geometric flows with discontinuous velocity

Nonlinear Analysis, 2020

We consider the singular limit of a bistable reaction diffusion equation in the case when the vel... more We consider the singular limit of a bistable reaction diffusion equation in the case when the velocity of the traveling wave solution depends on the space variable and converges to a discontinuous function. We show that the family of solutions converges to the stable equilibria off a front propagating with a discontinuous velocity. The convergence is global in time by applying the weak geometric flow uniquely defined through the theory of viscosity solutions and the level-set equation.

Research paper thumbnail of On Viscosity and Equivalent Notions of Solutions for Anisotropic Geometric Equations

Abstract and Applied Analysis, 2020

We prove that viscosity solutions of geometric equations in step two Carnot groups can be equival... more We prove that viscosity solutions of geometric equations in step two Carnot groups can be equivalently reformulated by restricting the set of test functions at the singular points. These are characteristic points for the level sets of the solutions and are usually difficult to deal with. A similar property is known in the Euclidian space, and in Carnot groups, it is based on appropriate properties of a suitable homogeneous norm. We also use this idea to extend to Carnot groups the definition of generalised flow, and it works similarly to the Euclidian setting. These results simplify the handling of the singularities of the equation, for instance, to study the asymptotic behaviour of singular limits of reaction diffusion equations. We provide examples of using the simplified definition, showing, for instance, that boundaries of strictly convex subsets in the Carnot group structure become extinct in finite time when subject to the horizontal mean curvature flow even if characteristic ...

Research paper thumbnail of A Comparison Principle for the Mean Curvature Flow Equation with Discontinuous Coefficients

International Journal of Differential Equations, 2016

We study the level set equation in a bounded domain when the velocity of the interface is given b... more We study the level set equation in a bounded domain when the velocity of the interface is given by the mean curvature plus a discontinuous velocity. We prove a comparison principle for the initial-boundary value problem whose consequence is uniqueness of continuous solutions and well- posedness of the level set method.

Research paper thumbnail of Cauchy problems for noncoercive Hamilton–Jacobi–Isaacs equations with discontinuous coefficients

Interfaces and Free Boundaries, 2010

We study the Cauchy problem for a homogeneous and not necessarily coercive Hamilton-Jacobi-Isaacs... more We study the Cauchy problem for a homogeneous and not necessarily coercive Hamilton-Jacobi-Isaacs equation with an x-dependent, piecewise continuous coefficient. We prove that under suitable assumptions there exists a unique and continuous viscosity solution. The result applies in particular to the Carnot-Carathéodory eikonal equation with discontinuous refraction index of a family of vector fields satisfying the Hörmander condition. Our results are also of interest in connection with geometric flows with discontinuous velocity in anisotropic media with a non-euclidian ambient space.

Research paper thumbnail of Some new results on reaction-diffusion equations and geometric flows

In this thesis we discuss the asymptotic behavior of the solutions of scaled reaction-diffusion e... more In this thesis we discuss the asymptotic behavior of the solutions of scaled reaction-diffusion equations in the unbounded domain R n × (0 + ∞), in the cases when such a behavior is described in terms of moving interfaces. As first class of asymptotic problems we consider the singular limit of bistable reaction-diffusion equations in the case when the velocity of the traveling wave equation depends on the space variable, i.e. c ε = c ε (x), and it satisfies, in some suitable sense, c ε /ε τ → α, as ε → 0 + , where α is a discontinuous function and τ is an integer that can be equal to 0 or 1. The second part of the thesis concerns semilinear reaction-diffusion equations with diffusion term of type tr(A ε (x)D 2 u ε), where tr denotes the trace operator, A ε = σ ε σ t ε for some matrix map σ ε : R n → R n×(m+n) and A ε converges to a degenerate matrix. In order to establish such results rigorously, we modify and adapt to our problems the "geometric approach" introduced by G. Barles and P. E. Souganidis for solving problems in R n , and then partially revisited by G. Barles and F. Da Lio for reaction-diffusion equations in bounded domains. When it is possible we always consider the question of the well posedness of the Cauchy problems governing the motion of the fronts that describe the asymptotics we consider. Sommario In questa tesi discutiamo il comportamento asintotico delle soluzioni di equazioni di reazionediffusione nel dominio illimitato R n × (0, +∞) nei casi in cui tale comportamento sia descritto da un'interfaccia in movimento. Come primo tipo di problemi asintotici consideriamo il limite singolare di equazioni di reazionediffusione bistabili nel caso in cui la velocità dell'onda viaggiante dipenda dalla variabile di stato, cioè c ε = c ε (x), e sia soddisfatto, al tendere di ε a zero e in qualche modo opportuno, c ε /ε τ → α, laddove αè una funzione discontinua e τè un intero che può essere uguale a 0 o a 1. La seconda parte della tesi riguarda equazioni di reazione-diffusione semilineari e aventi termini di diffusione del tipo tr(A ε (x)D 2 u ε), laddove tr denota l'operatore traccia, A ε = σ ε σ t ε per qualche funzione σ ε : R n → R n×(m+n) e A ε converge ad una matrice degenere. Al fine di provare tali risultati in modo rigoroso, abbiamo modificato e adattato "l'approccio geometrico" introdotto da G. Barles e P. E. Souganidis per risolvere problemi in R n e in seguito parzialmente rivisto dallo stesso G. Barles assieme a F. Da Lio per equazioni di reazione-diffusione in domini limitati. Laddove possibile abbiamo sempre considerato la questione della buona posizione dei problemi di Cauchy che governano il moto dei fronti che descrivono le asintotiche da noi considerate.

Research paper thumbnail of Singular limits of reaction diffusion equations and geometric flows with discontinuous velocity

arXiv (Cornell University), May 23, 2019

We consider the singular limit of a bistable reaction diffusion equation in the case when the vel... more We consider the singular limit of a bistable reaction diffusion equation in the case when the velocity of the traveling wave solution depends on the space variable and converges to a discontinuous function. We show that the family of solutions converges to the stable equilibria off a front propagating with a discontinuous velocity. The convergence is global in time by applying the weak geometric flow uniquely defined through the theory of viscosity solutions and the level-set equation.

Research paper thumbnail of Singular limits of reaction diffusion equations and geometric flows with discontinuous velocity

Nonlinear Analysis, 2020

We consider the singular limit of a bistable reaction diffusion equation in the case when the vel... more We consider the singular limit of a bistable reaction diffusion equation in the case when the velocity of the traveling wave solution depends on the space variable and converges to a discontinuous function. We show that the family of solutions converges to the stable equilibria off a front propagating with a discontinuous velocity. The convergence is global in time by applying the weak geometric flow uniquely defined through the theory of viscosity solutions and the level-set equation.

Research paper thumbnail of On Viscosity and Equivalent Notions of Solutions for Anisotropic Geometric Equations

Abstract and Applied Analysis, 2020

We prove that viscosity solutions of geometric equations in step two Carnot groups can be equival... more We prove that viscosity solutions of geometric equations in step two Carnot groups can be equivalently reformulated by restricting the set of test functions at the singular points. These are characteristic points for the level sets of the solutions and are usually difficult to deal with. A similar property is known in the Euclidian space, and in Carnot groups, it is based on appropriate properties of a suitable homogeneous norm. We also use this idea to extend to Carnot groups the definition of generalised flow, and it works similarly to the Euclidian setting. These results simplify the handling of the singularities of the equation, for instance, to study the asymptotic behaviour of singular limits of reaction diffusion equations. We provide examples of using the simplified definition, showing, for instance, that boundaries of strictly convex subsets in the Carnot group structure become extinct in finite time when subject to the horizontal mean curvature flow even if characteristic ...

Research paper thumbnail of A Comparison Principle for the Mean Curvature Flow Equation with Discontinuous Coefficients

International Journal of Differential Equations, 2016

We study the level set equation in a bounded domain when the velocity of the interface is given b... more We study the level set equation in a bounded domain when the velocity of the interface is given by the mean curvature plus a discontinuous velocity. We prove a comparison principle for the initial-boundary value problem whose consequence is uniqueness of continuous solutions and well- posedness of the level set method.

Research paper thumbnail of Cauchy problems for noncoercive Hamilton–Jacobi–Isaacs equations with discontinuous coefficients

Interfaces and Free Boundaries, 2010

We study the Cauchy problem for a homogeneous and not necessarily coercive Hamilton-Jacobi-Isaacs... more We study the Cauchy problem for a homogeneous and not necessarily coercive Hamilton-Jacobi-Isaacs equation with an x-dependent, piecewise continuous coefficient. We prove that under suitable assumptions there exists a unique and continuous viscosity solution. The result applies in particular to the Carnot-Carathéodory eikonal equation with discontinuous refraction index of a family of vector fields satisfying the Hörmander condition. Our results are also of interest in connection with geometric flows with discontinuous velocity in anisotropic media with a non-euclidian ambient space.

Research paper thumbnail of Some new results on reaction-diffusion equations and geometric flows

In this thesis we discuss the asymptotic behavior of the solutions of scaled reaction-diffusion e... more In this thesis we discuss the asymptotic behavior of the solutions of scaled reaction-diffusion equations in the unbounded domain R n × (0 + ∞), in the cases when such a behavior is described in terms of moving interfaces. As first class of asymptotic problems we consider the singular limit of bistable reaction-diffusion equations in the case when the velocity of the traveling wave equation depends on the space variable, i.e. c ε = c ε (x), and it satisfies, in some suitable sense, c ε /ε τ → α, as ε → 0 + , where α is a discontinuous function and τ is an integer that can be equal to 0 or 1. The second part of the thesis concerns semilinear reaction-diffusion equations with diffusion term of type tr(A ε (x)D 2 u ε), where tr denotes the trace operator, A ε = σ ε σ t ε for some matrix map σ ε : R n → R n×(m+n) and A ε converges to a degenerate matrix. In order to establish such results rigorously, we modify and adapt to our problems the "geometric approach" introduced by G. Barles and P. E. Souganidis for solving problems in R n , and then partially revisited by G. Barles and F. Da Lio for reaction-diffusion equations in bounded domains. When it is possible we always consider the question of the well posedness of the Cauchy problems governing the motion of the fronts that describe the asymptotics we consider. Sommario In questa tesi discutiamo il comportamento asintotico delle soluzioni di equazioni di reazionediffusione nel dominio illimitato R n × (0, +∞) nei casi in cui tale comportamento sia descritto da un'interfaccia in movimento. Come primo tipo di problemi asintotici consideriamo il limite singolare di equazioni di reazionediffusione bistabili nel caso in cui la velocità dell'onda viaggiante dipenda dalla variabile di stato, cioè c ε = c ε (x), e sia soddisfatto, al tendere di ε a zero e in qualche modo opportuno, c ε /ε τ → α, laddove αè una funzione discontinua e τè un intero che può essere uguale a 0 o a 1. La seconda parte della tesi riguarda equazioni di reazione-diffusione semilineari e aventi termini di diffusione del tipo tr(A ε (x)D 2 u ε), laddove tr denota l'operatore traccia, A ε = σ ε σ t ε per qualche funzione σ ε : R n → R n×(m+n) e A ε converge ad una matrice degenere. Al fine di provare tali risultati in modo rigoroso, abbiamo modificato e adattato "l'approccio geometrico" introdotto da G. Barles e P. E. Souganidis per risolvere problemi in R n e in seguito parzialmente rivisto dallo stesso G. Barles assieme a F. Da Lio per equazioni di reazione-diffusione in domini limitati. Laddove possibile abbiamo sempre considerato la questione della buona posizione dei problemi di Cauchy che governano il moto dei fronti che descrivono le asintotiche da noi considerate.