Martino Bardi

Martino Bardi
University of Padova | UNIPD · Department of Mathematics "Tullio Levi-Civita"

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106
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Publications

Publications (106)
Preprint
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The paper treats second order fully nonlinear degenerate elliptic equations having a family of subunit vector fields satisfying a full-rank bracket condition. It studies Liouville properties for viscosity sub- and supersolutions in the whole space, namely, that under a suitable bound at infinity from above and, respectively, from below, they must b...
Article
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In this paper we relax the current regularity theory for the eikonal equation by using the recent theory of set-valued iterated Lie brackets. We give sufficient conditions for small time local attainability of general, symmetric, nonlinear systems, which have as a consequence the Hölder regularity of the minimum time function in optimal control. We...
Article
Full-text available
This paper studies Liouville properties for viscosity sub- and supersolutions of fully nonlinear degenerate elliptic PDEs, under the main assumption that the operator has a family of generalized subunit vector fields that satisfy the Hörmander condition. A general set of sufficient conditions is given such that all subsolutions bounded above are co...
Article
For two classes of Mean Field Game systems we study the convergence of solutions as the interest rate in the cost functional becomes very large, modelling agents caring only about a very short time-horizon, and the cost of the control becomes very cheap. The limit in both cases is a single first order integro-partial differential equation for the e...
Preprint
Full-text available
This paper studies Liouville properties for viscosity sub- and supersolutions of fully nonlinear degenerate elliptic PDEs, under the main assumption that the operator has a family of generalized subunit vector fields that satisfy the H\"ormander condition. A general set of sufficient conditions is given such that all subsolutions bounded above are...
Preprint
Full-text available
For two classes of Mean Field Game systems we study the convergence of solutions as the interest rate in the cost functional becomes very large, modeling agents caring only about a very short time-horizon, and the cost of the control becomes very cheap. The limit in both cases is a single first order integro-partial differential equation for the ev...
Article
Full-text available
We introduce a notion of subunit vector field for fully nonlinear degenerate elliptic equations. We prove that an interior maximum of a viscosity subsolution of such an equation propagates along the trajectories of subunit vector fields. This implies strong maximum and minimum principles when the operator has a family of subunit vector fields satis...
Article
This paper deals with the periodic homogenization of nonlocal parabolic Hamilton–Jacobi equations with superlinear growth in the gradient terms. We show that the problem presents different features depending on the order of the nonlocal operator, giving rise to three different cell problems and effective operators. To prove the locally uniform conv...
Preprint
Full-text available
In this paper we relax the current regularity theory for the eikonal equation by using the recent theory of set-valued iterated Lie brackets. We give sufficient conditions for small time local attainability of general, symmetric, nonlinear systems, which have as a consequence the Hoelder regularity of the minimum time function in optimal control. W...
Preprint
Full-text available
We introduce a notion of subunit vector field for fully nonlinear degenerate elliptic equations. We prove that an interior maximum of a viscosity subsolution of such an equation propagates along the trajectories of subunit vector fields. This implies strong maximum and minimum principles when the operator has a family of subunit vector fields satis...
Preprint
This paper deals with the periodic homogenization of nonlocal parabolic Hamilton-Jacobi equations with superlinear growth in the gradient terms. We show that the problem presents different features depending on the order of the nonlocal operator, giving rise to three different limit problems.
Chapter
We study the uniqueness of solutions to systems of PDEs arising in Mean Field Games with several populations of agents and Neumann boundary conditions. The main assumption requires the smallness of some data, e.g., the length of the time horizon. This complements the existence results for MFG models of segregation phenomena introduced by the author...
Article
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We study the uniqueness of solutions to systems of PDEs arising in Mean Field Games with several populations of agents and Neumann boundary conditions. The main assumption requires the smallness of some data, e.g., the length of the time horizon. This complements the existence results for MFG models of segregation phenomena introduced by the author...
Article
This paper presents a class of evolutive Mean Field Games with multiple solutions for all time horizons T and convex but non-smooth Hamiltonian H, as well as for smooth H and T large enough. The phenomenon is analyzed in both the PDE and the probabilistic setting. The examples are compared with the current theory about uniqueness of solutions. In p...
Article
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In this paper we consider a class of quasilinear elliptic systems of PDEs which arise in the mean eld games theory of J-M Lasry and P-L. Lions. We provide a wide range of su cient conditions for existence of solutions to these systems: on one hand the Hamiltonians ( rst-order terms) need to be at most quadratic in the gradients, on the other they c...
Article
Full-text available
This paper introduces and analyses some models in the framework of Mean Field Games describing interactions between two populations motivated by the studies on urban settlements and residential choice by Thomas Schelling. For static games, a large population limit is proved. For differential games with noise, the existence of solutions is establish...
Article
For a class of Bellman equations in bounded domains we prove that sub-and supersolutions whose growth at the boundary is suitably controlled must be constant. The ellipticity of the operator is assumed to degenerate at the boundary and a condition involving also the drift is further imposed. We apply this result to stochastic control problems, in p...
Article
We prove some Liouville properties for sub- and supersolutions of fully nonlinear degenerate elliptic equations in the whole space. Our assumptions allow the coefficients of the first order terms to be large at infinity, provided they have an appropriate sign, as in Ornstein- Uhlenbeck operators. We give two applications. The first is a stabilizati...
Article
We study zero-sum dynamic games with deterministic transitions and alternating moves of the players. Player 1 aims at reaching a terminal set and minimizing a possibly discounted running and final cost. We propose and analyze an algorithm that computes the value function of these games extending Dijkstra’s algorithm for shortest paths on graphs. We...
Article
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We derive a Liouville type result for a Bellman operator in a bounded smooth domain. The ellipticity of the operator is assumed to degenerate at the boundary and a condition involving also the drift is further imposed. We apply this result to stochastic control problems, in particular to an exit problem and to the small discount limit related with...
Chapter
We consider optimal control problems where the dynamical system and the running cost are affected by fast periodic oscillations of the state variables. We show that, under suitable controllability and structure assumptions, it is possible to describe the limiting optimal control problem. The proofs make use of results in the theory of homogenizatio...
Article
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We consider stochastic control systems affected by a fast mean reverting volatility $Y(t)$ driven by a pure jump L\'evy process. Motivated by a large literature on financial models, we assume that $Y(t)$ evolves at a faster time scale $\frac{t}{\varepsilon}$ than the assets, and we study the asymptotics as $\varepsilon\to 0$. This is a singular per...
Article
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We consider the short time behaviour of stochastic systems affected by a stochastic volatility evolving at a faster time scale. We study the asymptotics of a logarithmic functional of the process by methods of the theory of homogenisation and singular perturbations for fully nonlinear PDEs. We point out three regimes depending on how fast the volat...
Article
Full-text available
We consider stochastic differential games with $N$ players, linear-Gaussian dynamics in arbitrary state-space dimension, and long-time-average cost with quadratic running cost. Admissible controls are feedbacks for which the system is ergodic. We first study the existence of affine Nash equilibria by means of an associated system of $N$ Hamilton-Ja...
Article
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We study properties of functions convex with respect to a given family X of vector fields, a notion that appears natural in Carnot-Caratheodory metric spaces. We define a suitable subdifferential and show that a continuous function is X-convex if and only if such subdifferential is nonempty at every point. For vector fields of Carnot type we deduce...
Conference Paper
We consider stochastic differential games with N players, linear-Gaussian dynamics in arbitrary state-space dimension, and long-time-average cost with quadratic running cost. Admissible controls are feedbacks for which the system is ergodic. We first study the existence of affine Nash equilibria by means of an associated system of N Hamilton-Jacobi...
Article
Full-text available
We study fully nonlinear partial differential equations of Monge-Ampere type involving the derivatives with respect to a family X of vector fields. The main result is a comparison principle among viscosity subsolutions, convex with respect to X, and viscosity supersolutions (in a weaker sense than usual), which implies the uniqueness of solution to...
Article
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We study the homogenization of Hamilton-Jacobi equations with oscillating initial data and non-coercive Hamiltonian, mostly of the Bellman-Isaacs form arising in optimal control and differential games. We describe classes of equations for which pointwise homogenization fails for some data. We prove locally uniform homogenization for various Hamilto...
Article
Full-text available
We consider N -person differential games involving linear systems affected by white noise, running cost quadratic in the control and in the displacement of the state from a reference position, and with long-time-average integral cost functional. We solve an associated system of Hamilton-Jacobi-Bellman and Kolmogorov-Fokker-Plank equations and find...
Article
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Given a family of vector fields we introduce a notion of convexity and of semiconvexity of a function along the trajectories of the fields and give infinitesimal characterizations in terms of inequalities in viscosity sense for the matrix of second derivatives with respect to the fields. We also prove that such functions are Lipschitz continuous wi...
Article
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We introduce a notion of convexity in the geometry of vector fields and we prove a PDE-characterization for such notion and related properties.
Article
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We study singular perturbations of a class of stochastic control problems under assumptions motivated by models of financial markets with stochastic volatilities evolving on a fast time scale. We prove the convergence of the value function to the solution of a limit (effective) Cauchy problem for a parabolic equation of HJB type. We use methods of...
Article
Full-text available
We study the Dirichlet problem for subelliptic partial differential equations of Monge-Ampere type involving the derivates with respect to a family X of vector fields of Carnot type. The main result is a comparison principle among viscosity subsolutions, convex with respect to X, and viscosity supersolutions (in a weaker sense than usual), which im...
Article
Full-text available
We model the parameters of a control problem as an ergodic diffusion process evolving at a faster time scale than the state variables. We study the asymptotics as the speed of the parameters gets large. We prove the convergence of the value function to the solution of a limit Cauchy problem for a Hamilton-Jacobi equation whose Hamiltonian is a suit...
Chapter
In this Chapter we consider several optimal control problems whose value function is defined and continuous on the whole space ℝ N . This setting is suitable for those problems where no a priori constraint is imposed on the state of the control system. For all the problems considered we establish the Dynamic Programming Principle and derive from it...
Chapter
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The paper deals with the ergodicity of deterministic zero-sum differential games with long-time-average cost. Some new sufficient conditions are given, as well as a class of games that are not ergodic. In particular, we settle the issue of ergodicity for the simple games whose associated Isaacs equation is a convex-concave eikonal equation.
Article
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We study singular perturbations of optimal stochastic control problems and differential games arising in the dimension reduction of system with multiple time scales. We analyze the uniform convergence of the value functions via the associated Hamilton-Jacobi-Bellman-Isaacs equations, in the framework of viscosity solutions. The crucial properties o...
Chapter
In this paper we prove an estimate of the rate of convergence of the approximation scheme for the nonlinear minimum time problem presented in [2]. The estimate holds provided the system have time-optimal controls with bounded variation. This estimate is of order v with respect to the discretization step in time, if the minimal time function is Höld...
Article
The paper is devoted to singular perturbation problems with a finite number of scales where both the dynamics and the costs may oscillate. Under some coercivity assumptions on the Hamiltonian, we prove that the value functions converge locally uniformly to the solution of an effective Cauchy problem for a limit Hamilton-Jacobi equation and that the...
Article
Full-text available
We compare a general controlled diffusion process with a deterministic system where a second controller drives the disturbance against the first controller. We show that the two models are equivalent with respect to two properties: the viability (or controlled invariance, or weak invariance) of closed smooth sets, and the existence of a smooth cont...
Article
Full-text available
We present two comparison principles for viscosity sub- and supersolutions of Monge-Ampere-type equations associated to a family of vector fields. In particular, we obtain the uniqueness of a viscosity solution to the Dirichlet problem for the equation of prescribed horizontal Gauss curvature in a Carnot group.
Article
We prove a general convergence result for singular perturbations with an arbitrary number of scales of fully nonlinear degenerate parabolic PDEs. As a special case we cover the iterated homogenization for such equations with oscillating initial data. Explicit examples, among others, are the two-scale homogenization of quasilinear equations driven b...
Chapter
We present and study a notion of ergodicity for deterministic zero-sum differential games that extends the one in classical ergodic control theory to systems with two conflicting controllers.We show its connections with the existence of a constant and uniform long-time limit of the value function of finite horizon games, and characterize this prope...
Article
Full-text available
We prove some comparison principles for viscosity solutions of fully nonlinear degenerate elliptic equations that satisfy conditions of partial non-degeneracy instead of the usual uniform ellipticity or strict monotonicity. These results are applied to the well-posedness of the Dirichlet problem under suitable conditions at the characteristic point...
Chapter
We propose a differential game as a model for the optimal control of systems affected by an unpredictable disturbance, when the controller follows a policy of total risk aversion. We study the problem of small disturbances of a given deterministic system, and estimate the difference between the value functions of the unperturbed and of the perturbe...
Chapter
In this paper we consider the classical problem of pursuit and evasion for continuous-time and discrete-time systems. We prove the convergence, as the time step goes to 0, of the upper and lower value functions of the discrete-time game to the upper and lower values of the differential game. This is done assuming a capturability condition either on...
Chapter
The capture time T of a pursuit-evasion game is considered. Under a "capturability" condition it is shown that T is the unique viscosity solution of the Isaacs equation coupled with certain singular boundary conditions. Moreover the upper and lower values of the game are compared by PDE methods, as well as the corresponding "capturability" sets, an...
Article
Full-text available
We develop a direct Lyapunov method for the almost sure open-loop stabilizability and asymptotic stabilizability of controlled degenerate diffusion processes. The infinitesimal decrease condition for a Lyapunov function is a new form of Hamilton--Jacobi--Bellman partial differential inequality of second order. We give local and global versions of t...
Article
Full-text available
We present a notion of ergodicity for deterministic zero-sum differential games that extends the one in classical ergodic control theory to systems with two conflicting controllers. We describe its connections with the existence of a constant and uniform long-time limit of the value function of finite horizon games, and characterize this property i...
Article
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We study Hamilton-Jacobi equations with upper semicontinuous initial data without convexity assumptions on the Hamiltonian. We analyse the behavior of generalized u.s.c. solutions at the initial time t = 0, and find necessary and sufficient conditions on the Hamiltonian such that the solution attains the initial data along a sequence (right accessi...
Article
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The main result of the paper is a general convergence theorem for the viscosity solutions of singular perturbation problems for fully nonlinear degenerate parabolic PDEs (partial differential equations) with highly oscillating initial data. It substantially generalizes some results obtained previously in [2]. Under the only assumptions that the Ham...
Article
Full-text available
We describe the set of propagation of maxima of upper semicontinuous viscosity subsolutions of fully nonlinear, degenerate elliptic Hamilton-Jacobi-Bellman equations in the con-cave case, i.e., for operators represented as the minimum of a parametrized family of 2 nd order linear operators. We show that the set where an interior maximum propagates...
Article
Full-text available
The direct method of Lyapunov to investigate the stability of dynamical systems is extended to the almost sure stability of Itô stochastic differential equations by means of semicontinuous Lyapunov functions satisfying a suitable system of partial differential inequalities in viscosity sense. The almost sure stabilizability of controlled degenerate...
Article
For a general controlled diffusion process and an arbitrary closed set K we study the viability, or weak invariance, or controlled invariance, of K, that is, the existence of a control for each initial point in K keeping the trajectory forever in K. By viscosity solutions methods we prove a simple necessary and sufficient condition involving only a...
Article
We study the boundary value problem for the Hamilton-Jacobi-Isaacs equation of pursuit-evasion differential games with state constraints. We prove existence of a continuous viscosity solution and a comparison theorem that we apply to establish uniqueness of such a solution and its uniform approximation by solutions of discretized equations.
Article
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Viscosity solutions methods are used to pass to the limit in some penalization problems for first order and second order, degenerate parabolic, Hamilton-Jacobi-Bellman equations. This characterizes the limit of the value functions of singularly perturbed optimal control problems for nonlinear deterministic systems and controlled degenerate diffusio...
Article
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We present a class of numerical schemes for the Isaacs equation of pursuit-evasion games. We consider continuous value functions, where the solution is interpreted in the viscosity sense, as well as discontinuous value functions, where the notion of viscosity envelope-solution is needed. The convergence of the approximation scheme to the value func...
Article
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We prove a strong maximum principle for semicontinuous viscosity subsolutions or supersolutions of fully nonlinear degenerate elliptic PDE's, which complements the results of [17]. Our assumptions and conclusions are different from those in [17], in particular our maximum principle implies the nonexistence of a dead core. We test the assumptions on...
Article
We study the behavior at boundary points of viscosity sub- and supersolutions of fully nonlinear degenerate elliptic equations and find a subset of the boundary where the equation is automatically satisfied. Then we prove comparison and uniqueness theorems where the solution is prescribed only on the complement Gamma of this part of the boundary. W...
Article
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: We study invariance and viability properties of a closed set for the trajectories of either a controlled diffusion process or a controlled deterministic system with disturbances. We use the value functions associated to suitable optimal control problems or differential games and analyze the related Dynamic Programming equation within the theory o...
Article
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. We study general 2nd order fully nonlinear degenerate elliptic equations on an arbitrary closed set with generalized Dirichlet boundary conditions in the viscosity sense. We prove some properties of the maximal subsolution and the minimal supersolution of the Dirichlet type problem. Under a sort of compatibility condition on the boundary data we...
Article
We study the singular perturbation of optimal control problems for nonlinear systems with constraints on the fast state variables and a cost functional either of Bolza type or involving the exit time of the system from a given domain. Under a controllability assumption on the fast variables, we show that these variables become controls in the limit...
Article
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Simple explicit estimates are presented for the viscosity solution of the Cauchy problem for the Hamilton–Jacobi equation where either the Hamiltonian or the initial data are the sum of a convex and a concave function. The estimates become equalities whenever a "minmax" equals a "maxmin" and thus a representation formula for the solution is obtaine...
Article
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We construct a generalized viscosity solution of the Dirichlet problem for fully nonlinear degenerate elliptic equations in general domains by the Perron-Wiener- Brelot method. The result is designed for the Hamilton-Jacobi-Bellman-Isaacs equations of time-optimal stochastic control and differential games with discon- tinuous value function. We stu...
Article
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We consider a class of finite horizon optimal control problems with unbounded data for nonlinear systems, which includes the Linear-Quadratic (LQ) problem. We give comparison results between the value function and viscosity sub- and supersolutions of the Bellman equation, and prove uniqueness for this equation among locally Lipschitz functions boun...
Article
In this chapter we consider some approximation and perturbation problems for Hamilton-Jacobi equations. The results are presented for the model case lu( x ) + sup{ - f( x,a ) ·Du( x ) - l( x,a ) } = 0 in \mathbbRN ,\lambda u\left( x \right) + sup\left\{ { - f\left( {x,a} \right) \cdot Du\left( x \right) - \ell \left( {x,a} \right)} \right\} = 0 i...
Article
In this chapter we extend the theory of continuous viscosity solutions developed in Chapter II to include solutions that are not necessarily continuous. This has two motivations. The first is that many optimal control problems have a discontinuous value function and we want to extend to these problems the results of Chapters III and IV, in particul...
Article
In this chapter we consider several asymptotic problems in optimal control. Our approach is to pass to the limit as the relevant parameter goes to zero in the Hamilton-Jacobi-Bellman equation satisfied by the value function and characterize the limit value function as the viscosity solution of the limit equation.
Article
The purpose of this introductory chapter is to motivate the relevance of the notion of viscosity solution of partial differential equations of the form F( x,u( x ),Du( x ) ) = 0F\left( {x,u\left( x \right),Du\left( x \right)} \right) = 0 in a Dynamic Programming approach to deterministic optimal control theory.
Book
Preface.- Basic notations.- Outline of the main ideas on a model problem.- Continuous viscosity solutions of Hamilton-Jacobi equations.- Optimal control problems with continuous value functions: unrestricted state space.- Optimal control problems with continuous value functions: restricted state space.- Discontinuous viscosity solutions and applica...
Chapter
In this chapter we consider two-person zero-sum differential games. Let us describe them.
Article
We describe an approximation scheme for the value function of general pursuit-evasion games and prove its convergence, in a suitable sense. The result works for problems with discontinuous value function as well, and it is new even for the case of a single player. We use some very recent results on generalized (viscosity) solutions of the Dirichlet...
Article
We consider the classical pursuit-evasion problem and an approximation scheme based on Dynamic Programming. We prove the convergence of the scheme to the value function of the game by using some recent results and methods of the theory of viscosity solutions to the Isaacs equations. The more restrictive assumption is the continuity of the value fun...
Article
Full-text available
For a general nonlinear system and closed target set ℐ we study the value functions ν and {Mathematical expression} of the control problems of reaching ℐ and, respectively, its interior, in minimum time. Under no controllability assumptions on the system, we characterize them as, respectively, the minimal viscosity supersolution and the maximal vis...
Article
For a large class of partial differential equations on exterior domains or on ℝN we show that any solution tending to a limit from one side as x goes to infinity satisfies the property of “asymptotic spherical symmetry”. The main examples are semilinear elliptic equations, quasilinear degenerate elliptic equations, and first-order Hamilton-Jacobi e...
Article
We study the stability of viscosity solutions of Dirichlet problems for HamiltonJacobi equations with respect to perturbations of the domain where the equation is set, of the boundary data and the Hamiltonian. We give estimates showing that the rate of convergence of the solutions depends only on the rate of convergence of the data and on the expon...
Article
We present two convergence theorems for Hamilton-Jacobi equations and we apply them to the convergence of approximations and perturbations of optimal control problems and of two-players zero-sum differential games. One of our results is, for instance, the following. LetT andT h be the minimal time functions to reach the origin of two control system...
Article
A class of Hamilton-Jacobi equations arising in generalized timeoptimal control problems and differential games is considered. The natural global boundary value problem for these equations has a singular boundary condition on a free boundary. The uniqueness of the continuous solution and of the free boundary is proved in the framework of viscosity...
Article
Full-text available
Simple inequalities are presented for the viscosity solution of a Hamilton-Jacobi equation in N space dimensions when neither the initial data nor the Hamiltonian need be convex (or concave). The initial data are uniformly Lipschitz and can be written as the sum of a convex function in a group of variables and a concave function in the remaining va...
Article
This paper presents an approximation scheme for the nonlinear minimum time problem with compact target. The cheme is derived from a discrete dynamic programming principle and the main convergence result is obtained by applying techniques related to discontinuous viscosity solutions for Hamilton-Jacobi equations. The convergence is proved under gene...
Article
The theory of phase transitions leads to consider semilinear elliptic equations of the form [Formula omitted], or, in the constrained case, [Formula omitted] Under suitable assumptions on f and M, it has been proved that converges pointwise to a function u0(x) defined for a given xo by [Formula omitted].
Article
A natural boundary value problem for the dynamic programming partial differential equation associated with the minimum time problem is proposed. The minimum time function is shown to be the unique viscosity solution of this boundary value problem.
Article
A general class of nonlinear degenerate parabolic equations in many space dimensions is considered and two main results concerning the free boundary are proved: (i) the eventual Lipschitz continuity in the space variable, (ii) the asymptotic spherical symmetry in a stronger sense than the almost radiality proved by Aronson & Caffarelli [2] for the...
Article
A nonlinear periodic functional differential equation with unbounded delay describing the growth of a single species with depensation is considered. The global bifurcation of positive periodic solutions from the null one is studied and the differences from logistic-type equations are shown, namely the multiplicity of non-trivial solutions and the o...
Article
The method of moving parallel planes, previously used for elliptic and parabolic PDE, is adapted to study solutions of the Cauchy problem for Hamilton-Jacobi equations. This is possible in the framework of the theory of viscosity solutions, using the comparison theorem for such solutions as a kind of maximum principle. One of the main results state...
Article
Various biological phenomena lead to single species models where the relative rate of increase is a non-monotone function of the density, i.e. depensation models. A brief survey of the literature and some new models are given. A nonlinear nonautonomous O.D.E. is proposed as a general depensation model in a periodically fluctuating environment. Resu...
Article
A class of ordinary or integrodifferential equations describing predator-prey dynamics is considered under the assumption that the coefficients are periodic functions of time. This class is characterized by the logistic behaviour of the prey in the absence of predators and it includes the Leslie model. We show that there exists a periodic solution...

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