Stochastic homogenization of Hamilton–Jacobi and degenerate Bellman equations in unbounded environments

We consider the homogenization of Hamilton–Jacobi equations and degenerate Bellman equations in stationary, ergodic, unbounded environments. We prove that, as the microscopic scale tends to zero, the equation averages to a deterministic Hamilton–Jacobi equation and study some properties of the effective Hamiltonian. We discover a connection between the effective Hamiltonian and an eikonal-type equation in exterior domains. In particular, we obtain a new formula for the effective Hamiltonian. To prove the results we introduce a new strategy to obtain almost sure homogenization, completing a program proposed by Lions and Souganidis that previously yielded homogenization in probability. The class of problems we study is strongly motivated by Sznitman?s study of the quenched large deviations of Brownian motion interacting with a Poissonian potential, but applies to a general class of problems which are not amenable to probabilistic tools.     

Envelopes and nonconvex Hamilton–Jacobi equations

This paper introduces a new representation formula for viscosity solutions of nonconvex Hamilton–Jacobi PDE using “generalized envelopes” of affine solutions. We study as well envelope and singular characteristic constructions of equivocal surfaces and discuss also differential game theoretic interpretations. In memory of Arik A. Melikyan.      

Solving a Hamilton–Jacobi–Bellman equation with constraints

Optimal investment strategies for an insurer with state-dependent constraints are computed via a recursive finite difference solution to the corresponding discretized Hamilton–Jacobi–Belman equation. Convergence is derived from viscosity solution arguments. For this, a comparison result is given which is similar to the result given by Azcue and Muler [Ann. Appl. Probab. 20 (2010), pp. 1253–1302].     

Singular boundary characteristics of the Hamilton–Jacobi equation

Situations exist in boundary value problems for first order partial differential equations arising in physics (the Hamilton–Jacobi equation), optimal control theory (the Bellman equation) and the theory of differential games (the Isaacs equation) when the value of the required function is not given on a part of the boundary or not at all, or it is not the limit of the (generalized) solution of the problem. Nevertheless, such conditions are required for constructing the solution (by the method of characteristics, for example). It is shown that the required boundary values can be exposed as a specific continuation of the conditions that are known in the boundary submanifolds of the given part of the boundary. This extension of the conditions is accomplished using the characteristic curves starting in a known submanifold of the boundary and running along the boundary. The characteristics are a generalization of the classical characteristics associated with a partial differential equation. They are called singular characteristics, and the theory of these has been developed in a number of the author's papers. After obtaining these “natural” boundary conditions, the solution is constructed using the conventional method of integrating the equations of the classical characteristics. Conditions of the Dirichlet and Neumann type are considered. The technique is illustrated using a numerical example from the theory of differential games containing a number of parameters.     

Lagrangian submanifolds and the Hamilton–Jacobi equation

Lagrangian submanifolds are becoming a very essential tool to generalize and geometrically understand results and procedures in the area of mathematical physics. The geometric version of the Hamilton–Jacobi equation in terms of Lagrangian submanifolds enables here some novel interesting applications of the Hamilton–Jacobi equation in holonomic, nonholonomic and time-dependent dynamics from a geometrical point of view.     

Discontinuous solutions of Hamilton–Jacobi–Bellman equation under state constraints

This article is devoted to the Hamilton–Jacobi partial differential equation $$\left\{\begin{array}{lll}\frac{\partial V}{\partial t} = H\left(t, x, - \frac{\partial V}{\partial x}\right) & \hbox{on} & [0, 1]\times {\overline{\Omega}}\\V(1, x) = g(x) & \hbox{on}& {\overline{\Omega}},\end{array}\right.$$ where the Hamiltonian ${{H:[0, 1] \times \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}}}$ is convex and positively homogeneous with respect to the last variable, ${{\Omega \subset \mathbb{R}^n}}$ is open and ${{g : \mathbb{R}^n \to \mathbb{R} \cup \{+ \infty\}}}$ is lower semicontinuous. Such Hamiltonians do arise in the optimal control theory. We apply the method of generalized characteristics to show uniqueness of lower semicontinuous solution of this first order PDE. The novelty of our setting lies in the fact that we do not ask regularity of the boundary of Ω and extend the Soner inward pointing condition in a nontraditional way to get uniqueness in the class of lower semicontinuous functions.     

Particular solutions of the Hamilton–Jacobi equation and their usage

We show the possibility of using particular solutions of the Hamilton–Jacobi equation in problems of qualitative analysis of Lagrangian systems with cyclic first integrals. We present a procedure for finding and studying invariant manifolds of such systems. The efficiency of the suggested approach is illustrated by examples of the solution of specific problems.     

A stochastic Hamilton–Jacobi equation with infinite speed of propagation

We give an example of a stochastic Hamilton–Jacobi equation $\mathrm{d}u=H(\mathrm{D}u)\mathrm{d}\xi$ which has an infinite speed of propagation as soon as the driving signal ξ is not of bounded variation.     

Self-similar extinction for a diffusive Hamilton–Jacobi equation with critical absorption

The behavior near the extinction time is identified for non-negative solutions to the diffusive Hamilton–Jacobi equation with critical gradient absorption

$$\begin{aligned} \partial _tu-\Delta _p u+|\nabla u|^{p-1}=0 \quad \hbox {in}~ (0,\infty )\times \mathbb {R}^N, \end{aligned}$$

and fast diffusion \(2N/(N+1)<p<2\). Given a non-negative and radially symmetric initial condition with a non-increasing profile which decays sufficiently fast as \(|x|\rightarrow \infty \), it is shown that the corresponding solution u to the above equation approaches a uniquely determined separate variable solution of the form

$$\begin{aligned} U(t,x)=(T_e-t)^{1/(2-p)}f_*(|x|), \quad (t,x)\in (0,T_e)\times \mathbb {R}^N, \end{aligned}$$

as \(t\rightarrow T_e\), where \(T_e\) denotes the finite extinction time of u. A cornerstone of the convergence proof is an underlying variational structure of the equation. Also, the selected profile \(f_*\) is the unique non-negative solution to a second order ordinary differential equation which decays exponentially at infinity. A complete classification of solutions to this equation is provided, thereby describing all separate variable solutions of the original equation. One important difficulty in the uniqueness proof is that no monotonicity argument seems to be available and it is overcome by the construction of an appropriate Pohozaev functional.

     

On the topology of the set of singularities of a solution to the Hamilton–Jacobi equation

We address the topology of the set of singularities of a solution to a Hamilton–Jacobi equation. For this, we will apply the idea of the first two authors (Cannarsa and Cheng, Generalized characteristics and Lax–Oleinik operators: global result, preprint, arXiv:1605.07581, 2016) to use the positive Lax–Oleinik semi-group to propagate singularities.     

Gradient blowup rate for a viscous Hamilton–Jacobi equation with degenerate diffusion

This paper is concerned with the gradient blowup rate for the one-dimensional p-Laplacian parabolic equation ${u_t=(|u_x|^{p-2} u_x)_x +|u_x|^q}$ with q > p > 2, for which the spatial derivative of solutions becomes unbounded in finite time while the solutions themselves remain bounded. We establish the blowup rate estimates of lower and upper bounds and show that in this case the blowup rate does not match the self-similar one.     

Approximation of a Multivalued Solution of the Hamilton–Jacobi Equation

Mathematical Notes - The paper deals with the construction of a multivalued solution of the Cauchy problem for the Hamilton–Jacobi equation with discontinuous Hamiltonian with respect to the...     

Probabilistic interpretation of a coupled system of Hamilton–Jacobi–Bellman equations

In this paper, we give a probabilistic interpretation for a coupled system of Hamilton–Jacobi–Bellman equations using the value function of a stochastic control problem. First we introduce this stochastic control problem. Then we prove that the value function of this problem is deterministic and satisfies a (strong) dynamic programming principle. And finally, the value function is shown to be the unique viscosity solution of the coupled system of Hamilton–Jacobi–Bellman equations.