We investigate a singular perturbation for Hamilton-Jacobi equations in an open subset of two dimensional Euclidean space, where the set is determined through a Hamiltonian and the Hamilton-Jacobi equations are the dynamic programming equations for optimal control of the Hamiltonian flow of the Hamiltonian. We establish the convergence of solutions of the Hamilton-Jacobi equations and identify the limit of the solutions as the solution of systems of ordinary differential equations on a graph. The perturbation is singular in the sense that the domain degenerates to the graph in the limiting process. Our result can be seen as a perturbation analysis, in the viewpoint of optimal control, of the Hamiltonian flow.for a system of odes on the graph. The result is stated in Theorem 3.5. The argument for establishing this result depends heavily on viscosity solution techniques including the perturbed test function method as well as representations, as value functions in optimal control, of solutions of (HJ ε ).In [2], authors treat a problem similar to the above. They consider general Hamilton-Jacobi equations in optimal control on an unbounded thin set converging to a graph, prove the convergence of the solutions, and identify the limit of the solutions.An interesting point of our result lies in that we have to treat a non-coercive Hamiltonian in Hamilton-Jacobi equation (HJ ε ). Many authors in their studies on Hamilton-Jacobi equations on graphs, including [2], assumes, in order to guarantee the existence of continuous solutions, a certain coercivity of the Hamiltonian in the Hamilton-Jacobi equations, which corresponds, in terms of optimal control, a certain controllability of the dynamics. In our result, we also make a coercivity assumption (see (G4) below) on the unperturbed Hamiltonian, called G in (HJ ε ), but, because of the term b · Du ε /ε in (HJ ε ), when ε > 0 is very small, the perturbed Hamiltonian becomes non-coercive and the controllability of the dynamics (1.1) breaks down. A crucial point in our study is that, when ε is very small, the perturbed term b · Du ε /ε makes the solution u ε nearly constant along the level set of the Hamiltonian H while the perturbed Hamiltonian −b(x) · p/ε + G(x, p) in (HJ ε ) is "coercive in the direction orthogonal to b(x)", that is the direction of the gradient DH(x). Heuristically at least, these two characteristics combined together allow us to analyze the asymptotic behavior of the solution u ε of (HJ ε ) as ε → 0+. This paper is organized as follows. In the next section, we describe precisely the Hamitonian H, the domain Ω as well as some relevant properties of the Hamiltonian system (HS), present the assumptions on the unperturbed Hamiltonian G used throughout the paper, and give a basic existence and uniqueness proposition (see Proposition 2.3) for (HJ ε ) and a proposition concerning the dynamic programming principle. Section 3, which is divided into two parts, is devoted to establishing Theorem 3.5. In the first part, we give some observations on the odes (3.1 i ) ...
We investigate the asymptotic behavior of solutions of Hamilton-Jacobi equations with large drift term in an open subset of two-dimensional Euclidean space. When the drift is given by ε −1 (H x 2 , −H x 1 ) of a Hamiltonian H, with ε > 0, we establish the convergence, as ε → 0+, of solutions of the Hamilton-Jacobi equations and identify the limit of the solutions as the solution of systems of ordinary differential equations on a graph. This result generalizes the previous one obtained by the author to the case where the Hamiltonian H admits a degenerate critical point and, as a consequence, the graph may have segments more than four at a node.
We consider the nonlinear Neumann problem for fully nonlinear elliptic PDEs on a quadrant. We establish a comparison theorem for viscosity sub and supersolutions of the nonlinear Neumann problem. The crucial argument in the proof of the comparison theorem is to build a C 1,1 test function which takes care of the nonlinear Neumann boundary condition. A similar problem has been treated on a general n-dimensional orthant by Biswas, Ishii, Subhamay, and Wang [SIAM J. Control Optim. 55 (2017), pp. 365-396], where the functions (H i in the main text) describing the boundary condition are required to be positively onehomogeneous, and the result in this paper removes the positive homogeneity in two-dimension. An existence result for solutions is also presented.
We study the asymptotic behavior of solutions to the Dirichlet problem for Hamilton-Jacobi equations with large drift terms, where the drift terms are given by divergence-free vector fields. This is an attempt to understand the averaging effect for fully nonlinear degenerate elliptic equations. In this work, we restrict ourselves to the case of Hamilton-Jacobi equations. The second author has already established averaging results for Hamilton-Jacobi equations with convex Hamiltonians (G below) under the classical formulation of the Dirichlet condition. Here we treat the Dirichlet condition in the viscosity sense and establish an averaging result for Hamilton-Jacobi equations with relatively general Hamiltonian G.
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