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Monge solutions for discontinuous Hamiltonians. (English) Zbl 1087.35023

The authors consider the Hamilton-Jacobi equation \[ H(x,Du)=0,\;x\in \Omega\subset\mathbb{R}^N \tag{1} \] where \(Du\) is the gradient of the unknown function \(u:\Omega\to\mathbb{R}\) and \(H:\overline\Omega\times \mathbb{R}^N\to\mathbb{R}\), \(H=H(x,p)\) is an Hamiltonian assumed to be only Borel measurable and quasi-convex in the \(p\)-variable for every \(x\in\overline \Omega\). The interest in this issue is easily motivated by the applications: Hamilton-Jacobi equations with discontinuous ingredients arise naturally in several models, as, for example, propagation of fronts in nonhomogeneous media, geometric optics in the presence of layers, shape-from-shading problems. In this work the authors want to extend the definition of Monge solution, introduced by Newcomb and Su, to equations of the more general form (1). The properties of Monge sub- and supersolutions strictly depend on the optical length function. This function is a geodesic, nonsymmetric distance. The definition of Monge solution reduces to the viscosity one when the Hamiltonian is continuous. The authors establish the comparison principle for Monge sub- and supersolutions, existence and uniqueness for equation (1) coupled with Dirichlet boundary conditions, and a stability result. The relation among Monge and Lipschitz subsolutions is also discussed. The paper ends with some examples. In particular, it is shown how Monge solutions of certain eikonal equations arise naturally as asymptotic limit of viscosity solutions of evolutive Hamilton-Jacobi equations with continuous ingredients.

MSC:

35F30 Boundary value problems for nonlinear first-order PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B50 Maximum principles in context of PDEs
35C15 Integral representations of solutions to PDEs
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
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