On viscosity solutions of Hamilton-Jacobi equations

By S. Zagatti. 2008

1,266 citations recorded by the source corpus.

We consider the Dirichlet problem for Hamilton-Jacobi equations and prove existence, uniqueness and continuous dependence on boundary data of Lipschitz continuous maximal viscosity solutions.

Subfields: Applied Mathematics, Computational Theory and Mathematics

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Manitest: Are classifiers really invariant? (2015, background): “Moreover, it was proved in [7] that the geodesic map u is the unique viscosity solution of the Eikonal equation, provided that τ → G ( τ ) is continuous.”
Multiplayer reach-avoid games via low dimensional solutions and maximum matching (2014, background): “Let Φ : R × [−T, 0] → R be the viscosity solution [23] to the constrained terminal value HJI PDE:”
Nonlinear Optimal Control Theory (2012, methodology): “5 The Value as Viscosity Solution The definition of viscosity solution of a nonlinear partial differential equation was first given by Crandall and Lions in [28].”
A review of numerical methods for nonlinear partial differential equations (2012, positive): “As an example, we mention developments of new geometric tools in nonlinear PDEs [98, 37, 210, 164], fully nonlinear elliptic equations, optimal transportation and Navier–Stokes equations [57, 29, 218, 139] as examples which should pave the way for future developments in numerical methods.”
Optimal Feedback Control: Foundations, Examples, and Experimental Results for a New Approach (2008, background): “This issue was resolved in 1983 by Crandall and Lions [18], who developed the notion of viscosity solutions to theHJB equation.”
On the Homogenization of the Hamilton-Jacobi Equation (2019, background): “We refer interested readers, for example, to [10, 21].”
Control: A perspective (2014, methodology): “The problem of existence of solutions to the Hamilton-Jacobi-Bellman equations was addressed by the viscosity approach [153,431,430,432].”
Generalized backward doubly stochastic differential equations and SPDEs with nonlinear Neumann boundary conditions (2007, methodology): “The notion of viscosity solution for PDEs was introduced by Crandall and Lions [3] for certain first-order Hamilton–Jacobi equations.”
Recent progresses in boundary layer theory (2015, background): “[53] and in the references therein the issue of boundary layers for hyperbolic equations; the issue already mentioned of the numerical approximation of singularly perturbed problems, in particular in the context of enriched spaces [18, 19, 60, 61, 62, 64, 65, 66, 67, 71, 74, 76, 85, 104, 105, 106, 1”
ASYMPTOTIC SOLUTIONS FOR LARGE-TIME OF HAMILTON-JACOBI EQUATIONS IN EUCLIDEAN $n$ SPACE(Viscosity Solution Theory of Differential Equations and its Developments) (2007, positive): “Lions [10], [11] introduced the notion of viscosity solution in the study of Hamilton–Jacobi equations.”