Viscosity solutions of Hamilton-Jacobi equations.

*(English)*Zbl 0599.35024The authors deal with the Dirichlet problem for the Hamilton-Jacobi equation, \(H(x,u,Du)=0\) in \(\Omega\) and \(u=z\) on \(\partial \Omega\), and with the corresponding Cauchy problem (for the equation \(u_ t+H(x,t,u,Du)=0)\). In general these problems do not have classical solutions, but under quite general conditions they possess generalized solutions, i.e. solutions which are locally Lipschitzian and satisfy the equation almost everywhere. Usually, the generalized solutions are not unique.

In the present paper the authors introduce a notion of weak solution (which they call a ”viscosity solution”) for which they establish uniqueness and stability in the sense of continuous dependence on data. A viscosity solution is by assumption continuous, but need not be differentiable anywhere. However, a viscosity solution which is locally Lipschitzian will satisfy the equation almost everywhere. Generalized solutions of the above problem that are obtained by the well-known method of vanishing viscosity belong to the class of viscosity solutions in the sense of the present paper. The class of weak solutions introduced in this paper is closely related to the class of weak solutions introduced by L. C. Evans [Isr. J. Math. 36, 225-247 (1980; Zbl 0454.35038)].

In the present paper the authors introduce a notion of weak solution (which they call a ”viscosity solution”) for which they establish uniqueness and stability in the sense of continuous dependence on data. A viscosity solution is by assumption continuous, but need not be differentiable anywhere. However, a viscosity solution which is locally Lipschitzian will satisfy the equation almost everywhere. Generalized solutions of the above problem that are obtained by the well-known method of vanishing viscosity belong to the class of viscosity solutions in the sense of the present paper. The class of weak solutions introduced in this paper is closely related to the class of weak solutions introduced by L. C. Evans [Isr. J. Math. 36, 225-247 (1980; Zbl 0454.35038)].

##### MSC:

35F20 | Nonlinear first-order PDEs |

35D05 | Existence of generalized solutions of PDE (MSC2000) |

35F30 | Boundary value problems for nonlinear first-order PDEs |

##### Keywords:

Dirichlet problem; Hamilton-Jacobi equation; Cauchy problem; generalized solutions; weak solution; viscosity solution
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\textit{M. G. Crandall} and \textit{P.-L. Lions}, Trans. Am. Math. Soc. 277, 1--42 (1983; Zbl 0599.35024)

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##### References:

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