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VARIATIONAL AND VISCOSITY OPERATORS FOR THE EVOLUTIVE HAMILTON-JACOBI EQUATION

Abstract : We study the Cauchy problem for the first order evolutive Hamilton-Jacobi equation with a Lipschitz initial condition. The Hamiltonian is not necessarily convex in the momentum variable and not a priori compactly supported. We build and study an operator giving a variational solution of this problem, and get local Lipschitz estimates on this operator. Iterating this variational operator we obtain the viscosity operator and extend the estimates to the viscosity framework. We also check that the construction of the variational operator gives the Lax-Oleinik semigroup if the Hamiltonian is convex or concave in the momentum variable.
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https://hal.archives-ouvertes.fr/hal-01637617
Contributor : Valentine Roos Connect in order to contact the contributor
Submitted on : Monday, January 29, 2018 - 2:39:06 PM
Last modification on : Thursday, March 17, 2022 - 10:08:19 AM
Long-term archiving on: : Monday, April 30, 2018 - 12:10:46 PM

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  • HAL Id : hal-01637617, version 1
  • ARXIV : 1801.09889

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Valentine Roos. VARIATIONAL AND VISCOSITY OPERATORS FOR THE EVOLUTIVE HAMILTON-JACOBI EQUATION. Communications in Contemporary Mathematics, In press. ⟨hal-01637617⟩

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