Uniqueness of the viscosity solution of a constrained Hamilton-Jacobi equation - Archive ouverte HAL
Article Dans Une Revue Calculus of Variations and Partial Differential Equations Année : 2020

Uniqueness of the viscosity solution of a constrained Hamilton-Jacobi equation

Résumé

In quantitative genetics, viscosity solutions of Hamilton-Jacobi equations appear naturally in the asymptotic limit of selection-mutation models when the population variance vanishes. They have to be solved together with an unknown function I(t) that arises as the counterpart of a non-negativity constraint on the solution at each time. Although the uniqueness of viscosity solutions is known for many variants of Hamilton-Jacobi equations, the uniqueness for this particular type of constrained problem was not resolved, except in a few particular cases. Here, we provide a general answer to the uniqueness problem, based on three main assumptions: convexity of the Hamiltonian function H(I, x, p) with respect to p, monotonicity of H with respect to I, and BV regularity of I(t).
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Dates et versions

hal-01873119 , version 1 (12-09-2018)

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Vincent Calvez, King-Yeung Lam. Uniqueness of the viscosity solution of a constrained Hamilton-Jacobi equation. Calculus of Variations and Partial Differential Equations, 2020, ⟨10.1007/s00526-020-01819-0⟩. ⟨hal-01873119v1⟩
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