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Uniqueness of the viscosity solution of a constrained Hamilton–Jacobi equation
Calculus of Variations and Partial Differential Equations ( IF 2.1 ) Pub Date : 2020-09-11 , DOI: 10.1007/s00526-020-01819-0
Vincent Calvez , King-Yeung Lam

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(Ixp) with respect to p, monotonicity of H with respect to I, and BV regularity of I(t).



中文翻译:

约束的Hamilton–Jacobi方程的粘度解的唯一性

在定量遗传学中,当种群方差消失时,汉密尔顿-雅各比方程的粘度解自然出现在选择突变模型的渐近极限中。它们必须与未知函数It)一起求解,该未知函数It)每次都对应于解决方案的非负约束。尽管对于Hamilton-Jacobi方程的许多变体,粘度解的唯一性是已知的,但除了少数特殊情况外,这种特殊类型的约束问题的唯一性还没有解决。在此,我们基于三个主要假设为唯一性问题提供一个通用答案:哈密顿函数HI,  x,  p)相对于p,的单调性ħ相对于,和BV规律性的)。

更新日期:2020-09-12
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