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Lipschitz dependence of viscosity solutions of Hamilton-Jacobi equations with respect to the parameter

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  • Let $M$ be a closed and smooth manifold and $H_\varepsilon:T^*M\to\mathbf{R}^1$ be a family of Tonelli Hamiltonians for $\varepsilon\geq0$ small. For each $\varphi\in C(M,\mathbf{R}^1)$, $T^\varepsilon_t\varphi(x)$ is the unique viscosity solution of the Cauchy problem \begin{align*} \left\{ \begin{array}{ll} d_tw+H_\varepsilon(x,d_xw)=0, & \ \mathrm{in}\ M\times(0,+\infty),\\ w|_{t=0}=\varphi, & \ \mathrm{on}\ M, \end{array} \right. \end{align*} where $T^\varepsilon_t$ is the Lax-Oleinik operator associated with $H_\varepsilon$. A result of Fathi asserts that the uniform limit, for $t\to+\infty$, of $T^\varepsilon_t\varphi+c_\varepsilon t$ exists and the limit $\bar{\varphi}_\varepsilon$ is a viscosity solution of the stationary Hamilton-Jacobi equation \begin{align*} H_\varepsilon(x,d_xu)=c_\varepsilon, \end{align*} where $c_\varepsilon$ is the unique $k$ for which the equation $H_\varepsilon(x,d_xu)=k$ admits viscosity solutions. In the present paper we discuss the continuous dependence of the viscosity solution $\bar{\varphi}_\varepsilon$ with respect to the parameter $\varepsilon$.
    Mathematics Subject Classification: Primary: 37J50; Secondary: 70H20.


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