Convergence/divergence phenomena in the vanishing discount limit of Hamilton-Jacobi equations
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910707147079680 |
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| author | Davini, Andrea Ni, Panrui Yan, Jun Zavidovique, Maxime |
| author_facet | Davini, Andrea Ni, Panrui Yan, Jun Zavidovique, Maxime |
| contents | We study the asymptotic behavior of solutions of an equation of the form \begin{equation}\label{abs}\tag{*} G\big(x, D_x u,λu(x)\big) = c_0\qquad\hbox{in $M$} \end{equation} on a closed Riemannian manifold $M$, where $G\in C(T^*M\times\mathbb{R})$ is convex and superlinear in the gradient variable, is globally Lipschitz but not monotone in the last argument, and $c_0$ is the critical constant associated with the Hamiltonian $H:=G(\cdot,\cdot,0)$. By assuming that $\partial_u G(\cdot,\cdot,0)$ satisfies a positivity condition of integral type on the Mather set of $H$, we prove that any equi-bounded family of solutions of \eqref{abs} uniformly converges to a distinguished critical solution $u_0$ as $λ\to 0^+$. We furthermore show that any other possible family of solutions uniformly diverges to $+\infty$ or $-\infty$. We then look into the linear case $G(x,p,u):=a(x)u + H(x,p)$ and prove that the family $(u_λ)_{λ\in (0,λ_0)}$ of maximal solutions to \eqref{abs} is well defined and equi-bounded for $λ_0>0$ small enough. When $a$ changes sign and enjoys a stronger localized positivity assumption, we show that equation \eqref{abs} does admit other solutions too, and that they all uniformly diverge to $-\infty$ as $λ\to 0^+$. This is the first time that converging and diverging families of solutions are shown to coexist in such a generality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_13780 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convergence/divergence phenomena in the vanishing discount limit of Hamilton-Jacobi equations Davini, Andrea Ni, Panrui Yan, Jun Zavidovique, Maxime Analysis of PDEs We study the asymptotic behavior of solutions of an equation of the form \begin{equation}\label{abs}\tag{*} G\big(x, D_x u,λu(x)\big) = c_0\qquad\hbox{in $M$} \end{equation} on a closed Riemannian manifold $M$, where $G\in C(T^*M\times\mathbb{R})$ is convex and superlinear in the gradient variable, is globally Lipschitz but not monotone in the last argument, and $c_0$ is the critical constant associated with the Hamiltonian $H:=G(\cdot,\cdot,0)$. By assuming that $\partial_u G(\cdot,\cdot,0)$ satisfies a positivity condition of integral type on the Mather set of $H$, we prove that any equi-bounded family of solutions of \eqref{abs} uniformly converges to a distinguished critical solution $u_0$ as $λ\to 0^+$. We furthermore show that any other possible family of solutions uniformly diverges to $+\infty$ or $-\infty$. We then look into the linear case $G(x,p,u):=a(x)u + H(x,p)$ and prove that the family $(u_λ)_{λ\in (0,λ_0)}$ of maximal solutions to \eqref{abs} is well defined and equi-bounded for $λ_0>0$ small enough. When $a$ changes sign and enjoys a stronger localized positivity assumption, we show that equation \eqref{abs} does admit other solutions too, and that they all uniformly diverge to $-\infty$ as $λ\to 0^+$. This is the first time that converging and diverging families of solutions are shown to coexist in such a generality. |
| title | Convergence/divergence phenomena in the vanishing discount limit of Hamilton-Jacobi equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2411.13780 |