A PDE formulation of Lyapunov stability for contact-type Hamilton-Jacobi equations
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| Format: | Preprint |
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2026
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| _version_ | 1866908995349905408 |
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| author | Ni, Panrui Yan, Jun |
| author_facet | Ni, Panrui Yan, Jun |
| contents | We study the Lyapunov stability of stationary solutions to contact-type Hamilton-Jacobi equations on a compact manifold. Previous works typically assume $C^3$ Tonelli Hamiltonians and characterize stability in terms of Mather measures. In this paper, we consider continuous, convex and coercive Hamiltonians and establish verifiable PDE-type criteria for both stability and instability. In particular, the dynamical conditions involving Mather measures are replaced by conditions expressed in terms of the critical value of the Hamiltonian and viscosity subsolutions. This provides a PDE-based framework for stability analysis and reveals connections with various asymptotic behaviors of viscosity solutions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_24329 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A PDE formulation of Lyapunov stability for contact-type Hamilton-Jacobi equations Ni, Panrui Yan, Jun Analysis of PDEs 35F21, 37J51, 35B40 We study the Lyapunov stability of stationary solutions to contact-type Hamilton-Jacobi equations on a compact manifold. Previous works typically assume $C^3$ Tonelli Hamiltonians and characterize stability in terms of Mather measures. In this paper, we consider continuous, convex and coercive Hamiltonians and establish verifiable PDE-type criteria for both stability and instability. In particular, the dynamical conditions involving Mather measures are replaced by conditions expressed in terms of the critical value of the Hamiltonian and viscosity subsolutions. This provides a PDE-based framework for stability analysis and reveals connections with various asymptotic behaviors of viscosity solutions. |
| title | A PDE formulation of Lyapunov stability for contact-type Hamilton-Jacobi equations |
| topic | Analysis of PDEs 35F21, 37J51, 35B40 |
| url | https://arxiv.org/abs/2604.24329 |