Regularity properties of viscosity solution of nonconvex Hamilton–Jacobi equations

ABSTRACT

Some properties of characteristic curves in connection with the viscosity solution of the Hamilton–Jacobi equation $(H,\mathit{\sigma })$ defined by the Hopf formula $u(t,x)={max}_{q\in {\mathbb{R}}^n}\{⟨x,q⟩-{\mathit{\sigma }}^{\ast }(q)-tH(q)\}$ are studied. We are concerned with the points at which the solution u(t, x) is differentiable, and the strip of the form $\mathcal{R}=(0,\mathit{\theta })\times {\mathbb{R}}^n$ in the domain $\mathrm{\Omega }$ where u(t, x) is of class $C^1(\mathcal{R}).$

KEYWORDS:

• Hamilton–Jacobi equation
• regularity
• Hopf formula
• viscosity solution

AMS SUBJECT CLASSIFICATIONS:

• 35D40
• 35F21
• 35F25

Acknowledgements

The author would like to thank the VIASM for financial support and hospitality.

Notes

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) [grant number 101.02-2013.09]. A part of this paper was done when the author was working at the Vietnam Institute for Advance Study in Mathematics (VIASM).