Derivation of the Aronsson equation for 𝐶¹ Hamiltonians

Crandall, Michael; Wang, Changyou; Yu, Yifeng

doi:10.1090/S0002-9947-08-04651-5

Derivation of the Aronsson equation for $C^1$ Hamiltonians
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by Michael G. Crandall, Changyou Wang and Yifeng Yu PDF

Trans. Amer. Math. Soc. 361 (2009), 103-124 Request permission

Abstract:

It is proved herein that any absolute minimizer $u$ for a suitable Hamiltonian $H\in C^1(\mathbb {R}^n \times \mathbb {R}\times U)$ is a viscosity solution of the Aronsson equation: \[ H_{p}(Du,u,x)\cdot (H (Du,u,x))_x=0 \quad \text {in} U. \] The primary advance is to weaken the assumption that $H\in C^2,$ used by previous authors, to the natural condition that $H\in C^1.$

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