Abstract

Suppose that $H \in C^{1,1} (\mathbb{R}^2)$ satisfies that $H$ is locally strongly convex in $\mathbb{R}^2$ and $H(0)=\min_{p\in\mathbb{R}^2}H(p)=0$. Let $\Omega\subset \mathbb{R}^2$ be any domain. For any $u$ absolute minimizer for $H$ in $\Omega$, or equivalently, for any viscosity solution to the Aronsson equation $\mathscr A_H[u]=\sum_{i,j=1}^2 H_{p_i}(Du) H_{p_j}(Du)u_{x_ix_j}=0\mbox{ in $Ømega$,}$ the following are proven in this paper: (i) We have $[H(Du)]^\alpha \in W^{1,2}_{\mathop\mathrm{\,loc\,}}(\Omega)$ whenever $\alpha>1/2-\tau_H(0)$; some quantitative upper bounds are also given. Here $\tau_H(0)=1/2$ when $H\in C^2(\mathbb{R}^2)$, and $0<\tau_H(0)\le 1/2$ in general. (ii) The distributional determinant $-{\rm det}D^2u\,dx$ is a nonnegative Radon measure in $\Omega$ and enjoys some quantitative lower/upper bounds. (iii) For all $\alpha>\frac12-\tau_H(0)$, we have $\mbox{$łangle D [H(Du )]^\alpha,D_p H(Du )\rangle=0 $ almost everywhere in $Ømega$}.$

Keywords

  1. $L^\infty$-variational problem
  2. absolute minimizer
  3. Aronsson equation
  4. viscosity solution
  5. Sobolev regularity

MSC codes

  1. 35J60
  2. 35J70
  3. 49K20

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Information & Authors

Information

Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 5792 - 5853
ISSN (online): 1095-7154

History

Submitted: 1 September 2021
Accepted: 7 August 2022
Published online: 1 November 2022

Keywords

  1. $L^\infty$-variational problem
  2. absolute minimizer
  3. Aronsson equation
  4. viscosity solution
  5. Sobolev regularity

MSC codes

  1. 35J60
  2. 35J70
  3. 49K20

Authors

Affiliations

Funding Information

National Natural Science Foundation of China https://doi.org/10.13039/501100001809 : 11522102, 11871088, 12001041

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