A quantative Sobolev regularity for absolute minimizers involving Hamiltonian $H(p)\in C^0 (\mathbb{R}^2)$ in plane

Abstract

Suppose that $H \in C^0 (\mathbb{R}^2)$ satisfies \begin{enumerate} \item[(H1)] $H$ is locally strongly convex and locally strongly concave in $\rr^2$, \item[(H2)] $H(0)=\min_{p\in\rr^2}H(p)=0$. \end{enumerate} Let $\Omega\subset \rr^2$ be any domain. For any $u$ absolute minimizer for $H$ in $\Omega$, or if $H\in C^1(\rr^2)$ additionally, for any viscosity solution to the Aronsson equation $$\mathscr A_H[u]=\sum_{i,j=1}^2 H_{p_i}(Du) H_{p_j}(Du)u_{x_ix_j}=0 \quad \mbox{ in $\Omega$,}$$ the following are proven in this paper: \begin{enumerate} \item[(i)] We have $[H(Du)]^\alpha\in W^{1,2}_\loc(\Omega)$ whenever $\alpha>1/2-\tau_H(0)$; some quantative upper bounds are also given. Here $\tau_H(0)=1/2$ when $H\in C^2(\rr^2)$, and $0< \tau_H(0)\le 1/2$ in general. \item[(ii)] If $H\in C^1(\rr^2)$, then the distributional determinant $-{\rm det}D^2u\,dx$ is a nonnegative Radon measure in $\Omega$ and enjoys some quantative lower/upper bounds. \item[(iii)] If $H\in C^1(\rr^2)$, then for all $\alpha>\frac12-\tau_H(0)$, we have $$\mbox{$\langle D [H(Du )]^\alpha ,D_p H(Du )\rangle=0 $ almost everywhere in $\Omega$}.$$ \end{enumerate}