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 $Ω\subset \rr^2$ 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 $$\mathscr A_H[u]=\sum_{i,j=1}^2 H_{p_i}(Du) H_{p_j}(Du)u_{x_ix_j}=0 \quad \mbox{ in $Ω$,}$$ the following are proven in this paper: \begin{enumerate} \item[(i)] We have $[H(Du)]^α\in W^{1,2}_\loc(Ω)$ 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 $-\det D^{2}udx$ 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 $\alpha >\frac{1}{2}-{\tau }_H(0)$, we have $$\mbox{$\langle D [H(Du )]^α,D_p H(Du )\rangle=0 $ almost everywhere in $Ω$}.$$ \end{enumerate}

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