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- W2912778283 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)] $H(0)=min_{pinrr^2}H(p)=0$. end{enumerate} Let $Omegasubset rr^2$ be any domain. For any $u$ absolute minimizer for $H$ in $Omega$, or if $Hin C^1(rr^2)$ 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 $[H(Du)]^alphain W^{1,2}_loc(Omega)$ whenever $alpha>1/2-tau_H(0)$; some quantative upper bounds are also given. Here $tau_H(0)=1/2$ when $Hin C^2(rr^2)$, and $0< tau_H(0)le 1/2$ in general. item[(ii)] If $Hin C^1(rr^2)$, 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 $Hin C^1(rr^2)$, 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}" @default.
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- W2912778283 date "2019-01-28" @default.
- W2912778283 modified "2023-09-27" @default.
- W2912778283 title "A quantative Sobolev regularity for absolute minimizers involving Hamiltonian $H(p)in C^0 (mathbb{R}^2)$ in plane" @default.
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