Calculus of Variations and Geometric Measure Theory

I. Fonseca - G. Leoni - R. Paroni

On Hessian Matrices in the Space $BH$

created on 17 Jul 2001
modified by leoni on 27 Jul 2011


Published Paper

Inserted: 17 jul 2001
Last Updated: 27 jul 2011

Journal: Commun. Contemp. Math.
Volume: 7
Number: 4
Pages: 401-420
Year: 2005


An extension of Alberti's result to second order derivatives is obtained. Precisely, if $\Omega$ is an open subset of $R^{N}$ and if $f\in L^{1}\left(\Omega;R^{N\times N}\right)$ is symmetric-valued, then there exist $u\in W^{1,1}\left( \Omega\right)$ with $\nabla u \in BV(\Omega;R^N)$ and a constant $C>0$ depending only on $N$ such that \[ D^{2}u=f\,\mathcal{L}^{N}\lfloor\,\Omega+[\nabla u]\otimes\nu_{\nabla u}\,\mathcal{H}^{N-1}\lfloor\,S(\nabla u), \] and \[ \int_{\Omega}\left
\nabla u\right
\,dx+\int_{S\left( \nabla u\right) \cap\Omega}\left
\left[ \nabla u\right] \right
\,d\mathcal{H}^{N-1}\leq C\int_{\Omega}\left
\,dx. \]

Keywords: functions of bounded variation, functions of bounded Hessian, special functions of bounded variation, special functions of bounded Hessian, triangulation