*Published Paper*

**Inserted:** 17 jul 2001

**Last Updated:** 27 jul 2011

**Journal:** Commun. Contemp. Math.

**Volume:** 7

**Number:** 4

**Pages:** 401-420

**Year:** 2005

**Abstract:**

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

u\right

+\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

f\right

\,dx.
\]

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

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