  We consider the problem of existence and uniqueness of strong solutions $u:
\Omega \subset \mathbb{R}^n \longrightarrow \mathbb{R}^N$ in $(H^{2}\cap
H^{1}_0)(\Omega)^N$ to the problem \[\label{1} \tag{1} \left\{ \begin{array}{l}
F(\cdot,D^2u ) \,=\, f, \ \ \text{ in }\Omega,\\ \hspace{31pt} u\,=\, 0, \ \
\text{ on }\partial \Omega, \end{array} \right. \] when $ f\in L^2(\Omega)^N$,
$F$ is a Carath\'eodory map and $\Omega$ is convex. \eqref{1} has been
considered by several authors, firstly by Campanato and under Campanato's
ellipticity condition. By employing a new weaker notion of ellipticity
introduced in recent work of the author [K2] for the respective global problem
on $\mathbb{R}^n$, we prove well-posedness of \eqref{1}. Our result extends
existing ones under hypotheses weaker than those known previously. An essential
part of our analysis in an extension of the classical Miranda-Talenti
inequality to the vector case of 2nd order linear hessian systems with rank-one
convex coefficients.
