This paper is concerned with a new type of minimization problem encountered, for example, in the study of perfectly plastic plate behaviour. It is required to minimize a functional over all pairs \((K,u)\), where \(K\) is a closed set in \(\mathbb{R}^ 2\) and \(u\) is a scalar- valued function which represents the transverse plate displacement. The domain of the plate is a bounded open set \(\Omega\) in \(\mathbb{R}^ 2\) and the set \(K\) represents, in the physical context, the region on which line discontinuities (yield lines) may occur. The kinds of singularities encountered in this class of problems are known as free discontinuities. Thus the functional contains terms involving second derivatives of the function \(u\), the Hausdorff measure of \(K\) as well as the jump in first derivative of \(u\) across \(K\). It is essential here to work within the appropriate functional framework, and for this purpose the authors devote considerable space to a discussion of the space \(\text{SBH}(\Omega)\) of functions with special bounded Hessian; this is a higher-dimensional version of the space \(\text{SBV}(\Omega)\) of functions of special bounded variation [see, for example, \textit{E. De Giorgi}, Frontiers in pure and applied mathematics, Coll. Pap. Ded. J.-L. Lions Occas. 60th Birthday, 55-62 (1991; Zbl 0758.49002)]. It is shown that local minimizers in SBH have good regularity properties: they are \(C^ 2\) outside \(K\). The existence of a minimizing pair is proved, under compatibility and safe load conditions on the data. Furthermore, the set \(K\) is shown to be rectifiable with respect to one-dimensional Hausdorff measure.