Existence theorems for multiple integrals of the calculus of variations for discontinuous solutions (Q1118156)

We prove existence theorems for the minimum of multiple integrals of the calculus of variations with constraints on the derivatives in classes of BV possibly discontinuous solutions. To this effect the integrals are written in the form proposed by \textit{J. Serrin} [Trans. Am. Math. Soc. 101, 139--167 (1961; Zbl 0102.04601)]. Usual convexity conditions are requested, but no growth condition. Preliminary closure and semicontinuity theorems are proved. Compactness in \(L_ 1\) of classes of BV functions with equibounded total variations is derived from Cafiero-Fleming theorems.