Seminormality conditions in the calculus of variations for \(BV\) solutions (Q1210164)

We discuss the closure in BV, with respect to \(L_ 1\)-convergence, of the orientor field (1) \((t,x(t))\in A\), \(Dx(t)\in Q(t,x(t))\) a.e., where \(A\subset \mathbb{R}^{\nu+n}\) and \(Q: A\to\mathbb{R}^ N\) is a given multi- function, \(\nu\), \(n\), \(N\geq 1\). As an application, we get the \(L_ 1\)- lower semicontinuity in BV of the integral functional \(\int_ G F_ 0(t,x(t),Dx(t))dt\) (where \(G\subset \mathbb{R}^ \nu\) is open and bounded) subject to constraints on the gradient of type (1). As it is well-known this last result is the key for the existence of absolute minima for the relaxed Serrin-functional over a class \(\Omega\subset\text{BV}\). The present paper extends and unifies previous analogous results of the authors [Arch. Ration. Mech. Anal. 98, 307-328 (1987; Zbl 0618.49004); Ann. Mat. Pura Appl., IV. Ser. 152, 95-121 (1988; Zbl 0668.49018)]. More precisely, for simple integrals, we have adopted the weak topology in BV (\(L_ 1\)-convergence of equiBV sequences) and have required the usual seminormality condition (Q), while, for multiple integrals, we have adopted the simple \(L_ 1\)-convergence but we have strengthened property (Q) by an additional seminormality condition (F). Here we introduce a new and more general ``weak condition \(F\)'' \((wF)\) which subsumes the above mentioned assumptions of the simple and multiple cases. We still prove closure, lower closure and lower semicontinuity theorems, with respect to \(L_ 1\)-convergence, which contains the analogous results of both above- mentioned papers. As an example, we also present applications to a relevant class of integrands introduced recently by \textit{L. Cesari} [Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 82, No. 4, 661-671 (1988; Zbl 0736.49029)] and show that the present results lead to a significant reduction of the assumptions.