Further results on \(\Gamma\)-convergence and lower semicontinuity of integral functionals depending on vector-valued functions. (English) Zbl 0735.49008

Carbone, Luciano; De Acangelis, Riccardo

Ric. Mat. 39, No. 1, 99-129 (1990).

Summary: We prove an integral representation result for the \(\Gamma^ -\)-limit, taken in the topology of \(L^ r(\Omega)\) with \(r\in[1,+\infty]\), of a sequence of integral functionals depending on vector-valued functions. An identity result for the \(\Gamma^ -\)-limits taken in \(L^ 1(\Omega)\) and \(L^ \infty(\Omega)\) topologies is also proved. The integrands are assumed to verify only partial convexity hypotheses, partial coerciveness conditions of order \(p\), global growth conditions of order \(q\) with \(p>q- 1\), and a geometric assumption that is, in some cases, necessary in order to have \(L^ \infty\)-lower semicontinuity. Some relaxation and lower semicontinuity results are also obtained.

MSC:

49J45

Methods involving semicontinuity and convergence; relaxation

Keywords:

integral representation; partial convexity; partial coerciveness; global growth conditions; Gamma-convergence; quasiconvexity