A necessary and sufficient condition for lower semicontinuity

作者：Kristensen Jan^*

来源：Nonlinear Analysis-Theory Methods & Applications, 2015, 120: 43-56.

DOI：10.1016/j.na.2015.02.018

• Quasiconvexity
• Compensated compactness
• Young measures
• Singular integrand

It is well-known that W-1,W- p quasiconvexity is a necessary condition for sequential weak lower semicontinuity of the variational integral
I[u, Omega] = integral(Omega) F(del u(x)) dx
on the Sobolev space W-1,W- p, and that it is sufficient too provided that the integrand F satisfies suitable growth conditions related to the exponent p. We show that for extended real-valued integrands a closely related convexity condition defined in terms of gradient Young measures is both necessary and sufficient for lower semicontinuity in the more general and flexible - setting of compensated compactness. Our main results identify the relaxation (lower semicontinuous envelope) in two related situations.

• 出版日期2015-6