On coercive variational integrals

Journal article

It is well-known that sequential weak lower semicontinuity of a variational integral

𝕱(u, Ω) = ∫Ω F (∇u(x)) dx

on the Sobolev space W^1,p (Ω, ℝ^N) under a p-growth condition on the integrand F is equivalent to quasiconvexity in the sense of Morrey. We show that coercivity on Dirichlet classes likewise is equivalent to a quasiconvexity condition. We also discuss more general notions of coercivity, and in the case of positively p-homogeneous integrands F we establish the existence of minimizers for a class of non-coercive quasiconvex variational integrals.

Authors: Kristensen, J; Chen, C

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Elsevier
• Journal: Nonlinear Analysis Theory Methods and Applications
• Volume: 153
• Pages: 213–229
• Publication date: 2016-10-01
• Acceptance date: 2016-09-16
• DOI: 10.1016/j.na.2016.09.011
• ISSN: 0362-546X