Convex integrand

Summary

A convex integrand is a Lagrangian $L(x, u, \xi)$ of an integral functional $\int_\Omega L(x, u, \nabla u) \, dx$ that is a convex function of the gradient variable $\xi$. Convexity in $\xi$ is the hypothesis that makes such a functional weakly lower semicontinuous on Sobolev spaces, for scalar functions $u$. It is therefore the basic structural assumption of existence theorems such as Tonelli's. For vector-valued $u$ it can be weakened to quasiconvexity.