Quasiconvexity (calculus of variations)

In the calculus of variations, a subfield of mathematics, quasiconvexity is a generalisation of the notion of convexity. It is used to characterise the integrand of a functional and related to the existence of minimisers. Under some natural conditions, quasiconvexity of the integrand is a necessary and sufficient condition for a functional $${\displaystyle {\mathcal {F}}:W^{1,p}(\Omega ,\mathbb {R} ^{m})\rightarrow \mathbb {R} \qquad u\mapsto \int _{\Omega }f(x,u(x),\nabla u(x))dx}$$ to be lower semi-continuous in the weak topology, for a sufficient regular domain ${\textstyle \Omega \subset \mathbb {R} ^{d}}$. By compactness arguments (Banach–Alaoglu theorem) the existence of minimisers of weakly lower semicontinuous functionals may then follow from the direct method. This concept was introduced by Morrey in 1952. This generalisation of convexity should not be confused with the polysemetic concept of a quasiconvex function.