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
to be lower semi-continuous in the weak topology, for a sufficient regular domain
By compactness arguments (Banach–Alaoglu theorem) the existence of minimisers of weakly lower semicontinuous functionals may then follow from the direct method.
[1] This concept was introduced by Morrey in 1952.
[2] This generalisation should not be confused with the same concept of a quasiconvex function which has the same name.
A locally bounded Borel-measurable function
is the Sobolev space of essentially bounded functions with essentially bounded derivative and vanishing trace.
[3] Quasiconvexity is a generalisation of convexity for functions defined on matrices, to see this let
The Riesz-Markov-Kakutani representation theorem states that the dual space of
can be identified with the space of signed, finite Radon measures on it.
We define a Radon measure
is a probability measure and its barycenter is given
If h is a convex function, then Jensens' Inequality gives
This holds in particular if V(x) is the derivative of
by the generalised Stokes' Theorem.
is an example of a quasiconvex function, which is not convex.
This shows that the determinant is not a quasiconvex function like in Game Theory and thus a distinct notion of convexity.
In the vectorial case of the Calculus of Variations there are other notions of convexity.
{\displaystyle f{\text{ convex}}\Rightarrow f{\text{ polyconvex}}\Rightarrow f{\text{ quasiconvex}}\Rightarrow f{\text{ rank-1-convex}}.}
Already in 1952, Morrey conjectured that rank-1-convexity does not imply quasiconvexity.
[10] This was a major unsolved problem in the Calculus of Variations, until Šverák gave an counterexample in 1993 for the case
is still an open problem, known as Morrey's conjecture.
[12] Under certain growth condition of the integrand, the sequential weakly lower semi-continuity (swlsc) of an integral functional in an appropriate Sobolev space is equivalent to the quasiconvexity of the integrand.
Acerbi and Fusco proved the following theorem: Theorem: If
is Carathéodory function and it holds
is swlsc in the Sobolev Space
[13] Other authors use different growth conditions and different proof conditions.
[14][15] The first proof of it was due to Morrey in his paper, but he required additional assumptions.