Generalisation of convexity
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.
Definition
A locally bounded Borel-measurable function is called quasiconvex if
for all
and all
, where
B(0,1) is the unit ball and
is the
Sobolev space of essentially bounded functions with essentially bounded derivative and vanishing
trace.
[3]
Properties of quasiconvex functions
- Quasiconvex functions are locally Lipschitz-continuous.[5]
- In the definition the space can be replaced by periodic Sobolev functions.[6]
Relations to other notions of convexity
Quasiconvexity is a generalisation of convexity for functions defined on matrices, to see this let and with
. 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
by
for
. It can be verified that
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.
[7]
The determinant is an example of a quasiconvex function, which is not convex.[8] To see that the determinant is not convex, consider
It then holds
but for
we have
. 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. For a function it holds that [9]
These notions are all equivalent if or . 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 and .[11]
The case or is still an open problem, known as Morrey's conjecture.[12]
Relation to weak lower semi-continuity
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 . Then the functional
is swlsc in the Sobolev Space
with
if and only if
is quasiconvex. Here
is a positive constant and
an integrable function.
[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.[16]
References
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- . Retrieved 2022-06-30.