Quasiconvexity (calculus of variations)

{{about|the generalisation of convexity used in the calculus of variations|the unrelated generalisation of convexity used in microeconomics|Quasiconvex function}}

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

$\mathcal{F}: W^{1,p}(\Omega, \mathbb{R}^m) \rightarrow \R \qquad u \mapsto \int_\Omega f(x, u(x), \nabla u(x)) dx$

to be lower semi-continuous in the weak topology, for a sufficient regular domain $\Omega \subset \mathbb{R}^d$ . By compactness arguments (Banach–Alaoglu theorem) the existence of minimisers of weakly lower semicontinuous functionals may then follow from the direct method.{{cite book

|last=Rindler

|first=Filip

|title=Calculus of Variations

|publisher=Springer International Publishing AG

|series=Universitext

|year=2018

|doi=10.1007/978-3-319-77637-8

|isbn=978-3-319-77636-1

|page=125

}}

This concept was introduced by Morrey in 1952.{{cite journal

|last1=Morrey

|first1=Charles B.

|author-link=Charles B. Morrey Jr.

|year=1952

|title=Quasiconvexity and the Lower Semicontinuity of Multiple Integrals

|journal=Pacific Journal of Mathematics

|volume=2

|issue=1

|pages=25–53

|publisher=Mathematical Sciences Publishers

|doi=10.2140/pjm.1952.2.25

|url=https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-2/issue-1/Quasi-convexity-and-the-lower-semicontinuity-of-multiple-integrals/pjm/1103051941.full

|access-date=2022-06-30

|doi-access=free

}} 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 $f:\mathbb{R}^{m\times d} \rightarrow \mathbb{R}$ is called quasiconvex if

$\int_{B(0,1)} \bigl(f(A + \nabla \psi(x)) - f(A)\bigr)dx \geq 0$

for all $A \in \mathbb{R}^{m\times d}$ and all $\psi \in W_0^{1,\infty}(B(0,1), \mathbb{R}^m)$ , where {{math|B(0,1)}} is the unit ball and $W_0^{1,\infty}$ is the Sobolev space of essentially bounded functions with essentially bounded derivative and vanishing trace.{{cite book

|last=Rindler

|first=Filip

|title=Calculus of Variations

|publisher=Springer International Publishing AG

|series=Universitext

|year=2018

|doi=10.1007/978-3-319-77637-8

|isbn=978-3-319-77636-1

|page=106

}}

Properties of quasiconvex functions

The domain {{math|B(0,1)}} can be replaced by any other bounded Lipschitz domain.{{cite book

|last=Rindler

|first=Filip

|title=Calculus of Variations

|publisher=Springer International Publishing AG

|series=Universitext

|year=2018

|doi=10.1007/978-3-319-77637-8

|isbn=978-3-319-77636-1

|page=108

}}

Quasiconvex functions are locally Lipschitz-continuous.{{cite book

|last=Dacorogna

|first=Bernard

|author-link=Bernard Dacorogna

|title=Direct Methods in the Calculus of Variations

|publisher=Springer Science+Business Media, LLC

|series= Applied mathematical sciences

|year=2008

|volume=78

|doi=10.1007/978-0-387-55249-1

|isbn=978-0-387-35779-9

|edition= 2nd

|page=159

}}

In the definition the space $W_0^{1,\infty}$ can be replaced by periodic Sobolev functions.{{cite book

|last=Dacorogna

|first=Bernard

|author-link=Bernard Dacorogna

|title=Direct Methods in the Calculus of Variations

|publisher=Springer Science+Business Media, LLC

|series= Applied mathematical sciences

|year=2008

|volume=78

|doi=10.1007/978-0-387-55249-1

|isbn=978-0-387-35779-9

|edition= 2nd

|page=173

}}

Relations to other notions of convexity

Quasiconvexity is a generalisation of convexity for functions defined on matrices, to see this let $A \in \mathbb{R}^{m\times d}$ and $V \in L^1(B(0,1), \mathbb{R}^m)$ with

$\int_{B(0,1)} V(x)dx = 0$ . The Riesz-Markov-Kakutani representation theorem states that the dual space of $C_0(\mathbb{R}^{m\times d})$ can be identified with the space of signed, finite Radon measures on it. We define a Radon measure $\mu$ by

$\langle h, \mu\rangle = \frac{1}$

B(0,1)

\int_{B(0,1)} h(A + V(x)) dx

for $h \in C_0(\mathbb{R}^{m\times d})$ . It can be verified that $\mu$ is a

probability measure and its barycenter is given

$[\mu] = \langle \operatorname{id}, \mu \rangle = A + \int_{B(0,1)} V(x) dx = A.$

If {{math|h}} is a convex function, then Jensens' Inequality gives

$h(A) = h([\mu]) \leq \langle h, \mu \rangle = \frac{1}$

B(0,1)

\int_{B(0,1)} h(A + V(x)) dx.

This holds in particular if {{math|V(x)}} is the derivative of $\psi \in W_0^{1,\infty}(B(0,1), \mathbb{R}^{m\times d})$ by the generalised Stokes' Theorem.{{cite book

|last=Rindler

|first=Filip

|title=Calculus of Variations

|publisher=Springer International Publishing AG

|series=Universitext

|year=2018

|doi=10.1007/978-3-319-77637-8

|isbn=978-3-319-77636-1

|page=107

}}

The determinant $\det \mathbb{R}^{d\times d} \rightarrow \mathbb{R}$ is an example of a quasiconvex function, which is not convex.{{cite book

|last=Rindler

|first=Filip

|title=Calculus of Variations

|publisher=Springer International Publishing AG

|series=Universitext

|year=2018

|doi=10.1007/978-3-319-77637-8

|isbn=978-3-319-77636-1

|page=105

}} To see that the determinant is not convex, consider

$A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \\ \end{pmatrix}
\quad \text{and} \quad
B = \begin{pmatrix} 0 & 0 \\ 0 & 1 \\ \end{pmatrix}.$

It then holds $\det A = \det B = 0$ but for $\lambda \in (0,1)$ we have

$\det (\lambda A + (1-\lambda)B) = \lambda(1-\lambda) > 0 = \max(\det A, \det B)$ . 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 $f: \mathbb{R}^{m\times d} \rightarrow \mathbb{R}$ it holds that {{cite book

|last=Dacorogna

|first=Bernard

|author-link=Bernard Dacorogna

|title=Direct Methods in the Calculus of Variations

|publisher=Springer Science+Business Media, LLC

|series= Applied mathematical sciences

|year=2008

|volume=78

|doi=10.1007/978-0-387-55249-1

|isbn=978-0-387-35779-9

|edition= 2nd

|page=159

}}

$f \text{ convex} \Rightarrow f \text{ polyconvex} \Rightarrow f \text{ quasiconvex} \Rightarrow
f \text{ rank-1-convex}.$

These notions are all equivalent if $d = 1$ or $m=1$ . Already in 1952, Morrey conjectured that rank-1-convexity does not imply quasiconvexity.{{cite journal

|last1=Morrey

|first1=Charles B.

|author-link=Charles B. Morrey Jr.

|year=1952

|title=Quasiconvexity and the Lower Semicontinuity of Multiple Integrals

|journal=Pacific Journal of Mathematics

|volume=2

|issue=1

|pages=25–53

|publisher=Mathematical Sciences Publishers

|doi=10.2140/pjm.1952.2.25

|url=https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-2/issue-1/Quasi-convexity-and-the-lower-semicontinuity-of-multiple-integrals/pjm/1103051941.full

|access-date=2022-06-30

|doi-access=free

}} This was a major unsolved problem in the Calculus of Variations, until Šverák gave an counterexample in 1993 for the case $d \geq 2$ and $m \geq 3$ .{{cite journal

|last1=Šverák

|first1=Vladimir

|author-link=Vladimír Šverák

|year=1993

|title=Rank-one convexity does not imply quasiconvexity

|journal=Proceedings of the Royal Society of Edinburgh Section A: Mathematics

|volume=120

|issue=1–2

|pages=185–189

|publisher= Cambridge University Press, Cambridge; RSE Scotland Foundation

|doi=10.1017/S0308210500015080

|s2cid=120192116

|url=https://www.cambridge.org/core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics/article/abs/rankone-convexity-does-not-imply-quasiconvexity/2E0BBDE4340DDDDE863F7C365C72C555

|access-date=2022-06-30

}}

The case $d = 2$ or $m = 2$ is still an open problem, known as Morrey's conjecture.{{cite journal

|title= Numerical Approaches for Investigating Quasiconvexity in the Context of Morrey's Conjecture

|journal=Journal of Nonlinear Science |date= 2022-01-17

|bibcode=2022JNS....32...77V |s2cid=246016000 }}

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 $f: \mathbb{R}^d \times \mathbb{R}^m \times \mathbb{R}^{d\times m} \rightarrow \mathbb{R}, (x,v,A) \mapsto f(x,v,A)$ is Carathéodory function and

it holds $0\leq f(x,v,A) \leq a(x) + C(|v|^p + |A|^p)$ . Then the functional

$\mathcal{F}[u] = \int_\Omega f(x, u(x),\nabla u(x)) dx$

is swlsc in the Sobolev Space $W^{1,p}(\Omega, \mathbb{R}^m)$ with $p > 1$ if and only if $f$ is quasiconvex. Here $C$ is a positive constant and $a(x)$ an integrable function.{{cite journal

|last1=Acerbi

|first1=Emilio

|last2=Fusco

|first2= Nicola

|author-link2=Nicola Fusco

|year=1984

|title=Semicontinuity problems in the calculus of variations

|journal=Archive for Rational Mechanics and Analysis

|volume=86

|issue=1–2

|pages=125–145

|publisher= Springer, Berlin/Heidelberg

|doi=10.1007/BF00275731

|bibcode=1984ArRMA..86..125A

|s2cid=121494852

|url=https://link.springer.com/article/10.1007/BF00275731

|access-date=2022-06-30

}}

Other authors use different growth conditions and different proof conditions.{{cite book

|last=Rindler

|first=Filip

|title=Calculus of Variations

|publisher=Springer International Publishing AG

|series=Universitext

|year=2018

|doi=10.1007/978-3-319-77637-8

|isbn=978-3-319-77636-1

|page=128

}}{{cite book

|last=Dacorogna

|first=Bernard

|author-link=Bernard Dacorogna

|title=Direct Methods in the Calculus of Variations

|publisher=Springer Science+Business Media, LLC

|series= Applied mathematical sciences

|year=2008

|volume=78

|doi=10.1007/978-0-387-55249-1

|isbn=978-0-387-35779-9

|edition= 2nd

|page=368

}} The first proof of it was due to Morrey in his paper, but he required additional assumptions.{{cite journal

|last1=Morrey

|first1=Charles B.

|author-link=Charles B. Morrey Jr.

|year=1952

|title=Quasiconvexity and the Lower Semicontinuity of Multiple Integrals

|journal=Pacific Journal of Mathematics

|volume=2

|issue=1

|pages=25–53

|publisher=Mathematical Sciences Publishers

|doi=10.2140/pjm.1952.2.25

|url=https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-2/issue-1/Quasi-convexity-and-the-lower-semicontinuity-of-multiple-integrals/pjm/1103051941.full

|access-date=2022-06-30

|doi-access=free

}}

References

Category:Calculus of variations