Quasiconvexity (calculus of variations) explained

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$$\backslash mathcal:\; W^(\backslash Omega,\; \backslash mathbb^m)\; \backslash rightarrow\; \backslash R\; \backslash qquad\; u\; \backslash mapsto\; \backslash int\_\backslash Omega\; f(x,\; u(x),\; \backslash nabla\; u(x))\; dx$$to be lower semi-continuous in the weak topology, for a sufficient regular domain $\backslash Omega\; \backslash subset\; \backslash mathbb^d$. 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 $f:\backslash mathbb^\; \backslash rightarrow\; \backslash mathbb$ is called quasiconvex if$$\backslash int\_\; \backslash bigl(f(A\; +\; \backslash nabla\; \backslash psi(x))\; -\; f(A)\backslash bigr)dx\; \backslash geq\; 0$$for all

A \in R ^{m x} and all \psi \in W ^{1 , inf ty} _{0} (B( 0 , 1 ), R ^{m)} , where is the unit ball and W ^{1 , inf ty} _{0} is the Sobolev space of essentially bounded functions with essentially bounded derivative and vanishing trace.^[3]

Properties of quasiconvex functions

• The domain can be replaced by any other bounded Lipschitz domain.^[4]
• Quasiconvex functions are locally Lipschitz-continuous.^[5]
• In the definition the space

W ^{1 , inf ty} _{0} 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

A \in R ^{m x} and V \in L ^{1(B(0,1),} R ^{m)} with\int _{B(0,1)} V(x)dx = 0 . The Riesz-Markov-Kakutani representation theorem states that the dual space of C ^{m x d} _{0 ( R} ) can be identified with the space of signed, finite Radon measures on it. We define a Radon measure \mu by$$\backslash langle\; h,\; \backslash mu\backslash rangle\; =\; \backslash frac$$ \int_ h(A + V(x)) dxfor h \in C ^{m x d} _{0 ( R} ) . It can be verified that \mu is aprobability measure and its barycenter is given$$[\backslash mu]\; =\; \backslash langle\; \backslash operatorname,\; \backslash mu\; \backslash rangle\; =\; A\; +\; \backslash int\_\; V(x)\; dx\; =\; A.$$If is a convex function, then Jensens' Inequality gives$$h(A)\; =\; h([\backslash mu])\; \backslash leq\; \backslash langle\; h,\; \backslash mu\; \backslash rangle\; =\; \backslash frac$$ \int_ h(A + V(x)) dx.This holds in particular if is the derivative of \psi \in W ^{1 , inf ty} _{0} (B( 0 , 1 ), R ^{m x} ) by the generalised Stokes' Theorem.^[7]

The determinant

\det R ^{d x} → R is an example of a quasiconvex function, which is not convex.^[8] To see that the determinant is not convex, considerIt then holds \det A = \det B = 0 but for λ \in ( 0 , 1 ) we have\det ( λ A + ( 1-λ )B) = λ ( 1-λ ) > 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: R ^{m x} → R it holds that ^[9] $$f\; \backslash text\; \backslash Rightarrow\; f\; \backslash text\; \backslash Rightarrow\; f\; \backslash text\; \backslash Rightarrowf\; \backslash text.$$

These notions are all equivalent if

d = 1 or m =1 . 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 d \geq 2 and m \geq 3 .^[11] The case d = 2 or m = 2 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

f: R ^{d} x R ^{m} x R ^{d x} → R , (x,v,A) \mapsto f(x,v,A) is Carathéodory function andit holds 0 \leq f(x,v,A) \leq a(x) + C(|v| ^{p} + |A| ^{p)} . Then the functional$$\backslash mathcal[u]\; =\; \backslash int\_\backslash Omega\; f(x,\; u(x),\backslash nabla\; u(x))\; dx$$is swlsc in the Sobolev Space W ^{1,p} (\Omega, R ^{m)} with p > 1 if and only if f is quasiconvex. Here C is a positive constant and a(x) 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]

Notes and References

1. Book: Rindler , Filip . Calculus of Variations . Springer International Publishing AG . Universitext . 2018 . 10.1007/978-3-319-77637-8 . 978-3-319-77636-1 . 125.
2. Morrey . Charles B. . Charles B. Morrey Jr. . 1952 . Quasiconvexity and the Lower Semicontinuity of Multiple Integrals . Pacific Journal of Mathematics . 2 . 1 . 25–53 . Mathematical Sciences Publishers . 10.2140/pjm.1952.2.25 . 2022-06-30. free .
3. Book: Rindler , Filip . Calculus of Variations . Springer International Publishing AG . Universitext . 2018 . 10.1007/978-3-319-77637-8 . 978-3-319-77636-1 . 106.
4. Book: Rindler , Filip . Calculus of Variations . Springer International Publishing AG . Universitext . 2018 . 10.1007/978-3-319-77637-8 . 978-3-319-77636-1 . 108.
5. Book: Dacorogna , Bernard . Bernard Dacorogna . Direct Methods in the Calculus of Variations . Springer Science+Business Media, LLC . Applied mathematical sciences . 2008 . 78 . 10.1007/978-0-387-55249-1 . 978-0-387-35779-9 . 2nd . 159.
6. Book: Dacorogna , Bernard . Bernard Dacorogna . Direct Methods in the Calculus of Variations . Springer Science+Business Media, LLC . Applied mathematical sciences . 2008 . 78 . 10.1007/978-0-387-55249-1 . 978-0-387-35779-9 . 2nd . 173.
7. Book: Rindler , Filip . Calculus of Variations . Springer International Publishing AG . Universitext . 2018 . 10.1007/978-3-319-77637-8 . 978-3-319-77636-1 . 107.
8. Book: Rindler , Filip . Calculus of Variations . Springer International Publishing AG . Universitext . 2018 . 10.1007/978-3-319-77637-8 . 978-3-319-77636-1 . 105.
9. Book: Dacorogna , Bernard . Bernard Dacorogna . Direct Methods in the Calculus of Variations . Springer Science+Business Media, LLC . Applied mathematical sciences . 2008 . 78 . 10.1007/978-0-387-55249-1 . 978-0-387-35779-9 . 2nd . 159.
10. Morrey . Charles B. . Charles B. Morrey Jr. . 1952 . Quasiconvexity and the Lower Semicontinuity of Multiple Integrals . Pacific Journal of Mathematics . 2 . 1 . 25–53 . Mathematical Sciences Publishers . 10.2140/pjm.1952.2.25 . 2022-06-30. free .
11. Šverák . Vladimir . Vladimír Šverák . 1993 . Rank-one convexity does not imply quasiconvexity . Proceedings of the Royal Society of Edinburgh Section A: Mathematics . 120 . 1–2 . 185–189 . Cambridge University Press, Cambridge; RSE Scotland Foundation . 10.1017/S0308210500015080 . 120192116 . 2022-06-30.
12. Numerical Approaches for Investigating Quasiconvexity in the Context of Morrey's Conjecture . Jendrik . Voss . Robert J. . Martin . Oliver . Sander . Siddhant . Kumar . Dennis M. . Kochmann . Patrizio . Neff . Journal of Nonlinear Science . 2022-01-17 . 32 . 6 . 10.1007/s00332-022-09820-x . 2201.06392. 246016000 .
13. Acerbi . Emilio . Fusco . Nicola . Nicola Fusco . 1984 . Semicontinuity problems in the calculus of variations . Archive for Rational Mechanics and Analysis . 86 . 1–2 . 125–145 . Springer, Berlin/Heidelberg . 10.1007/BF00275731 . 1984ArRMA..86..125A . 121494852 . 2022-06-30.
14. Book: Rindler , Filip . Calculus of Variations . Springer International Publishing AG . Universitext . 2018 . 10.1007/978-3-319-77637-8 . 978-3-319-77636-1 . 128.
15. Book: Dacorogna , Bernard . Bernard Dacorogna . Direct Methods in the Calculus of Variations . Springer Science+Business Media, LLC . Applied mathematical sciences . 2008 . 78 . 10.1007/978-0-387-55249-1 . 978-0-387-35779-9 . 2nd . 368.
16. Morrey . Charles B. . Charles B. Morrey Jr. . 1952 . Quasiconvexity and the Lower Semicontinuity of Multiple Integrals . Pacific Journal of Mathematics . 2 . 1 . 25–53 . Mathematical Sciences Publishers . 10.2140/pjm.1952.2.25 . 2022-06-30. free .