Quasiconvexity (calculus of variations)

Short description: 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 ℱ:W1,p(Ω,ℝm)->ℝu↦∫Ωf(x,u(x),∇u(x))dx to be lower semi-continuous in the weak topology, for a sufficient regular domain Ω⊂ℝd. 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.

## Contents

* 1 Definition
* 2 Properties of quasiconvex functions
* 3 Relations to other notions of convexity
* 4 Relation to weak lower semi-continuity
* 5 References

## Definition

A locally bounded Borel-measurable function f:ℝm×d->ℝ is called quasiconvex if ∫B(0,1)(f(A+∇ψ(x))−f(A))dx≥0 for all A∈ℝ and all ψ∈W01,∞(B(0,1),ℝm) , where B(0,1) is the unit ball and W01,∞ is the Sobolev space of essentially bounded functions with essentially bounded derivative and vanishing trace.[3]

## Properties of quasiconvex functions

* The domain B(0,1) can be replaced by any other bounded Lipschitz domain.[4]

* Quasiconvex functions are locally Lipschitz-continuous.[5]

* In the definition the space W01,∞ 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∈ℝm×d and V∈L1(B(0,1),ℝm) with ∫B(0,1)V(x)dx=0. The Riesz-Markov-Kakutani representation theorem states that the dual space of C0(ℝm×d) can be identified with the space of signed, finite Radon measures on it. We define a Radon measure μ by ⟨h,μ⟩=1|B(0,1)|∫B(0,1)h(A+V(x))dx for h∈C0(ℝm×d). It can be verfied that μ is a probability measure and its barycenter is given [μ]=⟨id⁡,μ⟩=A+∫B(0,1)V(x)dx=A. If h is a convex function, then Jensens' Inequality gives h(A)=h([μ])≤⟨h,μ⟩=1|B(0,1)|∫B(0,1)h(A+V(x))dx. This holds in particular if V(x) is the derivative of ψ∈W01,∞(B(0,1),ℝm×d) by the generalised Stokes' Theorem.[7]

The determinant det⁡ℝd×d->ℝ is an example of a quasiconvex function, which is not convex.[8] To see that the determinant is not convex, consider A=(1000)andB=(0001). It then holds det⁡A=det⁡B=0 but for λ∈(0,1) we have det⁡(λA+(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:ℝm×d->ℝ it holds that [9] f convex⇒f polyconvex⇒f quasiconvex⇒f 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.[10] This was a major unsolved problem in the Calculus of Variations, until Šverák gave an counterexample in 1993 for the case d≥2 and m≥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:ℝd×ℝm×ℝd×m->ℝ,(x,v,A)↦f(x,v,A) is Carathéodory function and it holds 0≤f(x,v,A)≤a(x)+C(|v|p+|A|p). Then the functional ℱ[u]=∫Ωf(x,u(x),∇u(x))dx is swlsc in the Sobolev Space W1,p(Ω,ℝm) 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]

## References

1. ↑ Rindler, Filip (2018). Calculus of Variations. Universitext. Springer International Publishing AG. p. 125. doi:10.1007/978-3-319-77637-8. ISBN 978-3-319-77636-1.
2. ↑ Morrey, Charles B. (1952). "Quasiconvexity and the Lower Semicontinuity of Multiple Integrals". Pacific Journal of Mathematics (Mathematical Sciences Publishers) 2 (1): 25–53. doi:10.2140/pjm.1952.2.25. https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-2/issue-1/Quasi-convexity-and-the-lower-semicontinuity-of-multiple-integrals/pjm/1103051941.full. Retrieved 2022-06-30.
3. ↑ Rindler, Filip (2018). Calculus of Variations. Universitext. Springer International Publishing AG. p. 106. doi:10.1007/978-3-319-77637-8. ISBN 978-3-319-77636-1.
4. ↑ Rindler, Filip (2018). Calculus of Variations. Universitext. Springer International Publishing AG. p. 108. doi:10.1007/978-3-319-77637-8. ISBN 978-3-319-77636-1.
5. ↑ Dacorogna, Bernard (2008). Direct Methods in the Calculus of Variations. Applied mathematical sciences. 78 (2nd ed.). Springer Science+Business Media, LLC. p. 159. doi:10.1007/978-0-387-55249-1. ISBN 978-0-387-35779-9.
6. ↑ Dacorogna, Bernard (2008). Direct Methods in the Calculus of Variations. Applied mathematical sciences. 78 (2nd ed.). Springer Science+Business Media, LLC. p. 173. doi:10.1007/978-0-387-55249-1. ISBN 978-0-387-35779-9.
7. ↑ Rindler, Filip (2018). Calculus of Variations. Universitext. Springer International Publishing AG. p. 107. doi:10.1007/978-3-319-77637-8. ISBN 978-3-319-77636-1.
8. ↑ Rindler, Filip (2018). Calculus of Variations. Universitext. Springer International Publishing AG. p. 105. doi:10.1007/978-3-319-77637-8. ISBN 978-3-319-77636-1.
9. ↑ Dacorogna, Bernard (2008). Direct Methods in the Calculus of Variations. Applied mathematical sciences. 78 (2nd ed.). Springer Science+Business Media, LLC. p. 159. doi:10.1007/978-0-387-55249-1. ISBN 978-0-387-35779-9.
10. ↑ Morrey, Charles B. (1952). "Quasiconvexity and the Lower Semicontinuity of Multiple Integrals". Pacific Journal of Mathematics (Mathematical Sciences Publishers) 2 (1): 25–53. doi:10.2140/pjm.1952.2.25. https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-2/issue-1/Quasi-convexity-and-the-lower-semicontinuity-of-multiple-integrals/pjm/1103051941.full. Retrieved 2022-06-30.
11. ↑ Šverák, Vladimir (1993). "Rank-one convexity does not imply quasiconvexity". Proceedings of the Royal Society of Edinburgh Section A: Mathematics (Cambridge University Press, Cambridge; RSE Scotland Foundation) 120 (1–2): 185–189. doi:10.1017/S0308210500015080. 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. Retrieved 2022-06-30.
12. ↑ Voss, Jendrik; Martin, Robert J.; Sander, Oliver; Kumar, Siddhant; Kochmann, Dennis M.; Neff, Patrizio (2022-01-17). "Numerical Approaches for Investigating Quasiconvexity in the Context of Morrey's Conjecture". Journal of Nonlinear Science 32 (6). doi:10.1007/s00332-022-09820-x.
13. ↑ Acerbi, Emilio; Fusco, Nicola (1984). "Semicontinuity problems in the calculus of variations". Archive for Rational Mechanics and Analysis (Springer, Berlin/Heidelberg) 86 (1–2): 125–145. doi:10.1007/BF00275731. Bibcode: 1984ArRMA..86..125A. https://link.springer.com/article/10.1007/BF00275731. Retrieved 2022-06-30.
14. ↑ Rindler, Filip (2018). Calculus of Variations. Universitext. Springer International Publishing AG. p. 128. doi:10.1007/978-3-319-77637-8. ISBN 978-3-319-77636-1.
15. ↑ Dacorogna, Bernard (2008). Direct Methods in the Calculus of Variations. Applied mathematical sciences. 78 (2nd ed.). Springer Science+Business Media, LLC. p. 368. doi:10.1007/978-0-387-55249-1. ISBN 978-0-387-35779-9.
16. ↑ Morrey, Charles B. (1952). "Quasiconvexity and the Lower Semicontinuity of Multiple Integrals". Pacific Journal of Mathematics (Mathematical Sciences Publishers) 2 (1): 25–53. doi:10.2140/pjm.1952.2.25. https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-2/issue-1/Quasi-convexity-and-the-lower-semicontinuity-of-multiple-integrals/pjm/1103051941.full. Retrieved 2022-06-30.

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