Quasiconvexity (calculus of variations)

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

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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

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  • 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

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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

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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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  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. 2 (1). Mathematical Sciences Publishers: 25–53. doi:10.2140/pjm.1952.2.25. 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. Vol. 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. Vol. 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. Vol. 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. 2 (1). Mathematical Sciences Publishers: 25–53. doi:10.2140/pjm.1952.2.25. 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. 120 (1–2). Cambridge University Press, Cambridge; RSE Scotland Foundation: 185–189. doi:10.1017/S0308210500015080. S2CID 120192116. 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). arXiv:2201.06392. doi:10.1007/s00332-022-09820-x. S2CID 246016000.
  13. ^ Acerbi, Emilio; Fusco, Nicola (1984). "Semicontinuity problems in the calculus of variations". Archive for Rational Mechanics and Analysis. 86 (1–2). Springer, Berlin/Heidelberg: 125–145. Bibcode:1984ArRMA..86..125A. doi:10.1007/BF00275731. S2CID 121494852. 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. Vol. 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. 2 (1). Mathematical Sciences Publishers: 25–53. doi:10.2140/pjm.1952.2.25. Retrieved 2022-06-30.