Quasiconvexity (calculus of variations)

Last updated July 14, 2024

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 ${\mathcal {F}}:W^{1,p}(\Omega ,\mathbb {R} ^{m})\rightarrow \mathbb {R} \qquad u\mapsto \int _{\Omega }f(x,u(x),\nabla u(x))dx$ to be lower semi-continuous in the weak topology, for a sufficient regular domain ${\textstyle \Omega \subset \mathbb {R} ^{d}}$ . By compactness arguments (Banach–Alaoglu theorem) the existence of minimisers of weakly lower semicontinuous functionals may then follow from the direct method.^[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 ${\textstyle f:\mathbb {R} ^{m\times d}\rightarrow \mathbb {R} }$ 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 $B (0,1)$ is the unit ball and $W_{0}^{1,\infty }$ is the Sobolev space of essentially bounded functions with essentially bounded derivative and vanishing trace.^[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 $W_{0}^{1,\infty }$ 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 $A\in \mathbb {R} ^{m\times d}$ 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 $h$ 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 $V (x)$ is the derivative of $\psi \in W_{0}^{1,\infty }(B(0,1),\mathbb {R} ^{m\times d})$ by the generalised Stokes' Theorem.^[7]

The determinant $\det \mathbb {R} ^{d\times d}\rightarrow \mathbb {R}$ is an example of a quasiconvex function, which is not convex.^[8] 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 ^[9] $f{\text{ convex}}\Rightarrow f{\text{ polyconvex}}\Rightarrow f{\text{ quasiconvex}}\Rightarrow f{\text{ rank-1-convex}}.$

These notions are all equivalent if $d=1$ or $m=1$ . Already in 1952, Morrey conjectured that rank-1-convexity does not imply quasiconvexity.^[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 $f:\mathbb {R} ^{d}\times \mathbb {R} ^{m}\times \mathbb {R} ^{d\times m}\rightarrow \mathbb {R} ,(x,v,A)\mapsto f(x,v,A)$ 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.^[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]

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References

↑ 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.
↑ 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.
↑ 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.
↑ 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.
↑ 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.
↑ 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.
↑ 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.
↑ 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.
↑ 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.
↑ 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.
↑ 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.
↑ 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.
↑ 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.
↑ 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.
↑ 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.
↑ 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.

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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.

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