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 [math]\displaystyle{ \mathcal{F}: W^{1,p}(\Omega, \mathbb{R}^m) \rightarrow \R \qquad u \mapsto \int_\Omega f(x, u(x), \nabla u(x)) dx }[/math] to be lower semi-continuous in the weak topology, for a sufficient regular domain [math]\displaystyle{ \Omega \subset \mathbb{R}^d }[/math]. 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 [math]\displaystyle{ f:\mathbb{R}^{m\times d} \rightarrow \mathbb{R} }[/math] is called quasiconvex if [math]\displaystyle{ \int_{B(0,1)} \bigl(f(A + \nabla \psi(x)) - f(A)\bigr)dx \geq 0 }[/math] for all [math]\displaystyle{ A \in \mathbb{R} }[/math] and all [math]\displaystyle{ \psi \in W_0^{1,\infty}(B(0,1), \mathbb{R}^m) }[/math] , where B(0,1) is the unit ball and [math]\displaystyle{ W_0^{1,\infty} }[/math] 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 [math]\displaystyle{ W_0^{1,\infty} }[/math] 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 [math]\displaystyle{ A \in \mathbb{R}^{m\times d} }[/math] and [math]\displaystyle{ V \in L^1(B(0,1), \mathbb{R}^m) }[/math] with [math]\displaystyle{ \int_{B(0,1)} V(x)dx = 0 }[/math]. The Riesz-Markov-Kakutani representation theorem states that the dual space of [math]\displaystyle{ C_0(\mathbb{R}^{m\times d}) }[/math] can be identified with the space of signed, finite Radon measures on it. We define a Radon measure [math]\displaystyle{ \mu }[/math] by [math]\displaystyle{ \langle h, \mu\rangle = \frac{1}{|B(0,1)|} \int_{B(0,1)} h(A + V(x)) dx }[/math] for [math]\displaystyle{ h \in C_0(\mathbb{R}^{m\times d}) }[/math]. It can be verfied that [math]\displaystyle{ \mu }[/math] is a probability measure and its barycenter is given [math]\displaystyle{ [\mu] = \langle \operatorname{id}, \mu \rangle = A + \int_{B(0,1)} V(x) dx = A. }[/math] If h is a convex function, then Jensens' Inequality gives [math]\displaystyle{ h(A) = h([\mu]) \leq \langle h, \mu \rangle = \frac{1}{|B(0,1)|} \int_{B(0,1)} h(A + V(x)) dx. }[/math] This holds in particular if V(x) is the derivative of [math]\displaystyle{ \psi \in W_0^{1,\infty}(B(0,1), \mathbb{R}^{m\times d}) }[/math] by the generalised Stokes' Theorem.^[7]

The determinant [math]\displaystyle{ \det \mathbb{R}^{d\times d} \rightarrow \mathbb{R} }[/math] is an example of a quasiconvex function, which is not convex.^[8] To see that the determinant is not convex, consider [math]\displaystyle{ A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \\ \end{pmatrix} \quad \text{and} \quad B = \begin{pmatrix} 0 & 0 \\ 0 & 1 \\ \end{pmatrix}. }[/math] It then holds [math]\displaystyle{ \det A = \det B = 0 }[/math] but for [math]\displaystyle{ \lambda \in (0,1) }[/math] we have [math]\displaystyle{ \det (\lambda A + (1-\lambda)B) = \lambda(1-\lambda) \gt 0 = \max(\det A, \det B) }[/math]. 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 [math]\displaystyle{ f: \mathbb{R}^{m\times d} \rightarrow \mathbb{R} }[/math] it holds that ^[9] [math]\displaystyle{ f \text{ convex} \Rightarrow f \text{ polyconvex} \Rightarrow f \text{ quasiconvex} \Rightarrow f \text{ rank-1-convex}. }[/math]

These notions are all equivalent if [math]\displaystyle{ d = 1 }[/math] or [math]\displaystyle{ m=1 }[/math]. 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 [math]\displaystyle{ d \geq 2 }[/math] and [math]\displaystyle{ m \geq 3 }[/math].^[11] The case [math]\displaystyle{ d = 2 }[/math] or [math]\displaystyle{ m = 2 }[/math] 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 [math]\displaystyle{ 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) }[/math] is Carathéodory function and it holds [math]\displaystyle{ 0\leq f(x,v,A) \leq a(x) + C(|v|^p + |A|^p) }[/math]. Then the functional [math]\displaystyle{ \mathcal{F}[u] = \int_\Omega f(x, u(x),\nabla u(x)) dx }[/math] is swlsc in the Sobolev Space [math]\displaystyle{ W^{1,p}(\Omega, \mathbb{R}^m) }[/math] with [math]\displaystyle{ p \gt 1 }[/math] if and only if [math]\displaystyle{ f }[/math] is quasiconvex. Here [math]\displaystyle{ C }[/math] is a positive constant and [math]\displaystyle{ a(x) }[/math] 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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