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 $${\displaystyle {\mathcal {F}}:W^{1,p}(\Omega ,\mathbb {R} ^{m})\rightarrow \mathbb {R} \qquad u\mapsto \int _{\Omega }f(x,u(x),\nabla u(x))dx}$$ 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.

A locally bounded Borel-measurable function ${\textstyle f:\mathbb {R} ^{m\times d}\rightarrow \mathbb {R} }$ is called quasiconvex if $${\displaystyle \int _{B(0,1)}{\bigl (}f(A+\nabla \psi (x))-f(A){\bigr )}dx\geq 0}$$ for all ${\displaystyle A\in \mathbb {R} ^{m\times d}}$ and all ${\displaystyle \psi \in W_{0}^{1,\infty }(B(0,1),\mathbb {R} ^{m})}$ , where B(0,1) is the unit ball and ${\displaystyle W_{0}^{1,\infty }}$ is the Sobolev space of essentially bounded functions with essentially bounded derivative and vanishing trace.^[3]

• 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 ${\displaystyle W_{0}^{1,\infty }}$ can be replaced by periodic Sobolev functions.^[6]

Quasiconvexity is a generalisation of convexity for functions defined on matrices, to see this let ${\displaystyle A\in \mathbb {R} ^{m\times d}}$ and ${\displaystyle V\in L^{1}(B(0,1),\mathbb {R} ^{m})}$ with ${\displaystyle \int _{B(0,1)}V(x)dx=0}$. The Riesz-Markov-Kakutani representation theorem states that the dual space of ${\displaystyle C_{0}(\mathbb {R} ^{m\times d})}$ can be identified with the space of signed, finite Radon measures on it. We define a Radon measure ${\displaystyle \mu }$ by $${\displaystyle \langle h,\mu \rangle ={\frac {1}{|B(0,1)|}}\int _{B(0,1)}h(A+V(x))dx}$$ for ${\displaystyle h\in C_{0}(\mathbb {R} ^{m\times d})}$. It can be verified that ${\displaystyle \mu }$ is a probability measure and its barycenter is given $${\displaystyle =\langle \operatorname {id} ,\mu \rangle =A+\int _{B(0,1)}V(x)dx=A.}$$ If h is a convex function, then Jensens' Inequality gives $${\displaystyle h(A)=h()\leq \langle h,\mu \rangle ={\frac {1}{|B(0,1)|}}\int _{B(0,1)}h(A+V(x))dx.}$$ This holds in particular if V(x) is the derivative of ${\displaystyle \psi \in W_{0}^{1,\infty }(B(0,1),\mathbb {R} ^{m\times d})}$ by the generalised Stokes' Theorem.^[7]

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

These notions are all equivalent if ${\displaystyle d=1}$ or ${\displaystyle m=1}$. 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 ${\displaystyle d\geq 2}$ and ${\displaystyle m\geq 3}$.^[11] The case ${\displaystyle d=2}$ or ${\displaystyle m=2}$ is still an open problem, known as Morrey's conjecture.^[12]

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