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Quasiconvexity is a generalisation of convexity in the Calculus of Variations to characterise the integrand of a functional for the existence of minimizers. This means to find necessary and sufficient conditions for a functional $${\mathcal {F}}:W_{g}^{1,p}(\Omega ,\mathbb {R} ^{m})\rightarrow \mathbb {R} \qquad u\mapsto \int _{\Omega }f(x,u(x),\nabla u(x))$$ for a sufficient regular domain ${\textstyle \Omega \subset \mathbb {R} ^{d}}$ with described boundary data g and the integrand $f:\Omega \times \mathbb {R} ^{m}\times \mathbb {R} ^{m\times d}$, to be lower semi-continuous. By compactness arguments (Banach–Alaoglu theorem) the existence of minimizers follows from the direct method. This concept was introduced by Morrey in 1952.

Definition

A locally bounded Borel-measurable function ${\textstyle f:\mathbb {R} ^{m\times d}\rightarrow \mathbb {R} }$ is called quasiconvex if $$\int _{B(0,1)}f(A+\nabla \psi (x))-f(A)\geq 0$$ for all $A\in \mathbb {R}$ 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 essiantially bounded functions with vanishing trace.

Properties of quasiconvex functions

• The domain B(0,1) can be replaced by any other bounded Lipschitz domain.

• Quasiconvex functions are locally Lipschitz-continuous.

• In the definition the space $W_{0}^{1,\infty }$ can be replaced by periodic Sobolev functions.

Relations to other notions of convexity

Quasiconvexity is a generalisation of convexity, 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 by $$\langle h,\mu \rangle ={\frac {1}{|B(0,1)|}}\int _{B(0,1)}h(A+V(x))dx$$ for $C_{0}(\mathbb {R} ^{m\times d})$. It can be verfied 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 especially 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.

The determinant $\det \mathbb {R} ^{d\times d}\rightarrow \mathbb {R}$ is a quasiconvex function, which is not convex. The determinant is an counterexample to, since for $$A={\begin{pmatrix}1&0\\0&0\\\end{pmatrix}}B={\begin{pmatrix}0&0\\0&1\\\end{pmatrix}}$$ it holds $\det A=\det B=0$ but for $\lambda \in (0,1)$ it is $\det(\lambda A+(1-\lambda )B)=\lambda (1-\lambda )>1$.

However this should not be confused with the notion of quasiconvexity in Game Theory.

In the vectorial case of the Calculus of Variations there are other notions of convexity. For a function $f:\mathbb {R} ^{m\times d}\rightarrow \mathbb {R}$ holds $$f{\text{ convex}}\Rightarrow f{\text{ polyconvex}}\Rightarrow f{\text{ quasiconvex}}\Rightarrow f{\text{ rank-1-convex}}.$$

These notions are all equivalent if $d=m=1$. Already in 1952, Morrey conjectured that rank-1-convexity does not imply quasiconvexity. 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 $m\geq 3$.. The case $d=m=2$ is still an open problem, known as Morrey's conjecture.

Relation to weak lower semi-continuity

Under certain growth condition of the integrand, the sequential weakly lower semi-continuity (swlsc) of the functional is equivalent to the quasiconvexity of the integrand. Acerbi and Fusco proved the following theorem:

Theorem: If $f:\mathbb {R} ^{d}\times \mathbb {R} ^{m}\times \mathbb {R} ^{d\times m},(x,v,A)\mapsto f(x,v,A)$ is measurable in v and continuous in A 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 if and only if f is quasiconvex. Here C is a positive constant and a(x) an integrable function.

Other authors use different growth conditions and different proof conditions . The first proof of it was due to Morrey in his paper, but he required additional assumptions.