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 $${\displaystyle {ℱ}:W^{1,p}(\mathrm{\Omega },{\mathbb{ℝ}}^m)\to \mathbb{ℝ}u\mapsto \underset{\mathrm{\Omega }}{\int }f(x,u(x),\mathrm{\nabla }u(x))dx}$$ to be lower semi-continuous in the weak topology, for a sufficient regular domain $\mathrm{\Omega }\subset {\mathbb{ℝ}}^d$. 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 $f:{\mathbb{ℝ}}^{m\times d}\to \mathbb{ℝ}$ is called quasiconvex if $${\displaystyle \underset{B(0,1)}{\int }(f(A+\mathrm{\nabla }\psi (x))-f(A))dx\ge 0}$$ for all ${\displaystyle A\in \mathbb{ℝ}}$ and all ${\displaystyle \psi \in W_0^{1,\mathrm{\infty }}(B(0,1),{\mathbb{ℝ}}^m)}$ , where B(0,1) is the unit ball and ${\displaystyle W_0^{1,\mathrm{\infty }}}$ 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 ${\displaystyle W_0^{1,\mathrm{\infty }}}$ 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 ${\displaystyle \underset{B(0,1)}{\int }V(x)dx=0}$. 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 $${\displaystyle ⟨h,\mu ⟩=\frac{1}{|B(0,1)|}\underset{B(0,1)}{\int }h(A+V(x))dx}$$ for ${\displaystyle h\in C_0({\mathbb{ℝ}}^{m\times d})}$. It can be verfied that ${\displaystyle \mu }$ is a probability measure and its barycenter is given $${\displaystyle [\mu ]=⟨\mathrm{id},\mu ⟩=A+\underset{B(0,1)}{\int }V(x)dx=A.}$$ If h is a convex function, then Jensens' Inequality gives $${\displaystyle h(A)=h([\mu ])\le ⟨h,\mu ⟩=\frac{1}{|B(0,1)|}\underset{B(0,1)}{\int }h(A+V(x))dx.}$$ This holds in particular if V(x) is the derivative of ${\displaystyle \psi \in W_0^{1,\mathrm{\infty }}(B(0,1),{\mathbb{ℝ}}^{m\times d})}$ by the generalised Stokes' Theorem.^[7]

The determinant ${\displaystyle \det {\mathbb{ℝ}}^{d\times d}\to \mathbb{ℝ}}$ is an example of a quasiconvex function, which is not convex.^[8] To see that the determinant is not convex, consider $${\displaystyle A=\left(\begin{array}{cc}1& 0\\ 0& 0\\ \end{array}\right)\text{and}B=\left(\begin{array}{cc}0& 0\\ 0& 1\\ \end{array}\right).}$$ 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 ${\displaystyle f:{\mathbb{ℝ}}^{m\times d}\to \mathbb{ℝ}}$ 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 ${\displaystyle d\ge 2}$ and ${\displaystyle m\ge 3}$.^[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 ${\displaystyle f:{\mathbb{ℝ}}^d\times {\mathbb{ℝ}}^m\times {\mathbb{ℝ}}^{d\times m}\to \mathbb{ℝ},(x,v,A)\mapsto f(x,v,A)}$ is Carathéodory function and it holds ${\displaystyle 0\le f(x,v,A)\le a(x)+C(|v{|}^p+|A{|}^p)}$. Then the functional $${\displaystyle {ℱ}[u]=\underset{\mathrm{\Omega }}{\int }f(x,u(x),\mathrm{\nabla }u(x))dx}$$ is swlsc in the Sobolev Space ${\displaystyle W^{1,p}(\mathrm{\Omega },{\mathbb{ℝ}}^m)}$ 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]

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