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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

for a sufficient regular domain $ \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 $ 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.