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
F : W 1 , p ( Ω , R m ) → R u ↦ ∫ Ω f ( x , u ( x ) , ∇ u ( x ) ) d x {\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 Ω ⊂ R d {\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. This concept was introduced by Morrey in 1952. This generalisation of convexity should not be confused with the polysemetic concept of a quasiconvex function.
Definition A locally bounded Borel-measurable function f : R m × d → R {\textstyle f:\mathbb {R} ^{m\times d}\rightarrow \mathbb {R} } is called quasiconvex if
∫ B ( 0 , 1 ) ( f ( A + ∇ ψ ( x ) ) − f ( A ) ) d x ≥ 0 {\displaystyle \int _{B(0,1)}{\bigl (}f(A+\nabla \psi (x))-f(A){\bigr )}dx\geq 0}
for all A ∈ R m × d {\displaystyle A\in \mathbb {R} ^{m\times d}} and all ψ ∈ W 0 1 , ∞ ( B ( 0 , 1 ) , R m ) {\displaystyle \psi \in W_{0}^{1,\infty }(B(0,1),\mathbb {R} ^{m})} , where B(0,1) is the unit ball and W 0 1 , ∞ {\displaystyle W_{0}^{1,\infty }} is the Sobolev space of essentially bounded functions with essentially bounded derivative and 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 , ∞ {\displaystyle W_{0}^{1,\infty }} can be replaced by periodic Sobolev functions.
Relations to other notions of convexity Quasiconvexity is a generalisation of convexity for functions defined on matrices, to see this let A ∈ R m × d {\displaystyle A\in \mathbb {R} ^{m\times d}} and V ∈ L 1 ( B ( 0 , 1 ) , R m ) {\displaystyle V\in L^{1}(B(0,1),\mathbb {R} ^{m})} with
∫ B ( 0 , 1 ) V ( x ) d x = 0 {\displaystyle \int _{B(0,1)}V(x)dx=0} . The Riesz-Markov-Kakutani representation theorem states that the dual space of C 0 ( R m × d ) {\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 μ {\displaystyle \mu } by
⟨ h , μ ⟩ = 1 | B ( 0 , 1 ) | ∫ B ( 0 , 1 ) h ( A + V ( x ) ) d x {\displaystyle \langle h,\mu \rangle ={\frac {1}{|B(0,1)|}}\int _{B(0,1)}h(A+V(x))dx}
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