n-polyconvexity: a new generalized semiconvexity which contains both poly- and rank-one convexity

It is well known that for real-valued functions on matrices polyconvexity implies quasiconvexity implies rank-one convexity. While quasiconvexity arises as a natural condition in the Calculus of Variations it is also the most difficult to verify and instead the only accessible path to many problems is to work with the more tractable notions of poly- and rank-one convexity. In this talk we will define the new concept of n-polyconvexity. For $f:ℝ^{d×D}→ℝ∪{+∞}$ n-polyconvexity unifies polyconvexity and rank-one convexity in the following sense: If $n=d∧D:=min{d,D}$ then $(d∧D)$-polyconvexity is equivalent to polyconvexity and if n=1 then 1-polyconvexity is equivalent to rank-one convexity. Additionally one gains the new convexities for $n=(d∧D-1)...2$ in weakening order. We will discuss some basic properties of n-polyconvexity, including non-locality, subdifferentiability and the formation of generalized versions of $T_k$-configurations. Furthermore the concept of n-polyconvexity may prove to be useful in relation to quasiconvexity the same way as poly- and rank-one convexity do and we will in particular discuss whether in $ℝ^{3×3}$ quasiconvexity implies or is implied by the previously unknown concept of 2-polyconvexity or not.