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Talk

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

  • Sandra Kabisch (University of Surrey)
Felix-Klein-Hörsaal Universität Leipzig (Leipzig)

Abstract

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.

Katja Heid

MPI for Mathematics in the Sciences Contact via Mail

Upcoming Events of This Seminar

  • Mar 12, 2024 tba with Theresa Simon
  • Mar 26, 2024 tba with Phan Thành Nam
  • Mar 26, 2024 tba with Dominik Schmid
  • May 7, 2024 tba with Manuel Gnann
  • May 14, 2024 tba with Barbara Verfürth
  • May 14, 2024 tba with Lisa Hartung
  • Jun 25, 2024 tba with Paul Dario
  • Jul 16, 2024 tba with Michael Loss