Nonlocal supremals: relaxation for arbitrary dimensions
A talk in the Nonlocal Equations: Analysis and Numerics series by
Antonella Ritorto from Ingolstadt
| Abstract: | In this talk, we introduce a new notion of convexity: Cartesian convexity, which coincides with separate
convexity in the scalar case and it is strictly stronger in higher dimensions. We prove that Cartesian
convexity is the necessary and sufficient condition for weakly* lower semicontinuity of nonlocal supremals.
When it fails, we determine when there is structure-preservation during relaxation and provide a closed
representation formula where the supremand is the Cartesian level convex envelope of the (diagonalized)
original one. Unlike the scalar case where the relaxed functional is always structure-preserved, we present an
example in the vectorial setting where the nonlocal supremal form is lost during relaxation. The results rely
on a characterization of the asymptotic behavior of nonlocal inclusions, a theoretical result of independent
interest, along with recent developments from [Kreisbeck and Zappale, Calc. Var. PDE, 2020].
Joint work with Carolin Kreisbeck (KU Eichstätt - Ingolstadt) and Elvira Zappale (Sapienza University of Rome). Within the CRC this talk is associated to the project(s): A7 |