Wednesday, February 23, 2022 - 16:15 in H7 + Zoom
Quasiconvexity in the fractional calculus of variations: Characterization of lower semicontinuity and relaxation
A talk in the Nonlocal Equations: Analysis and Numerics series by
Hidde Schönberger from Ingolstadt
| Abstract: |
Based on recent developments in the theory of fractional Sobolev spaces, an interesting new class of nonlocal variational problems has emerged in the literature. These problems, which are the focus of this talk, involve integral functionals that depend on Riesz fractional gradients instead of ordinary gradients and are considered subject to a complementary-value condition. With the aim of establishing existence of minimizers, we provide a full characterization for the weak lower semicontinuity of these functionals under suitable growth assumptions on the integrands. In doing so, we surprisingly identify quasiconvexity, which is intrinsic to the standard vectorial calculus of variations, as the natural notion also in the fractional setting. In the absence of quasiconvexity, we determine a representation formula for the corresponding relaxed functionals, obtained via partial quasiconvexification outside the region where complementary values are prescribed. Our proofs rely crucially on an inherent relation between classical and fractional gradients, which we extend to Sobolev spaces, enabling us to transition between the two settings. This is based on a joint work with Carolin Kreisbeck (KU Eichstätt-Ingolstadt).
Within the CRC this talk is associated to the project(s): A7 |
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