Wednesday, March 8, 2023 - 11:15 in H2
Variational analysis of integral functionals involving nonlocal gradients on bounded domains
A talk in the Nonlocal Equations: Analysis and Numerics 2023 series by
Carolin Kreisbeck from Eichstätt-Ingolstadt
| Abstract: |
The focus of this talk lies on variational problems with integral functionals
depending on nonlocal gradients that correspond to truncated versions of the Riesz fractional gradient. We discuss several new aspects regarding the existence theory of these problems and the study of their asymptotic behavior. Our analysis is based on suitable translation operators that allow us to switch between the three types of gradients: classical, fractional, and nonlocal. These provide helpful technical tools for transferring results from one setting to the other. Based on this approach, we show that quasiconvexity, which is the natural convexity notion in the classical and fractional calculus of variations, gives a necessary and sufficient condition for the weak lower semicontinuity of the nonlocal functionals as well. As a consequence of a general Gamma-convergence statement, we obtain relaxation and homogenization results. The analysis of the limiting behavior for varying fractional parameters yields, in particular, a rigorous localization with a classical local limit model.
Within the CRC this talk is associated to the project(s): A7 |
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