##### Department of Mathematics,

University of California San Diego

****************************

### MATH 248 Seminar In Real Analysis

## Dr. Javier Cueto

#### University of Nebraska-Lincoln

## A framework for variational problems based on nonlocal gradients on bounded domains inspired by Peridynamics

##### Abstract:

Inspired by the rise on the interest for nonlocal models, mainly Peridynamics, we decided to study a functional framework suitable for (variational) nonlocal models, such as that of nonlocal hyperelasticity. This has lead to a nonlocal framework based on truncated fractional gradients (i.e. nonlocal gradients with a fractional singularity defined over bounded domains), in which continuous and compact embeddings and, in particular, nonlocal PoincarÃ© inqualities has been obtained thanks to a nonlocal version of the fundamental theorem of calculus. As a consequence, the existence of minimizers of nonlocal polyconvex vectorial functionals is obtained, and more recently quasiconvex functionals. Some of these last results have been obtained from a result that relates nonlocal gradients with classical ones and vice-versa. These results are accompanied by a study of the localization (recovering of the classical model) when s goes to 1 (actually, continuity on s with s being the fractional index of differentiability).

Host: Prof. Xiaochuan Tian

### February 28, 2023

### 11:00 AM

APM 7321

****************************