Dev-Div- and DevSym-DevCurl-inequalities for incompatible square tensor fields with mixed boundary conditions

TL;DR

This paper introduces new inequalities for bounding the norm of square tensor fields with mixed boundary conditions, which are useful for establishing coercivity in mathematical models involving such tensors.

Contribution

The paper derives novel Dev-Div and DevSym-Curl inequalities for incompatible square tensor fields with mixed boundary conditions, expanding the theoretical toolkit for tensor analysis.

Findings

Inequalities relate tensor norms to divergence, curl, and trace-free parts.

Results applicable to coercivity in tensor-based models.

Extensions for n-dimensional bounded domains.

Abstract

For an n-dimensional bounded domain we derive some inequalities bounding the norm of a square tensor field. Concerning the Div-Dev-inequality the bound is given by the trace-free part and the divergence and the tensor. In the case of the DevSym-Curl-inequality the bound is given by the trace-free and symmetric part and the curl of the tensor. For n=3 the bound is given by the trace-free symmetric part of the tensor and the trace-free part of the curl of the tensor. Some prototype applications are presented in which the new inequalities may be used to derive the coercivity of the models.