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

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.