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

Fakultät für Mathematik, Universität Duisburg-Essen, Campus Essen, Thea-Leymann-Str. 9, 45127 Essen, Germany.
sebastian.bauer.seuberlich@uni-due.de

Received: 5 February 2014
Revised: 29 July 2014

Abstract

Let Ω ⊂ ℝⁿ, n ≥ 2, be a bounded Lipschitz domain and 1 <q< ∞. We prove the inequality being valid for tensor fields T:Ω → ℝ^{n × n} with a normal boundary condition on some open and non-empty part Γ_ν of the boundary ∂Ω. Here dev T = T - 1/n tr(T)·Id denotes the deviatoric part of the tensor T and Div is the divergence row-wise. Furthermore, we prove being valid for tensor fields T with a tangential boundary condition on some open and non-empty part Γ_τ of ∂Ω. Here, sym T = 1/2(T + T^⊤) denotes the symmetric part of T and Curl is the rotation row-wise.