DEV-DIV- AND DEVSYM-DEVCURL-INEQUALITIES FOR INCOMPATIBLE SQUARE TENSOR FIELDS WITH MIXED BOUNDARY CONDITIONS

Let Omega subset of R-n, n >= 2, be a bounded Lipschitz domain and 1 < q < infinity. We prove the inequality parallel to T parallel to(Lq(Omega)) <= C-DD (parallel to dev T parallel to(Lq(Omega)) + parallel to Div T parallel to(Lq(Omega))) being valid for tensor fields T : Omega --> R-nxn with a normal boundary condition on some open and non- empty part Gamma(nu) of the boundary partial derivative Omega. Here dev T = T - 1/n tr (T) center dot Id denotes the deviatoric part of the tensor T and Div is the divergence row-wise. Furthermore, we prove parallel to T parallel to(L2(Omega)) <= C-DSC (parallel to dev sym T parallel to(L2(Omega)) + parallel to Curl T parallel to(Lq(Omega))) if n >= 3, parallel to T parallel to(L2(Omega)) <= C-DSDC (parallel to dev sym T parallel to(L2(Omega)) + parallel to dev Curl T parallel to(L2(Omega))) if n = 3, being valid for tensor fields T with a tangential boundary condition on some open and non- empty part Gamma(tau) of partial derivative Omega. Here, sym T = 1/2 (T + T-inverted perpendicular) denotes the symmetric part of T and Curl is the rotation row-wise.