On tensor functions whose gradients have some skew-symmetries (Q1192791)

Let \(V_ n\) be a real inner product space of dimension \(n\) and let \(T^ p_ q(V_ n)\) be the vector space of tensors of type \(p\choose q\) on \(V_ n\). The author considers the tensor functions \[ Q: U_ \tau\subset T^ 0_ \tau(V_ n)\to T^ \nu_ 0(V_ n) \] (\(U_ \tau\) -- an open connected subset) verifying the condition that the tensor gradient \(\text{Grad }Q\) is skew-symmetric in some indexes. This condition leads to a first order symmetric system of homogeneous partial differential equations with constant coefficients. The author characterizes the \(C^ 2\)-solutions of this system, which in the cases \(n=3\), \(\nu,\tau\in\{1,2\}\) allow to study the indeterminateness of the constitutive equations for various continuous media.