Quality properties of solutions of differential inequalities of a special form (Q1204721)

This article deals with functions \(f(x)= (f_ 1(x), \dots, f_ n(x))\in W_{\text{loc}}^ 1({\mathcal D})\), \({\mathcal D}\) is a domain in \(\mathbb{R}^ M\), \(1\leq n\leq M\), \(M\geq 2\), for which \[ \nu_ 1| \nabla f|^ \alpha\leq *(df_ 1\wedge \dots\wedge df_ n\wedge \omega_ f), \qquad |\omega_ f|\leq \nu_ 2| \nabla f|^{\alpha-n} \quad (\alpha>1) \] where \(\omega\) is a closed differential form of degree \(M-n\) with locally integrable coefficients (the sign \(*\) means the orthogonal complement for the corresponding differential form); such inequalities appear in the theory of different types of nonlinear elliptic systems. The main results are the maximum principle for solutions to the inequalities above, integral estimates for \(\nabla f\) (the Saint-Venant principle) and then analogs of Liouville and Phragmen-Lindelöf theorems.