On the regularity of solutions in convex integration theory (Q1174476)

This article gives a negative answer to a question raised by \textit{M. L. Gromov} in Section 2.4 of his book Partial differential relations (1986; Zbl 0651.53001) about the regularity of solutions of certain undetermined systems. The counterexamples are ``triangular'' systems of order \(r\) with arbitrarily smooth coefficients, for which Gromov's convex integration theory [op. cit.] provides ``many'' solutions of class \(C^ r\), but which have ``few'' \(C^{r+1}\) solutions. The simplest example is the following: (*) \(\| \partial_ x f\|^ 2= \| \partial_ y f\|^ 2= A(x,y,f)\), for a map \(f:[0,1]\times [0,1]\to\mathbb{R}^ 2\), where \(A\) is \(C^ 1\), positive, and satisfies \(\partial_ x f\pm \partial_ y f\neq 0\) everywhere. Using convex integration theory, the author shows that \(C^ 1\) solutions of (*) are \(C^ 0\) dense in the space \({\mathcal S}\) of ``short'' maps, that is, maps which satisfy: \(\| \partial_ x f\|^ 2<A\), \(\|\partial_ y f\|^ 2<A\). On the other hand, it is easy to see that a \(C^ 2\) solution is necessarily an immersion, so that such solutions are not \(C^ 0\) dense in \({\mathcal S}\): more precisely, a short map whose Jacobian changes sign cannot be \(C^ 0\) approximated by \(C^ 2\) solutions of (*).