R-manifolds and multi-valued solutions of PDE. Part 2

Exercises and examination problems

Vladimir Chetverikov

Exercise 11. Prove that the surface S introduced in lecture determines a multi-valued solution of the eikonal equation.

Problem 12. Let θ ∈ singV, W = T_{θ}(V), P = (π_{k,k−1})_{∗}(W), P_{0} = ker(π_{k,k−1}|_{V})_{∗,θ}. Prove that P_{0} ⊂ T_{θ}(l(P)), where l(P) is the s–ray, s = dim P.

Problem 13. Let an equation E ⊂ J^{2}(2, 1) be determined by the equality

F (x_{1}, x_{2}, u, u_{1}, u_{2}, u_{11}, u_{12}, u_{22}) = 0.

Find 1) elliptic points; 2) hyperbolic points; 3) parabolic points of this equation.

Problem 14. Find the multi-valued solution of the Cauchy–Riemann system corresponding to the analytic function w = 2/3z^{3/2}. Show that 1) this solution has one singular point; 2) the type of this singular point is 2.

Problem 15. Find a finite intrinsic symmetry of the Cauchy–Riemann system E that maps vertical fibers of E to horizontal submanifolds. Also, find the image under this symmetry of the solution from the previous problem.

Exercise 12 Prove the theorem about the relation characteristic covectors of an equation E with points of its associated 1–singularity equation E_{[1]}.

Exercise 13. Let E = {F = 0} ⊂ J^{1}(n, 1). Show that the vector field

Y_{F} = ∑_{i=1}^{n}(−\frac{∂F}{∂p_{i}} \frac{∂}{∂x_{i}} + (\frac{∂F}{∂x_{i}} + p_{i} \frac{∂F}{∂u} )\frac{∂}{∂p_{i}} ) − (∑_{i=1}^{n} p_{i} \frac{∂F}{∂p_{i}} )\frac{∂}{∂u}

is a characteristic symmetry of E.