Solving a PDE with constraints

General
1

Hello,
I want to solve a system of PDEs in a two-dimensional domain \Omega. The unknowns are the three components of a vector field X^\alpha(x), where \alpha = 1,2,3 and x = {x^1,x^2} are the coordinates on \Omega.

The system of PDEs is given by

  • One fourth-order PDE
  • Two constraints for \partial X, which enforce that the mapping x → X is conformal:
    • {\bf e}_1 \cdot {\bf e}_1 = {\bf e}_2 \cdot {\bf e}_2
    • {\bf e}_1 \cdot {\bf e}_2 = 0
      where {\bf e}_i \equiv \partial_i X

I can derive the variational formulation of the first PDE. What is te correct way to formulate in a variational way the two additional constraints ?

Thank you

Solving a PDE with a non-linear constraint