  In this paper, we study the well-posedness of boundary value problems for a
special class of degenerate elliptic equations coming from geometry. Such
problems is intimately tied to rigidity problem arising in infinitesimal
isometric deformation, The characteristic form of this class of equations is
changing its signs in the domain. Therefore the well-posedness of these above
problems deserve to make a further discussion. Finally, we get the existence
and uniqueness of $H^1$ solution for such boundary value problems.
