  The main objective of this paper is to study the regularity and stability for
solutions to the conductivity problems with degenerate coefficients in the
presence of two rigid conductors, as one conductor keeps motionless and another
conductor moves in some direction by a sufficiently small translational
distance. We will show that the solutions are smooth and stable with respect to
the small translational distance. Our results contain the following three
cases: two perfect conductors, two insulators, a perfect conductor and an
insulator. Further, we extend the results to the elasticity problem modeled by
the Lam\'{e} system with partially infinite coefficients.
