  We investigate a level-set type method for solving ill-posed problems, with
the assumption that the solutions are piecewise, but not necessarily constant
functions with unknown level sets and unknown level values. In order to get
stable approximate solutions of the inverse problem we propose a Tikhonov-type
regularization approach coupled with a level set framework. We prove the
existence of generalized minimizers for the Tikhonov functional. Moreover, we
prove convergence and stability of the regularized solutions with respect to
the noise level, characterizing the level-set approach as a regularization
method for inverse problems. We also show the applicability of the proposed
level set method in some interesting inverse problems arising in elliptic PDE
models.
  Keywords: Level Set Methods, Regularization, Ill-Posed Problems, Piecewise
Non-Constant Coefficients
