Lipschitz continuity theorems for free discontinuity problem in several variables

In this paper, following the method in [S. Solimini, Simplified excision techniques for free discontinuity problems in several variables, J. Funct. Anal. 151 (1997) 1–34], we prove a regularity of the function in minimizer for free discontinuity problem. Namely, we prove that the function is globally Lipschitz continuous out of a small neighborhood of the singular set.