## Differential and Integral Equations

### Higher regularity of solutions of free discontinuity problems

#### Abstract

In this paper we continue the analysis, started in [6],[7], of the regularity of solutions of free discontinuity problems. We choose as a model problem the minimization of the Mumford-Shah functional. Assuming that in some region the optimal discontinuity set $\Gamma$ is the graph of a $C^{1,\rho}$ function, we look for conditions ensuring the higher regularity of $\Gamma$. Our results are optimal in the two dimensional case. As an application, we prove that in the case of the Mumford-Shah functional and in similar problems the Lavrentiev phenomenon does not occur.

#### Article information

Source
Differential Integral Equations, Volume 12, Number 4 (1999), 499-520.

Dates
First available in Project Euclid: 29 April 2013