Differential and Integral Equations

Higher regularity of solutions of free discontinuity problems

Luigi Ambrosio, Nicola Fusco, and Diego Pallara

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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

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

First available in Project Euclid: 29 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 49N60: Regularity of solutions
Secondary: 35R35: Free boundary problems 49K20: Problems involving partial differential equations


Ambrosio, Luigi; Fusco, Nicola; Pallara, Diego. Higher regularity of solutions of free discontinuity problems. Differential Integral Equations 12 (1999), no. 4, 499--520. https://projecteuclid.org/euclid.die/1367267005

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