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.
Citation
Luigi Ambrosio. Nicola Fusco. Diego Pallara. "Higher regularity of solutions of free discontinuity problems." Differential Integral Equations 12 (4) 499 - 520, 1999. https://doi.org/10.57262/die/1367267005
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