The $\Gamma$-limit for singularly perturbed functionals of Perona-Malik type in arbitrary dimension - Archive ouverte HAL Access content directly
Journal Articles Mathematical Models and Methods in Applied Sciences Year : 2014

## The $\Gamma$-limit for singularly perturbed functionals of Perona-Malik type in arbitrary dimension

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Giovanni Bellettini
• Function : Author
Antonin Chambolle
Michael Goldman

#### Abstract

In this paper we generalize to arbitrary dimensions a one-dimensional equicoerciveness and $\Gamma$-convergence result for a second derivative perturbation of Perona-Malik type functionals. Our proof relies on a new density result in the space of special functions of bounded variation with vanishing diffuse gradient part. This provides a direction of investigation to derive approximation for functionals with discontinuities penalized with a ''cohesive'' energy, that is, whose cost depends on the actual opening of the discontinuity.

#### Domains

Mathematics [math] Analysis of PDEs [math.AP]

### Dates and versions

hal-00779609 , version 1 (22-01-2013)

### Identifiers

• HAL Id : hal-00779609 , version 1
• ARXIV :

### Cite

Giovanni Bellettini, Antonin Chambolle, Michael Goldman. The $\Gamma$-limit for singularly perturbed functionals of Perona-Malik type in arbitrary dimension. Mathematical Models and Methods in Applied Sciences, 2014. ⟨hal-00779609⟩

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