Smooth and Broken Minimizers of Some Free Discontinuity Problems

2017 ◽  
pp. 431-468 ◽  
Author(s):  
Danilo Percivale ◽  
Franco Tomarelli
Keyword(s):  
Free Discontinuity Problems

scholarly journals Finite-difference approximation of free-discontinuity problems

2001 ◽  
Vol 131 (3) ◽  
pp. 567-595 ◽  
Author(s):  
Massimo Gobbino ◽  
Maria Giovanna Mora
Keyword(s):  
Finite Difference ◽  
Finite Differences ◽  
Pointwise Convergence ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Compactness Result ◽  
Free Discontinuity Problems ◽  
Non Local ◽  
Γ Convergence

We approximate functionals depending on the gradient of u and on the behaviour of u near the discontinuity points by families of non-local functionals where the gradient is replaced by finite differences. We prove pointwise convergence, Γ-convergence and a compactness result, which implies, in particular, the convergence of minima and minimizers.

Free discontinuity problems generated by singular perturbation: the n−dimensional case

2000 ◽  
Vol 130 (3) ◽  
pp. 449-469 ◽  
Author(s):  
R. Alicandro ◽  
M. S. Gelli
Keyword(s):  
Singular Perturbation ◽  
Hessian Matrix ◽  
Higher Order ◽  
Dimensional Case ◽  
Free Discontinuity Problems ◽  
Limiting Behaviour ◽  
Image Position

We provide an approximation of some free discontinuity problems by local functionals with a singular perturbation of higher order. More precisely, we study the limiting behaviour of energies of the form where Hu denotes the Hessian matrix of u.

FREE DISCONTINUITY PROBLEMS WITH UNBOUNDED DATA

1994 ◽  
Vol 04 (06) ◽  
pp. 843-855 ◽  
Author(s):  
ANTONIO LEACI
Keyword(s):  
Pattern Recognition ◽  
Surface Energy ◽  
Variational Problem ◽  
Mathematical Physics ◽  
Closed Set ◽  
Free Discontinuity Problems ◽  
Volume Energy ◽  
Discontinuity Problem

We prove the existence of a minimizing pair for a free discontinuity problem, i.e. a variational problem in which the unknowns are a closed set K and a function suitably smooth outside K. Examples of such problems come from pattern recognition and mathematical physics, when both “volume” energy and “surface” energy are present.

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