scholarly journals The Lavrentiev Phenomenon for Free Discontinuity Problems

1995 ◽  
Vol 127 (1) ◽  
pp. 1-20
Author(s):  
A. Braides
Keyword(s):  
Lavrentiev Phenomenon ◽  
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.

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