We analyze a nonlinear discrete scheme depending on second-order finite differences. This is the two-dimensional analog of a scheme which in one dimension approximates a free-discontinuity energy proposed by Blake and Zisserman as a higher-order correction of the Mumford and Shah functional. In two dimension we give a compactness result showing that the continuous problem approximating this difference scheme is still defined on special functions with bounded hessian, and we give an upper and a lower bound in terms of the Blake and Zisserman energy. We prove a sharp bound by exhibiting the discrete-to-continuous Γ-limit for a special class of functions, showing the appearance new ‘shear' terms in the energy, which are a genuinely two-dimensional effect.
@article{M2AN_2012__46_2_389_0,
author = {Braides, Andrea and Defranceschi, Anneliese and Vitali, Enrico},
title = {A compactness result for a second-order variational discrete model},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {46},
year = {2012},
pages = {389-410},
doi = {10.1051/m2an/2011043},
mrnumber = {2855647},
zbl = {1272.49095},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2012__46_2_389_0}
}
Braides, Andrea; Defranceschi, Anneliese; Vitali, Enrico. A compactness result for a second-order variational discrete model. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 46 (2012) pp. 389-410. doi : 10.1051/m2an/2011043. http://gdmltest.u-ga.fr/item/M2AN_2012__46_2_389_0/
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