@article {2013, title = {Stable regular critical points of the Mumford-Shah functional are local minimizers}, journal = {Annales de l{\textquoteright}Institut Henri Poincare (C) Non Linear Analysis}, volume = {32}, number = {SISSA preprint;SISSA 33/2013/MATE}, year = {2015}, pages = {533-570}, publisher = {SISSA}, chapter = {533}, abstract = {

In this paper it is shown that any regular critical point of the Mumford{\textendash}Shah functional, with positive definite second variation, is an isolated local minimizer with respect to competitors which are sufficiently close in the $L^1$
-topology. A global\ minimality\ result in small tubular neighborhoods of the\ discontinuity\ set is also established.

}, keywords = {Mumford-Shah functional}, doi = {10.1016/j.anihpc.2014.01.006}, url = {https://www.sciencedirect.com/science/article/pii/S0294144914000171}, author = {Marco Bonacini and Massimiliano Morini} } @article {2013, title = {Local and global minimality results for a nonlocal isoperimetric problem on R^N}, journal = { SIAM Journal on Mathematical Analysis}, volume = {46}, number = {Siam journal on mathematical analysis}, year = {2014}, pages = {2310-2349}, publisher = {SIAM Publications}, abstract = {

We consider a nonlocal isoperimetric problem defined in the whole space R^N, whose nonlocal part is given by a Riesz potential with exponent $\alpha\in(0, N-1)$. We show that critical configurations with positive second variation are local minimizers and satisfy a quantitative inequality with respect to the L^1-norm. This criterion provides the existence of a (explicitly determined) critical threshold determining the interval of volumes for which the ball is a local minimizer, and allows to address several global minimality issues.

}, keywords = {Nonlocal isoperimetric problem}, doi = {10.1137/130929898}, url = {http://hdl.handle.net/1963/6984}, author = {Marco Bonacini and Riccardo Cristoferi} } @article {2013, title = {Stability of equilibrium configurations for elastic films in two and three dimensions}, journal = {Advances in Calculus of Variations}, volume = {8}, number = {SISSA preprint;SISSA 38/2013/MATE}, year = {2014}, month = {01/2014}, pages = {117-153}, publisher = {SISSA}, chapter = {117}, abstract = {

We establish a local minimality sufficiency criterion, based on the strict positivity of the second variation, in the context of a variational model for the epitaxial growth of elastic films. Our result holds also in the three-dimensional case and for a general class of nonlinear elastic energies. Applications to the study of the local minimality of flat morphologies are also shown.

}, keywords = {Epitaxially strained elastic films}, issn = {Online: 1864-8266; Print: 1864-8258}, doi = { https://doi.org/10.1515/acv-2013-0018}, url = {https://www.degruyter.com/view/j/acv.2015.8.issue-2/acv-2013-0018/acv-2013-0018.xml}, author = {Marco Bonacini} } @article {2013, title = {Epitaxially strained elastic films: the case of anisotropic surface energies}, journal = {ESAIM Control. Optim. Calc. Var. 19 (2013) 167-189}, number = {SISSA;31/2011/M}, year = {2013}, publisher = {EDP Sciences}, abstract = {

In the context of a variational model for the epitaxial growth of strained elastic films, we study the effects of the presence of anisotropic surface energies in the determination of equilibrium configurations. We show that the threshold effect that describes the stability of flat morphologies in the isotropic case remains valid for weak anisotropies, but is no longer present in the case of highly anisotropic surface energies, where we show that the flat configuration is always a local minimizer of the total energy. The main tool used to obtain these results is a minimality criterion based on the positivity of the second variation.

}, doi = {10.1051/cocv/2012003}, url = {http://hdl.handle.net/1963/4268}, author = {Marco Bonacini} } @mastersthesis {2013, title = {Minimality and stability results for a class of free-discontinuity and nonlocal isoperimetric problems}, year = {2013}, school = {SISSA}, keywords = {free-discontinuity problems}, author = {Marco Bonacini} }