02147nas a2200157 4500008004100000245008700041210006900128260001000197300001200207490000700219520161800226653002801844100002001872700002501892856007201917 2015 en d00aStable regular critical points of the Mumford-Shah functional are local minimizers0 aStable regular critical points of the MumfordShah functional are bSISSA a533-5700 v323 a

In this paper it is shown that any regular critical point of the Mumford–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.

10aMumford-Shah functional1 aBonacini, Marco1 aMorini, Massimiliano uhttps://www.sciencedirect.com/science/article/pii/S029414491400017101060nas a2200157 4500008004100000245008400041210006900125260002200194300001400216490000700230520054900237653003500786100002000821700002500841856003600866 2014 en d00aLocal and global minimality results for a nonlocal isoperimetric problem on R^N0 aLocal and global minimality results for a nonlocal isoperimetric bSIAM Publications a2310-23490 v463 a

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.

10aNonlocal isoperimetric problem1 aBonacini, Marco1 aCristoferi, Riccardo uhttp://hdl.handle.net/1963/698400979nas a2200157 4500008004100000022004000041245009000081210006900171260001900240300001200259490000600271520039700277653003900674100002000713856008800733 2014 en d aOnline: 1864-8266; Print: 1864-825800aStability of equilibrium configurations for elastic films in two and three dimensions0 aStability of equilibrium configurations for elastic films in two bSISSAc01/2014 a117-1530 v83 a

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.

10aEpitaxially strained elastic films1 aBonacini, Marco uhttps://www.degruyter.com/view/j/acv.2015.8.issue-2/acv-2013-0018/acv-2013-0018.xml01025nas a2200109 4500008004100000245008100041210006900122260001700191520065100208100002000859856003600879 2013 en d00aEpitaxially strained elastic films: the case of anisotropic surface energies0 aEpitaxially strained elastic films the case of anisotropic surfa bEDP Sciences3 a

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.

1 aBonacini, Marco uhttp://hdl.handle.net/1963/426800512nas a2200109 4500008004100000245010700041210006900148260001000217653003200227100002000259856012300279 2013 en d00aMinimality and stability results for a class of free-discontinuity and nonlocal isoperimetric problems0 aMinimality and stability results for a class of freediscontinuit bSISSA10afree-discontinuity problems1 aBonacini, Marco uhttps://math.sissa.it/publication/minimality-and-stability-results-class-free-discontinuity-and-nonlocal-isoperimetric