In this paper it is shown that any regular critical point of the Mumfordâ€“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.

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.

PB - SIAM Publications VL - 46 UR - http://hdl.handle.net/1963/6984 IS - 4 U1 - 6976 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - JOUR T1 - Stability of equilibrium configurations for elastic films in two and three dimensions JF - Advances in Calculus of Variations Y1 - 2014 A1 - Marco Bonacini KW - Epitaxially strained elastic films AB -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.

PB - SISSA VL - 8 UR - https://www.degruyter.com/view/j/acv.2015.8.issue-2/acv-2013-0018/acv-2013-0018.xml IS - 2 U1 - 6997 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - JOUR T1 - Epitaxially strained elastic films: the case of anisotropic surface energies JF - ESAIM Control. Optim. Calc. Var. 19 (2013) 167-189 Y1 - 2013 A1 - Marco Bonacini AB -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.

PB - EDP Sciences UR - http://hdl.handle.net/1963/4268 U1 - 3999 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - THES T1 - Minimality and stability results for a class of free-discontinuity and nonlocal isoperimetric problems Y1 - 2013 A1 - Marco Bonacini KW - free-discontinuity problems PB - SISSA U1 - 7204 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER -