A Quantitative Second Order Minimality Criterion for Cavities in Elastic Bodies

Abstract

Abstract. We consider a functional which models an elastic body with a cavity. We show that if a critical point has positive second variation then it is a strict local minimizer. We also provide a quantitative estimate.

Local and Global Minimality Results for a Nonlocal Isoperimetric Problem on $\mathbb{R}^N$

“…As anticipated above, our methods follow a second variation approach which has been recently developed and applied to different variational problems, whose common feature is the fact that the energy functionals are characterized by the competition between bulk energies and surface energies (see [14] in the context of epitaxially strained elastic films, [5,4] for the Mumford-Shah functional, [6] for a variational model for cavities in elastic bodies). In particular we stress the attention on [1], which deals with energies in the form (1.1) in a periodic setting (see also [21], where the same problem is considered in an open set with Neumann boundary conditions).…”

Minimality via second variation for microphase separation of diblock copolymer melts

“…Indeed it has successfully been applied to prove stability and minimality criteria in several contexts (see e.g. [1,3,5,10]). Directly related to our problem is the work by Ren and Wei and by Choksi and Sternberg (cf.…”

“…The proof of the main result, Theorem 1.1, is presented in Sections 5 and 6. The scheme follows a well-established path (see for instance [1,3,5,10]). First we use the general second variation formula from Proposition 4.1 to prove the local minimality among regular sets which are close to the critical one in the W 2;p -topology and satisfy the orthogonality condition (1.7) (cf.…”

Motion of three-dimensional elastic films by anisotropic surface diffusion with curvature regularization

“…In this paper we study the morphologic evolution of anisotropic epitaxially strained films, driven by stress and surface mass transport in three dimensions. This can be viewed as the evolutionary counterpart of the static theory developed in [11,23,25,22,9,15] in the two-dimensional case and in [10] in three dimensions. The two dimensional formulation of the same evolution problem has been addressed in [24] (see also [32] for the case of motion by evaporation-condensation).…”