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Gamma-convergence results for phase-field approximations of the 2D-Euler Elastica Functional

Luca Mugnai — 2013

ESAIM: Control, Optimisation and Calculus of Variations

We establish some new results about the -limit, with respect to the -topology, of two different (but related) phase-field approximations { } , { ˜ } ℰ ε ε ,   x10ff65; ℰ ε ε of the so-called Euler’s Elastica Bending Energy for curves in the plane. In particular we characterize the-limit as  → 0 of ℰ, and show that in general the -limits of ℰand ˜ x10ff65; ℰ ε do not coincide on indicator functions of sets with non-smooth boundary. More precisely we show that the domain of the-limit of...

Anisotropic geometric functionals and gradient flows

Giovanni BellettiniLuca Mugnai — 2009

Banach Center Publications

We survey some recent results on the gradient flow of an anisotropic surface energy, the integrand of which is one-homogeneous in the normal vector. We discuss the reasons for assuming convexity of the anisotropy, and we review some known results in the smooth, mixed and crystalline case. In particular, we recall the notion of calibrability and the related facet-breaking phenomenon. Minimal barriers as weak solutions to the gradient flow in case of nonsmooth anisotropies are proposed. Furthermore,...

Some aspects of the variational nature of mean curvature flow

Giovanni BellettiniLuca Mugnai — 2008

Journal of the European Mathematical Society

We show that the classical solution of the heat equation can be seen as the minimizer of a suitable functional defined in space-time. Using similar ideas, we introduce a functional on the class of space-time tracks of moving hypersurfaces, and we study suitable minimization problems related with . We show some connections between minimizers of and mean curvature flow.

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