Singular vanishing-viscosity limits of gradient flows: the finite-dimensional case

In this note we study the singular vanishing-viscosity limit of a gradient flow set in a finite-dimensional Hilbert space and driven by a smooth, but possibly non convex, time-dependent energy functional. We resort to ideas and techniques from the variational approach to gradient flows and rate-independent evolution to show that, under suitable assumptions, the solutions to the singularly perturbed problem converge to a curve of stationary points of the energy, whose behavior at jump points is characterized in terms of the notion of Dissipative Viscosity solution. We also provide sufficient conditions under which Dissipative Viscosity solutions enjoy better properties, which turn them into Balanced Viscosity solutions. Finally, we discuss the generic character of our assumptions.Comment: 27 page

Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature

In this paper we consider complete noncompact Riemannian manifolds $(M, g)$ with nonnegative Ricci curvature and Euclidean volume growth, of dimension $n \geq 3$. We prove a sharp Willmore-type inequality for closed hypersurfaces $\partial \Omega$ in $M$, with equality holding true if and only if $(M{\setminus}\Omega, g)$ is isometric to a truncated cone over $\partial\Omega$. An optimal version of Huisken's Isoperimetric Inequality for $3$-manifolds is obtained using this result. Finally, exploiting a natural extension of our techniques to the case of parabolic manifolds, we also deduce an enhanced version of Kasue's non existence result for closed minimal hypersurfaces in manifolds with nonnegative Ricci curvature.Comment: Any comment is welcome

Dimension reduction via Gamma convergence for soft active materials

We present a rigorous derivation of dimensionally reduced theories for thin sheets of nematic elastomers, in the finite bending regime. Focusing on the case of twist nematic texture, we obtain 2D and 1D models for wide and narrow ribbons exhibiting spontaneous flexure and torsion. We also discuss some variants to the case of twist nematic texture, which lead to 2D models with different target curvature tensors. In particular, we analyse cases where the nematic texture leads to zero or positive Gaussian target curvature, and the case of bilayers. \ua9 2017 Springer Science+Business Media Dordrech

Shape programming for narrow ribbons of nematic elastomers

Using the theory of Γ-convergence, we derive from three-dimensional elasticity new one-dimensional models for non-Euclidean elastic ribbons, i.e., ribbons exhibiting spontaneous curvature and twist. We apply the models to shape-selection problems for thin films of nematic elastomers with twist and splay-bend texture of the nematic director. For the former, we discuss the possibility of helicoid-like shapes as an alternative to spiral ribbons

Second order approximations of quasistatic evolution problems in finite dimension

In this paper, we study the limit, as epsilon goes to zero, of a particular solution of the equation epsilon(2) A(sic)(epsilon)(t) + epsilon B(u) over dot(epsilon) (t) + del(x)f(t, u(epsilon)(t)) = 0, where f (t, x) is a potential satisfying suitable coerciveness conditions. The limit u (t) of u(epsilon)(t) is piece-wise continuous and verifies del(x)f(t, u (t)) = 0. Moreover, certain jump conditions characterize the behaviour of u (t) at the discontinuity times. The same limit behaviour is obtained by considering a different approximation scheme based on time discretization and on the solutions of suitable autonomous systems