Related papers: The {\L}ojasiewicz-Simon gradient inequality for o…
We consider a parabolic obstacle problem for Euler's elastic energy of graphs with fixed ends. We show global existence, well-posedness and subconvergence provided that the obstacle and the initial datum are suitably 'small'. For symmetric…
In this paper, we deals with isoperimetric-type inequalities for closed convex curves in the Euclidean plane R^2. We derive a family of parametric inequalities involving the following geometric functionals associated to a given convex curve…
We introduce a non-local $L^2$-gradient flow for the Willmore energy of immersed surfaces which preserves the isoperimetric ratio. For spherical initial data with energy below an explicit threshold, we show long-time existence and…
In this paper we consider a control system of the form $\dot x = F(x)u$, linear in the control variable $u$. Given a fixed starting point, we study a finite-horizon optimal control problem, where we want to minimize a weighted sum of an…
We present variational approximations of boundary value problems for curvature flow (curve shortening flow) and elastic flow (curve straightening flow) in two-dimensional Riemannian manifolds that are conformally flat. For the evolving open…
In this work, we show that a family of non-linear mean-field equations on discrete spaces can be viewed as a gradient flow of a natural free energy functional with respect to a certain metric structure we make explicit. We also prove that…
In this note, we study an obstacle problem for the elastic flow. We prove the local-in-time existence of weak solutions and discuss their relation to classical solutions when additional regularity is obtained. Related results concerning…
We prove a Lojasiewicz-Simon inequality $$ \left| E(u) - 4\pi n \right| \leq C \| \mathcal{T}(u) \|^\alpha $$ for maps $u \in W^{2,2}\left( S^2, S^2 \right).$ The inequality holds with $\alpha = 1$ in general and with $\alpha > 1$ unless…
We analyze the long-time behavior of solutions to semilinear parabolic equations in Euclidean space that arise as gradient flows of an energy functional. We prove that, for general initial data (including data without compact support) the…
We study the gradient flow of the potential energy on the infinite-dimensional Riemannian manifold of spatial curves parametrized by the arc length, which models overdamped motion of a falling inextensible string. We prove existence of…
We compute the relaxation of the total energy related to a variational model for nematic elastomers, involving a nonlinear elastic mechanical energy depending on the orientation of the molecules of the nematic elastomer, and a nematic…
We study families of smooth, embedded, regular planar curves $ \alpha : \left [-1,1 \right ]\times \left [0,T \right )\to \mathbb{R}^{2}$ with generalised Neumann boundary conditions inside cones, satisfying three variants of the…
Due to its applications in many different places in machine learning and other connected engineering applications, the problem of minimization of a smooth function that satisfies the Polyak-{\L}ojasiewicz condition receives much attention…
We investigate equilibrium configurations for surface energies which contain the squared $L^2$ norm of the difference of the mean curvature H and the spontaneous curvature $c_o$ coupled with the elastic energy of the boundary curve, which…
We investigate several possibilities of obtaining a {\L}ojasiewicz inequality for definable multifunctions and give some examples of applications thereof. In particular, we prove that the Hausdorff distance and its extension to closed sets…
In this paper we prove some geometric inequalities for closed surfaces in Euclidean three-space. Motivated by Gage's inequality for convex curves, we first verify that for convex surfaces the Willmore energy is bounded below by some…
In this letter we study the proximal gradient dynamics. This recently-proposed continuous-time dynamics solves optimization problems whose cost functions are separable into a nonsmooth convex and a smooth component. First, we show that the…
We consider the evolution of open planar curves by the steepest descent flow of a geometric functional, under different boundary conditions. We prove that, if any set of stationary solutions with fixed energy is finite, then a solution of…
We consider a non-negative and one-homogeneous energy functional $\mathcal J$ on a Hilbert space. The paper provides an exact relation between the solutions of the associated gradient-flow equations and the energetic solutions generated via…
Very little is yet known regarding the Willmore flow of surfaces with Dirichlet boundary conditions. We consider surfaces with a rotational symmetry as initial data and prove a global existence and convergence result for solutions of the…