Related papers: Structure and classification results for the $\inf…
We associate curves of isotropic, Lagrangian and coisotropic subspaces to higher order, one parameter variational problems. Minimality and conjugacy properties of extremals are described in terms of these curves.
We prove quantitative versions for several results from geometric partial differential equations. Firstly, we obtain a double stability theorem for Serrin's overdetermined problem in spaceforms. Secondly, we prove stability theorems for…
In this paper we study the $L^2$-gradient flow of the penalized elastic energy on networks of $q$-curves in $\R^{n}$ for $q \geq 3$. Each curve is fixed at one end-point and at the other is joint to the other curves at a movable…
This paper is devoted to analyse the Dirichlet problem for a nonlinear elliptic equation involving the $1$--Laplacian and a total variation term, that is, the inhomogeneous case of the equation arising in the level set formulation of the…
The classical Euler's problem on optimal configurations of elastic rod in the plane with fixed endpoints and tangents at the endpoints is considered. The global structure of the exponential mapping that parameterises extremal trajectories…
The aim of the paper is to show that the solutions to variational problems with non-standard growth conditions satisfy a corresponding variational inequality without any smallness assumptions on the gap between growth and coercitivity…
Let $S$ be a complete flat surface, such as the Euclidean plane. We obtain direct characterizations of the connected components of the space of all curves on $S$ which start and end at given points in given directions, and whose curvatures…
Given a functional for a one-dimensional physical system, a classical problem is to minimize it by finding stationary solutions and then checking the positive definiteness of the second variation. Establishing the positive definiteness is,…
We consider the problem of prescribing the $Q_{\ gamma}$ curvature on $\mathbb{S}^n$. Using a perturbation method, we obtain existence results for curvatures close to a positive constant.
In this paper, we classify the class of constant weighted curvature curves in the plane with a log-linear density, or in other words, classify all traveling curved fronts with a constant forcing term in $\Bbb R^2.$ The classification gives…
A short historical account of the curves related to the two-dimensional floating bodies of equilibrium and the bicycle problem is given. Bor, Levi, Perline and Tabachnikov found, quite a number had already been described as Elastica by…
In this article we study constrained variational problems in one independent variable defined on the space of integral curves of a Frenet system in a homogeneous space G/H. We prove that if the Lagrangian is G-invariant and coisotropic then…
We extend some results known for the K\"ahler-Ricci flow to the Chern-Ricci flow regarding the independence of singularity types for long-time solutions. Specifically, we show that if a solution to the Chern-Ricci flow exists with uniformly…
We study horizontal differentiation when the set of feasible products is a structured subset of the Lancasterian characteristics space. Modeling this set as a compact Riemannian manifold, we show that intrinsic geometry governs…
Several integrability problems of differential equations are addressed by using the concept of $\mathcal{C}^{\infty}$-structure, a recent generalization of the notion of solvable structure. Specifically, the integration procedure associated…
We introduce novel finite element schemes for curve diffusion and elastic flow in arbitrary codimension. The schemes are based on a variational form of a system that includes a specifically chosen tangential motion. We derive optimal $L^2$-…
We study the systems of ordinary differential equations which are implicit with respect to the higher derivatives, appearing in the linear form, and their solutions near the singular points. The invertibility of the higher derivatives…
In this paper, we study geometric rigidity of Riemannian manifolds admitting stable solutions of certain elliptic problems (stability in a variational sense), that is, under suitable hypotheses, we are able to characterize the Riemannian…
We use the solution set of a real ordinary differential equation which has order n which is at least 2 to construct a smooth curve C in R^n. We describe when C is a proper embedding of infinite length with finite total first curvature.
In this paper the problems of the retrospective analysis of models with time-varying structure are considered. These models include contamination models with randomly switching parameters and multivariate classification models with an…