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Luca Mugnai

Publications and source records attributed to Luca Mugnai.

8 recordsLinked to original sources
arXiv2026

Confined elastic wires of large length: the sharp second-order asymptotics

We study the least bending energy $m_L$ of a closed embedded wire of prescribed length $L$ confined to the closed unit disk. It is classical that the bending energy of any closed curve in the disk is at least its length, with equality only for multiply covered unit circles. Such a circle is the energy limit of a stack of disjoint embedded circles, but not of a single embedded loop, and the size of the resulting defect $m_L-L$ has remained open. We prove that \[ m_L \;=\; L+c_1\,L^{4/9}+O\bigl(L^{1/3}\bigr)\qquad\text{as }L\to\infty , \] with a sharp constant $c_1=5.223049\ldots$ given in closed form by two one-dimensional model problems. In particular $m_L-L\asymp L^{4/9}$, which improves the previously known upper bound of order $\sqrt{L}$ and provides the first lower bound beyond the trivial one.

math.AP↗
arXiv2015

Global solutions to the volume-preserving mean-curvature flow

In this paper, we construct global distributional solutions to the volume-preserving mean-curvature flow using a variant of the time-discrete gradient flow approach proposed independently by Almgren, Taylor and Wang (SIAM J. Control Optim. 31(2): 387- 438, 1993) and Luckhaus and Sturzenhecker (Calc. Var. Partial Differential Equations 3(2): 253-271, 1995).

math.AP↗
arXiv2013

A phase field model for the optimization of the Willmore energy in the class of connected surfaces

We consider the problem of minimizing the Willmore energy connected surfaces with prescribed surface area which are confined to a finite container. To this end, we approximate the surface by a phase field function $u$ taking values close to +1 on the inside of the surface and -1 on its outside. The confinement of the surface is now simply given by the domain of definition of $u$. A diffuse interface approximation for the area functional, as well as for the Willmore energy are well known. We address the topological constraint of connectedness by a nested minimization of two phase fields, the second one being used to identify connected components of the surface. In this article, we provide a proof of Gamma-convergence of our model to the sharp interface limit.

math.AP↗
arXiv2010

Confined elastic curves

We consider the problem of minimizing Euler's elastica energy for simple closed curves confined to the unit disk. We approximate a simple closed curve by the zero level set of a function with values +1 on the inside and -1 on the outside of the curve. The outer container now becomes just the domain of the phase field. Diffuse approximations of the elastica energy and the curve length are well known. Implementing the topological constraint thus becomes the main difficulty here. We propose a solution based on a diffuse approximation of the winding number, present a proof that one can approximate a given sharp interface using a sequence of phase fields, and show some numerical results using finite elements based on subdivision surfaces.

math.AP↗
arXiv2009

Convergence of perturbed Allen-Cahn equations to forced mean curvature flow

We study perturbations of the Allen-Cahn equation and prove the convergence to forced mean curvature flow in the sharp interface limit. We allow for perturbations that are square-integrable with respect to the diffuse surface area measure. We give a suitable generalized formulation for forced mean curvature flow and apply previous results for the Allen-Cahn action functional. Finally we discuss some applications.

math.AP↗
arXiv2007

The Allen-Cahn Action functional in higher dimensions

The Allen-Cahn action functional is related to the probability of rare events in the stochastically perturbed Allen-Cahn equation. Formal calculations suggest a reduced action functional in the sharp interface limit. We prove in two and three space dimensions the corresponding lower bound. One difficulty is that diffuse interfaces may collapse in the limit. We therefore consider the limit of diffuse surface area measures and introduce a generalized velocity and generalized reduced action functional in a class of evolving measures. As a corollary we obtain the Gamma convergence of the action functional in a class of regularly evolving hypersurfaces.

math.AP↗