[00426] Variational methods for thin structures and free-boundary problems
Session Date & Time :
00426 (1/3) : 4C (Aug.24, 13:20-15:00)
00426 (2/3) : 4D (Aug.24, 15:30-17:10)
00426 (3/3) : 4E (Aug.24, 17:40-19:20)
Type : Proposal of Minisymposium
Abstract : Thin structures are classically studied using variational methods and PDEs, and they may be described both by surfaces and by free interfaces. On one hand, surfaces appear for modeling soap films and biological membranes minimizing suitable energy functionals, like area and Canham-Helfrich functionals. On the other hand, free interfaces separate a domain whose boundary is free: it is not known a priori and it depends on the solution of a PDE. Such a free-boundary problems naturally arise in many different models in Physics and Engineering, like for instance the Bernoulli one-phase problem.
Salvatore Stuvard (Università degli Studi di Milano)
Riccardo Tione (MPI MiS Leipzig)
Bozhidar Velichkov (Università di Pisa)
Hui Yu (University of Singapore)
Talks in Minisymposium :
[02176] On Stationary Points of Polyconvex Functionals
Author(s) :
Riccardo Tione (MPI MiS Leipzig)
Camillo De Lellis (Institute for Advanced Study)
Guido De Philippis (Courant Institute of Mathematics)
Antonio De Rosa (University of Maryland)
Jonas Hirsch (Universität Leipzig)
Bernd Kirchheim (Universität Leipzig)
Abstract : Quasi- and polyconvex energies arise naturally in modeling physical phenomena related to elasticity. From the mathematical viewpoint, a challenging question concerns the regularity of critical/stationary points and minimizers of these energies. My talk focuses on recent results in this direction concerning stationary points, i.e. critical points subject to outer and inner variations. I also address the application of this question to geometric measure theory.
[02741] Long time behavior and stability of surface diffusion flow
Author(s) :
Antonia Diana (Scuola Superiore Meridionale )
Abstract : We present a long-time existence and stability result for the surface diffusion flow in the flat torus.
According to this flow, smooth hypersurfaces move with the outer normal velocity given by the Laplacian of their mean curvature.
We show that if the initial set is sufficiently ``close’’ to a stable critical set for the volume-constrained Area functional, then the flow exists for all times and asymptotically converges to a ``translation’’ of the critical set.
[02967] Rectifiability for flat singularities of higher codimension area minimizers
Author(s) :
Camillo De Lellis (Institute for Advanced Study)
Paul Minter (Princeton University)
Anna Skorobogatova (Princeton University)
Abstract : Integral currents provide a natural setting in which to study the Plateau problem, but permit the formation of singularities in area-minimizers. The problem of determining the size and structure of the interior singular set of an area-minimizer in this setting has been studied in great detail since the 1960s, with many ground-breaking contributions. When the codimension is higher than 1, due to the presence of singular points with high multiplicity flat tangent cones, little progress has been made since Almgren's celebrated (m-2)-Hausdorff dimension bound on the singular set, the proof of which has since been simplified by De Lellis and Spadaro.
In this talk I will discuss joint work with Camillo De Lellis and Paul Minter, in which we achieve (m-2)-rectifiability for the singular set.
[03181] A capillarity theory approach to the analysis of soap films
Author(s) :
Salvatore Stuvard (University of Milan)
Darren King (New York University)
Francesco Maggi (University of Texas at Austin)
Antonello Scardicchio (Abdus Salam ICTP)
Abstract : I will present a variational model, based on Gauss' theory of capillarity, which describes soap films as sets of finite perimeter enclosing a prescribed volume of fluid and satisfying a spanning condition of homotopic type, rather as minimal surfaces. I will discuss the corresponding existence theory, the sharp regularity properties of the minimizers, their asymptotic behavior in the vanishing volume limit, and I will attempt a qualitative description of their local and global geometry.
[03306] Transport of currents and geometric Rademacher-type theorems
Author(s) :
Paolo Bonicatto (University of Warwick)
Abstract : Given a vector field $b$ on $\mathbb R^d$, one usually studies the transport/continuity equation drifted by $b$ looking for solutions in the class of functions or at most in the class of measures. I will talk about recent efforts, motivated by the modeling of defects in crystalline materials, aimed at extending the previous theory to the case when the unknown is instead a family of k-currents in $\mathbb R^d$.
[03455] Interior regularity for stationary two-dimensional multivalued maps
Author(s) :
Jonas Hirsch (University of Leipzig)
Luca Spolaor (UCSD)
Abstract : $Q$-valued maps minimizing a suitably defined Dirichlet energy were introduce by Almgren in his proof of the optimal regularity of area minimizing currents in any dimension and codimension. In this talk I will discuss the extension of Almgren's result to stationary $Q$-valued maps in dimension $2$. This is joint work with Jonas Hirsch (Leipzig).
[03469] On the Kircchoff-Plateau problem: critical points and regularity
Author(s) :
Giulia Bevilacqua (Università di Pisa)
Abstract : In this talk I will discuss some generalization of the Plateau problem, which in its classical form asks if it exists a surface of minimal area spanning a given boundary. First-order necessary conditions and regularity properties are studied when a thick rod (with non vanishing thickness) and/or an elastic curve are assigned as the boundary spanned by the surface.
[03626] Graphical solutions to one-phase problems
Author(s) :
Hui Yu (National University of Singapore)
Max Engelstein (University of Minnesota-Twin Cities)
Xavier Fernandez-Real (Ecole Polytechnique Federale de Lausanne)
Abstract : The free boundaries of solutions (critical points of the functional) to the one-phase problem can have rich geometry. Such richness can be reduced by imposing the graphical condition.
In this talk, we show that if homogeneous minimizers are trivial in dimension k, then graphical solutions are trivial in dimension k+1. This works for both the classical one-phase problem as well as its thin counterpart.
This talk is based on a joint work with Max Engelstein (Minnesota) and Xavier Fernandez-Real (EPFL).
[04117] Stable Möbius bands obtained by isometrically deforming circular helicoids
Author(s) :
Eliot Fried (Okinawa Institute of Science and Technology)
Vikash Chaurasia (Okinawa Institute of Science and Technology)
Abstract : We consider a variational problem for finding an isometric deformation from a (circular helicoid to a stable Möbius band. Helicoids with certain specific numbers of turns yield stable bands with \(n=2k+1\), \(k\ge1\), half twists and \(n\)-fold rotational symmetry. Each such band has the least energy of any stable competitor with the same number of half twists. Helicoids with other numbers of turns yield two stable bands with equal energy but different numbers of half twists.
[04199] Minimization of the Canham-Helfrich within generalised Gauss graphs
Author(s) :
Anna Kubin (Politecnico di Torino)
Luca Lussardi (Politecnico di Torino)
Marco Morandotti (Politecnico di Torino)
Abstract : The Canham-Helfrich functional is the most widely used functional to study the equilibrium of biological membranes as a result of the competition between mean curvature and Gaussian curvature.
In this talk, we review some approaches to the minimisation problem for this functional and present novel results in the setting of generalised Gauss graphs.
This is joint work with Anna Kubin and Luca Lussardi.
[04579] Min-max minimal surfaces with contact angle conditions
Author(s) :
Luigi De Masi (University of Padova)
Abstract : Existence and regularity of minimal surfaces (i.e. stationary points for area functional) has been an active topic of research for the last decades. When minimizing strategies produce just trivial solutions, min-max methods may be successfully used.
In this talk, based on a joint work with G. De Philippis, I will discuss min-max construction of a minimal surface $\Sigma$ in a container $\mathcal{M} \subset \mathbb{R}^3$ which meets $\partial \mathcal{M}$ with a fixed angle.
[04656] Regularity of the optimal shapes for a class of integral functionals
Author(s) :
Bozhidar Velichkov (Università di Pisa)
Abstract : This talk is dedicated to the regularity of solutions to the following variational problem
$$\min\Big\{J(\Omega)\ :\ \Omega\subset D\Big\},$$
where $D\subset \mathbb{R}^d$ is a given bounded open set and $J$ is a functional of the form
$$J(\Omega)=\int_\Omega j(x,u_\Omega)\,dx+|\Omega|\,,$$
where $j:D\times \mathbb{R}\to\mathbb{R}$ is a given "cost function" and $u_\Omega$ is the solution to
$$-\Delta u_\Omega=f(x)\quad\text{in}\quad\Omega\ ,\qquad u_\Omega\in H^1_0(\Omega)\,,$$
where $f:D\to\mathbb{R}$ is a fixed function. We will discuss the case $j(x,u_\Omega)=-g(x)u_\Omega$, which leads to a free boundary system of the form
\begin{align*}
-\Delta u_\Omega=f(x)&\quad\text{in}\quad\Omega\\
-\Delta v_\Omega=g(x)&\quad\text{in}\quad\Omega\\
u_\Omega=v_\Omega=0&\quad\text{on}\quad\partial \Omega\\
|\nabla u_\Omega||\nabla v_\Omega|=1&\quad\text{on}\quad\partial \Omega
\end{align*}
We will show that, if the functions $f$ and $g$ are positive and comparable, and if the dimension $d$ of the ambient space is $2$, $3$, or $4$, then any optimal set $\Omega$ is $C^{1,\alpha}$ smooth (and solves the above system in the classical sense). The talk is based on joint works with Giorgio Tortone (University of Pisa) and Francesco Paolo Maiale (Scuola Normale Superiore), and on a joint work with Giuseppe Buttazzo (University of Pisa), Francesco Paolo Maiale (Scuola Normale Superiore), Dario Mazzoleni (University of Pavia), and Giorgio Tortone (University of Pisa).
