Geometry, Analysis, and Approximation of Variational Problems



Seminarraum 226, 2nd floor, Hermann-Herder-Str. 10, 79104 Freiburg (map).


Monday (May 14)Tuesday (May 15)Wednesday (May 16)
9:30-10:20M. RumpfE. OudetP. Reiter
10:20-11:10R. NürnbergM. MoraM. Röger
11:40-12:30P. PozziL. DieningL. Mari
14:00-14:25M. JensenV. BangertEnd
14:25-14:50M. RuggeriA. Pluda
14:50-15:15D. GallistlA. Malchiodi
15:15-15:40A. PapathanassopoulosM. Novaga
15:40-16:10CoffeeM. Krepela
16:10-17:00B. StinnerFree

Welcome Reception:

Participants are invited to join for an informal dinner on Sunday, May 13, between 7pm and 10pm at Pizzeria Ochsenbrugg (map).


Monday: Schlossbergrestaurant Dattler (map).
Tuesday: Paradies Freiburg (map).


University Mensa Institutsviertel (map) during breaks 12:30-14:00.


Victor Bangert (Freiburg): An upper area bound for surfaces in Riemannian manifolds

Consider a sequence of complete, immersed surfaces in a compact Riemannian manifold M. Assume that the areas of the surfaces tend to infinity, while the \(L^2\)-norms of their second fundamental forms remain bounded. We prove that there exists a totally geodesic immersion of a complete surface into M. So, if M does not contain any totally geodesic surface, then an upper bound on the \(L^2\)-norm of the second fundamental form of a complete surface in M implies an upper bound on its area. Note that generic Riemannian manifolds (of dimension at least three) do not contain any totally geodesic surfaces. (Joint work with Ernst Kuwert)

Lars Diening (Bielefeld): Linearization of the p-Poisson equation

This is a joint work with Massimo Fornasier and Maximilian Wank. In this talk we propose a iterative method to solve the non-linear \(p\)-Poisson equation. The method is derived from a relaxed energy by an alternating direction method. We are able to show algebraic convergence of the iterates to the solution. However, our numerical experiments based on finite elements indicate optimal, exponential convergence.

Dietmar Gallistl (Twente): Rayleigh-Ritz approximation of the inf-sup constant for the divergence

This contribution proposes a compatible finite element discretization for the approximation of the inf-sup constant for the divergence. The new approximation replaces the \(H^{-1}\) norm of a gradient by a discrete \(H^{-1}\) norm which behaves monotonically under mesh-refinement. By discretizing the pressure space with piecewise polynomials, upper bounds to the inf-sup constant are obtained. The scheme enables an approximation with arbitrary polynomial degrees. It can be viewed as a Rayleigh-Ritz method and it gives monotonically decreasing approximations of the inf-sup constant under mesh refinement. In particular, the computed approximations are guaranteed upper bounds for the inf-sup constant. The novel error estimates prove convergence rates for the approximation of the inf-sup constant provided it is an isolated eigenvalue of the corresponding non-compact eigenvalue problem; otherwise, plain convergence is achieved. Numerical computations on uniform and adaptive meshes are presented.

Max Jensen (Sussex): Dynamic Programming for Finite Ensembles of Nanomagnetic Particles

The stochastic Landau-Lifshitz-Gilbert equation describes magnetization dynamics in ferromagnetic materials in a thermal bath. In this presentation I discuss the optimal control of a finite spin system governed by the stochastic Landau-Lifshitz-Gilbert equation in order to guide the configuration optimally into a target state. At the heart of our analysis is the problem statement with a Bellman PDE on the state manifold. Through the Hopf-Cole transformation, an equivalent linear variational formulation is available, through which we show wellposedness as well as regularity of the value function and the optimal controls. The seminar is based on joint work with Ananta Majee and Andreas Prohl (Universität Tuebingen).

Martin Krepela (Freiburg): A counterexample related to the regularity of the p-Stokes problem.

It was expected that the natural regularity of a solution to the p-Stokes problem could be proven by using the Korn inequality in weighted Lebesgue spaces. Unfortunately, such an approach is not possible, as will be shown in this talk by constructing an appropriate counterexample.

Andrea Malchiodi (Pisa): Prescribing Gaussian and Geodesic curvature on surfaces with boundary

We consider the classical problem of finding conformal metrics on a surface such that both the Gaussian and the geodesic curvatures are assigned functions. We use variational methods and blow-up analysis to find existence of solutions under suitable assumptions. A peculiar aspect of the problem is that there are blow-up profiles with infinite volume that have to be taken care of. This is joint work with R. Lopez-Soriano and D. Ruiz.

Luciano Mari (Pisa): On some Liouville-Bernstein theorems for graphs with prescribed mean curvature

The core of this talk is the investigation of Bernstein-Liouville type theorems for entire graphs \(u : M \rightarrow \mathbb{R}\) over a complete Riemannian manifold, under the condition that the mean curvature of the graph is a prescribed function. Emphasis will be put on minimal graphs, on solitons for the mean curvature flow and on capillary graphs. After a review of some classical results to stress the peculiar features of the mean curvature operator, we will focus on recent contributions based on joint works with B. Bianchini, M. Rigoli, P. Pucci, G. Colombo, M. Magliaro and J.H. de Lira. In most cases, the geometry of the underlying manifold is taken into account just via the growth of the volume of its geodesic balls. The main Liouville type theorems are also applicable to more general quasilinear inequalities of the type $$ \mathrm{div} \big( A(|\nabla u|) \nabla u\big) \ge b(x) f(u) l(|\nabla u|), $$ a model that includes operators of interest like the one of exponentially harmonic functions, and the \(p\) and \((p,q)\)-Laplacians.

Maria-Giovanna Mora (Pavia): The equilibrium measure for a nonlocal dislocation energy

In this talk I will discuss the minimization problem for a nonlocal energy, that describes the interaction of positive edge dislocations in the plane. The interaction kernel is given by the sum of the Coulomb potential and of an anisotropic term, that makes the potential non-radially symmetric. The purely logarithmic potential has been studied in a variety of contexts (Ginzburg-Landau vortices, Coulomb gases, random matrices, Fekete sets) and in this case it is well known that the equilibrium measure is given by the celebrated circle law. I will show how the anisotropy of the potential changes dramatically the nature of the equilibrium measure, which turns out to be supported on the vertical axis and distributed according to Wigner’s semi-circle law. This result is one of the few examples where the minimizer of a nonlocal energy is explicitly computed and the first one in the case of anisotropic kernels. Moreover, it gives a positive answer to the conjecture that positive dislocations tend to arrange themselves in vertical walls.

Matteo Novaga (Pisa): Convergence of tessellations

I will discuss a discrete to continuous problem for the Heitmann-Radin energy, and a related Gamma-convergence result for optimal tessellations in two-dimensions.

Robert Nürnberg (London): Variational approximation of axisymmetric formulations for geometric evolution equations

We present variational formulations of mean curvature flow and surface diffusion for axisymmetric hypersurfaces in R^3. On recalling important properties of the schemes introduced by the authors for the corresponding geometric evolution equations for closed curves in the plane, we introduce suitable finite element approximations, and investigate their stability and vertex distribution properties. (Joint work with John W. Barrett and Harald Garcke)

Edouard Oudet (Grenoble): Investigating one dimensional structures

We focus our attention on shape optimization problems in which one dimensional connected objects are involved. Very old and classical problems in calculus of variation are of this kind: euclidean Steiner's tree problem, optimal irrigation networks, cracks propagation, etc. In a first part we quickly recall some previous work in collaboration with F. Santambrogio related to the functional relaxation of the irrigation cost. We establish a \(\Gamma\)-convergence of Modica and Mortola's type and illustrate its efficiency from a numerical point of view by computing optimal networks associated to simple sources/sinks configurations. We also present more evolved situations with non Dirac sinks in which a fractal behavior of the optimal network is expected. In the second part of the talk we restrict our study to the euclidean Steiner's tree problem. We recall recent numerical approach which have been developed the last five years to approximate optimal trees: partitioning formulation, relaxation with geodesic distance terms and energetic constraints. We describe the first results obtained in collaboration with A. Massaccesi and B. Velichkov to certify the optimality of a given tree. With our discrete parametrization of generalized calibration, we are able to recover the theoretical optimal matrix fields which certify the optimality of simple trees associated to the vertices of regular polygons. Finally, we focus on the delicate problem of the identification of the optimal structure. Based on a recent approach obtained in collaboration with G. Orlandi and M. Bonafini, we describe the first convexification framework associated to the Euclidean Steiner tree problem which provide relevant tools from a numerical point of view.

Alexis Papathanassopoulos (Freiburg): Numerical computation of minimizers for nonlinear isometric elastic energy functionals

Minimizing the nonlinear elastic energy of thin plates consists in an appropriate treatment of bending and stretching terms. The model we use is given by the reduced two-dimensional limit of an abstract three-dimensional hyperelastic formulation. We consider different approaches for the numerical approximation of isometry for such an energy and present some results for the special application of thin films on compliant substrates, which exhibit the patterns that were analytically predicted by identifying the correct energy scaling laws.

Alessandra Pluda (Regensburg): Calibrations for minimal Steiner networks

Abstract: The Steiner problem in it classical formulation reads as follows: find the shortest network that connects n given points in the plane. Although existence and regularity of minimizers is well known, in general finding explicitly a solution is extremely challenging, even numerically. For this reason every method to determine solutions is welcome. A possible tool is the notion of calibrations, introduced in the context of minimal surfaces by Harvey and Lawson in their paper “Calibratied Geometries” and adapted to the case of Steiner in several variants. In this talk I will describe the approach to the Steiner problem via covering spaces by Brakke and Amato-Bellettini-Paolini and I will define calibrations in this setting. I will also give some examples of both existence and non-existence of calibrations and to overcome this second unlucky case I will introduce the notion of calibration in families. If time allows, I will compare the different definitions of calibrations presented in the literature. This is an ongoing project with Marcello Carioni (Universität Graz).

Paola Pozzi (Duisburg-Essen): On anisotropic curvature flow

In this talk I will discuss anisotropic curvature motion for planar immersed curves and give a short-time existence result that holds for general anisotropies. This is joint work with G. Mercier and M. Novaga

Philipp Reiter (Georgia): Approximating self-avoiding inextensible curves

In order to minimize the bending energy of curves within isotopy classes, we model impermeability by considering a linear combination of the bending energy and a self-repulsive functional, namely the tangent-point potential. We discuss a semi-implicit numerical scheme for a gradient flow and present a stability result which guarantees both energy decay during the evolution and maintenance of the arclength constraint. Numerical simulations indicate a complex energy landscape which makes it a challenging task to identify global minimizers. (This is joint work with Sören Bartels)

Matthias Röger (Dortmund): Phase-field approximations of the Willmore functional: non-smooth configurations and interacting planar interfaces

A modification of De Giorgis phase field approximation of the Willmore energy is widely used in many applications and has been rigorously justified for smooth phase boundaries and small space dimensions. We investigate the behavior of the phase field approximation for non-smooth limit configurations. We also discuss the slow motion of interacting 1D phase boundaries. (This is joint work with Carsten Zwilling, Dortmund)

Michele Ruggeri (Wien): Convergent finite element methods for the Landau-Lifshitz-Gilbert equation

We consider the numerical approximation of the Landau-Lifshitz-Gilbert (LLG) equation, which describes the dynamics of the magnetization in ferromagnetic materials. The numerical integration of the LLG equation poses several challenges: strong nonlinearities, a nonconvex pointwise constraint, an intrinsic energy law, and the presence of nonlocal field contributions, which prescribe the coupling with other partial differential equations. We discuss numerical integrators, based on lowest-order finite elements in space, that are proven to be (unconditionally) convergent towards a weak solution of the problem. The talk is based on recent work with C.-M. Pfeiler, D. Praetorius, and B. Stiftner (TU Wien).

Martin Rumpf (Bonn): Variational time discretization of Riemannian splines

The talk discusses a generalization of cubic splines to Riemannian manifolds. Spline curves are defined as minimizers of the spline energy - a combination of the Riemannian path energy and the time integral of the squared covariant derivative of the path velocity - under suitable interpolation conditions. A variational time discretization for the spline energy leads to a constrained optimization problem over discrete paths on the manifold. Existence of continuous and discrete spline curves is established using the direct method in the calculus of variations. Furthermore, the convergence of discrete spline paths to a continuous spline curve follows from the \(\Gamma\)-convergence of the discrete to the continuous spline energy. Finally, selected example settings are discussed, including splines on embedded finite-dimensional manifolds, on a high-dimensional manifold of discrete shells with applications in surface processing, and on the infinite-dimensional shape manifold of viscous rods. This is joint work with Behrend Heeren and Benedikt Wirth.

Björn Stinner (Warwick): Finite element approximation of geometric PDEs coupled with surface PDEs

We consider problems where a geometric equation for an evolving surface or curve is coupled with a system of reaction-diffusion equations on the thus described moving manifold. Such type of problems are motivated by applications in cell biology. We present some finite element schemes for such systems and report on some recent convergence results, which cover the one-dimensional graph case for the elastic flow with forcing, coupled with lateral diffusion for a surface field. Building up on previous findings for the geometric equation the main challenge is to obtain better control of the length element that is required to treat the diffusion equation. Numerical simulation results support the theoretical findings but also show that the approach is effective beyond the parameter regime that has been analysed.