Category Archives: Abstracts

A phase field model for Willmore’s energy with topological constraint

Patrick Dondl (Freiburg, Germany)

We consider the problem of minimizing Willmore’s energy on confined and connected surfaces with prescribed surface area. To this end, we approximate the surface by a level set function u admitting the value +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 Willmore’s energy are well known. We address the main difficulty, namely the topological constraint of connectedness by a penalization of a geodesic distance which is chosen to be sensitive to connected components of the phase field level sets and provide a proof of Gamma-convergence of our model to the sharp interface limit. Furthermore, we show some numerical results. This is joint work with Stephan Wojtowytsch (Durham University) and Antoine Lemenant (Universit Paris 7).

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Pathwise Well-Posedness of the Fast Diffusion Equation with Affine Multiplicative Noise

Ben Fehrman (MPI-MIS Leipzig)

In this talk, which describes joint work with Benjamin Gess, I will discuss the existence and uniqueness of pathwise entropy solutions for the fast diffusion equation driven by affine multiplicative noise.  The theory of such solutions is motivated by the study of stochastic viscosity solutions, and was first developed in the context of scalar conservation laws with rough fluxes by Lions, Perthame and Souganidis, and later extended by Gess and Souganidis.  Their approach is based upon the kinetic formulation of the equation, and involves testing the solution against data propagating along the corresponding path-dependent characteristics.  I hope to describe the analogous theory in the context of the fast-diffusion equation, and to explain how it can be used to establish the well-posedness of pathwise entropy solutions in this setting.

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Some results on stochastic homogenization of non-convex Hamilton-Jacobi equations

William Feldman (Chicago, USA)

I will discuss some progress about the homogenization of non-convex Hamilton-Jacobi equations in random media.  I will revisit the recent counter-example of Ziliotto who constructed a coercive but non-convex Hamilton-Jacobi equation with stationary ergodic random potential field for which homogenization does not hold.  We have extended this result showing that for any Hamiltonian with a strict saddle-point there is a random stationary ergodic potential field V so that homogenization does not hold for the Hamiltonian H = h(p)-V(x).  I will also discuss a positive result,  under a finite range of dependence assumption we show that homogenization holds for Hamiltonians with strictly star-shaped sub-level sets.

This talk is based on joint work with P. Souganidis.

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Hölder continuity for solutions of eikonal equations

Ermal Feleqi (Cardiff, UK)

I will talk about results  on Hölder continuity of viscosity solutions of eikonal PDEs

|∇X u| = f    in Ω

u = 0              on ∂Ω

structured on possibly  degenerate and  nonsmooth systems of vector  fields X = (X1 , . . . , Xp ) .

The  typical result  goes as follows:  if the  given vector  fields satisfy  Hör- mander’s  condition of at most  step  k at each  interior point of Ω and  if at each  boundary point of Ω a Lie bracket of degree  at most  k can by found not  tangential (i.e.,  transversal) to ∂Ω, then  the  solution  of (P)  is (1/k)- Hölder continuous. The proof relies on representing u as the value function of an optimal control  problem:  actually, when f ≡ 1, as the minimum time to  reach  the  exterior of Ω by  X -trajectories, that is, concatenations of a finite number of integral curves  of the  vector  fields ±Xi , i = 1, . . . , p.

The  smoothness assumptions on vector  fields and  ∂Ω are  reduced  to  a “bare  minimum”.    For  vector  fields  this  is made  possible  by  introducing a  set-valued notion   of iterated Lie  bracket which  makes  sense  for  quite nonsmooth vector  fields.   Concerning the  regularity of Ω, we require  for it to satisfy  an exterior cone condition. Then  the  transversality condition is expressed  by requiring that all the  vectors  of a set-valued bracket point toward an exterior cone.  When  Ω is of class C 1  or possesses Bony normals the  transversality condition can  be phrased in more  natural terms  (every vector  of the  bracket should  not  be orthogonal to the  normal). We  can  cover  also  boundaries with  isolated points.  At those  isolated points  the  trensversality condition is expressed  by requiring that Hörman- der’s condition be satisfied  therein.

Joint work with Franco Rampazzo, Martino Bardi  and Pierpaolo Soravia

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On a class of Optimal Transport problems with repulsive cost functions

Augusto Gerolin (Bath, UK)

The goal of this communication is to present a new class of optimal transport problems with repulsive cost functions motivated by Quantum Mechanics (computation of the ground state energy of an Electronic Schrödinger Equation). I want highlight some issues and point out our progress regarding the existence of Monge-type solutions in this setting.

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Convergence along mean flows

Harsha Hutridurga (Cambridge, UK)

This talk shall address the homogenization of parabolic equations of convection-diffusion type with large drift and with locally periodic rapidly oscillating coefficients. We answer an outstanding open problem in the theory of homogenization of parabolic problems. We shall develop a technique of multiple scale asymptotic expansions along mean flows and a corresponding notion of weak multiple scale convergence. Crucial to our analysis is the introduction of a fast time variable. We shall prove that the solution family taken along a particularly chosen rapidly moving coordinate system converges to the solution of a diffusion equation. The effective diffusion coefficient is expressed in terms of the average of Eulerian cell solutions along the orbits of the mean flow in the fast time variable. To make this notion rigorous, we use the theory of ergodic algebras with mean value. This is a joint work with Thomas Holding (Cambridge) and Jeffrey Rauch (Michigan).

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Higher order L∞ variational problems and the ∞-Polylaplacian

Nikos Katzourakis (Reading, UK)

Calculus of Variations in L∞ is a relatively new field initiated by Aronsson in the 1960s which is under active research since. Minimising the supremum of a function of the gradient is very challenging because the equations arising as the analogues of the Euler-Lagrange equations are non-divergence and highly degenerate. However, it provides more realistic models than the classical average functionals (integrals). In this talk I will discuss a very recent advance made jointly with T. Pryer (Reading, UK), where we initiated the study of 2nd order variational problems in L∞ , seeking to minimise the L∞ norm of a function of the hessian. We also derived and studied the associated PDE. The latter is fully nonlinear and of 3rd order. Special cases arise when the function is the Euclidean length of either the full hessian or of the Laplacian, leading to the ∞-Polylaplacian and the ∞-Bilaplacian respectively. Our analysis relies heavily on the recently proposed by the speaker theory of D-solutions, a general duality-free notion of generalised solutions for fully nonlinear PDE systems which do not support integration-by-parts.

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Uniaxial versus Biaxial Character of Landau-de Gennes Minimizers in Three Dimensions

Apala Majumdar (Bath, UK)

We study global minimizers of the Landau-de Gennes energy functional for nematic liquid crystals, on arbitrary three-dimensional simply connected geometries with topologically non-trivial and physically relevant Dirichlet boundary conditions. Our results are specific to an asymptotic limit defined in terms of a re-scaled reduced temperature, t. We prove (i) that (re-scaled) global LdG minimizers converge uniformly to a (minimizing) limiting harmonic map, away from the singular set of the limiting map; (ii) we have points of maximal biaxiality and uniaxiality near each singular point of the limiting map (this improves recent results of Contreras and Lamy); (iii) estimates for the size of “strongly biaxial” regions in terms of the reduced temperature t. This is joint work with Duvan Henao and Adriano Pisante.

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Mean Field Games on Networks

Claudio Marchi (Padova, Italy)

We consider stationary Mean Field Games (briefly, MFG) defined on a network. In this framework, the transition conditions at the vertices play a crucial role: the ones here considered are based on the optimal  control interpretation of the problem.

First, we prove separately the well-posedness of each of the two equations  composing the MFG system. After we prove existence and uniqueness of the  solution to the MFG system.

Finally, we propose some numerical methods, proving the well-posedness and  the converging of the scheme.

These are joint works with F. Camilli and S. Cacace.

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