Geometric Analysis Free Boundary Problems and Measure Theory: MPI MIS

The conference will start on Monday at 10 am and close on Wednesday at 5 pm.

University of Cambridge

Goethe-Universität Frankfurt

Écoles Normales Supérieures Lyon

The University of Texas at Austin

University of Firenze

University of Prague

University of Cambridge

University of Texas at Austin

Max-Planck-Institut für Gravitationsphysik, Golm and Universität Potsdam

Laboratoire J.-L. Lions, UPMC Paris 6

University of Warwick

Universität Düsseldorf

University of Fribourg

Nonlocal minimal surfaces naturally appear when studying the structure of interphases that arise in classical phase field models with very long space correlations. These surfaces are boundaries of sets whose characteristic functions minimize a fractional Sobolev norm, and they generalize the classical notion of minimal surfaces in geometric measure theory. In this talk we’ll explain and compare the general regularity theory for both local and non-local minimal surfaces, and discuss several recent developments and open problems.

We will discuss generalizations of Weiss's homogeneity improvement approach to the thin and fractional obstacle problems. The main ingredients are an epiperimetric inequality and a monotonicity formula, which give a powerful combination in the analysis of free boundaries and establish the

C^{1, α}

regularity of the regular set. The advantage of this method is that it is purely energy based and allows generalization to the case of thin obstacles living on codimension one

C^{1, 1}

manifolds, or more generally, the thin obstacle problem for the divergence form operators with Lipschitz coefficients. The method can also be used in the study of the obstacle problem for the fractional Laplacian with drift, when the fractional order is greater that one half. Based on joint works with Nicola Garofalo, Camelia Pop, and Mariana Smit Vega Garcia.

We study maps minimising the energy

\int_{D} (| \nabla u |^{2} + 2 | u |) d x,

which, due to Lipschitz character of the integrand, gives rise to the singular Euler equations

Δ u = \frac{u}{| u |} χ_{{| u | > 0}}, u = (u_{1}, \dots, u_{m}) .

Our results here concern regularity of the solution as well as that of the free boundary. The main ingredient consists in an epiperimetric inequality.(The result is a joint work with J. Andersson, H. Shahgholian and N. Uraltseva, accepted for publication in Advances in Mathematics)

We discuss an epsilon-regularity result at the endpoint of connected arcs for 2-dimensional Mumford-Shah minimizers obtained in a joint work with C. De Lellis (U. Zuerich). As an outcome of our analysis, if in a ball

B_{r} (x)

the jump set of a given Mumford-Shah minimizer is sufficiently close in the Hausdorff distance to a radius of

B_{r} (x)

, then in a smaller ball the jump set is a connected arc terminating at some interior point and

C^{1, α}

up to the tip.

We show that it is possible to construct a homeomorphism f in the Sobolev space W^{1,1}([0,1]^4,R^4) such that there are no smooth (or piecewise affine) homeomorphisms f_k that converge to f in W^{1,1} norm. This is joint result with Benjamin Vejnar.

Alexandrov's theorem asserts that a (bounded, embedded) constant mean curvature (cmc) hypersurface must be a sphere. It is well-known that if this condition is relaxed and the mean curvature is just assumed to be close to a constant, then the corresponding hypersurfaces does not need to be close to a sphere. Indeed any family of nearby spheres with equal radii connected by short catenoidal necks can be slightly perturbed to obtain examples of almost-cmc hypersurfaces (Kapouleas, Butscher, Mazzeo). We show that these examples actually capture the only possible behavior of almost-cmc hypersurfaces, by proving various quantitative bounds on the distance between an almost-cmc hypersurface and a collection of tangent spheres of equal radii in terms of their mean curvature oscillation. This is a joint work with G. Ciraolo (U Palermo). We next discuss these issues for the nonlocal mean curvature introduced by Caffarelli and Souganidis, showing in particular a remarkable rigidity property of the nonlocal problem which prevents bubbling phenomena, in other words, every nonlocal almost-cmc hypersurface must be close to a single sphere. This is a joint work with G. Ciraolo, A. Figalli (UT Austin) and M. Novaga (U Pisa).

In this talk a theory of weakly differentiable functions on rectifiable varifolds with locally bounded first variation will be presented. The concept proposed here is defined by means of integration by parts identities for certain compositions with smooth functions. Results include a variety of Sobolev Poincaré type embeddings, embeddings into spaces of continuous and sometimes Hölder continuous functions, pointwise differentiability results both of approximate and integral type as well as coarea formulae. As applications the finiteness of the geodesic distance associated to varifolds with suitable summability of the mean curvature and a characterisation of curvature varifolds are obtained.

I will describe a result obtained with Sylvia Serfaty about asymptotic limits of particle systems in

R^{d}

with pairwise interactions modeled by Riesz kernels

| x |^{- s}

for

s \in [max {0, d - 2}, d [

. Motivations for such choices of

s

arise in several fields, including Coulomb gases, eigenvalues for some random matrix ensembles, Fekete sets and spherical designs from approximation theory and the physics of seminconductors immersed in strong magnetic fields. I will briefly recall how the first order term in the asymptotic expansion of the equilibrium energy (the mean field limit) can be obtained. Then I will show how to study and control a next order "renormalized energy" that governs microscopic patterns of points and is an energy of infinite configurations of points. In the second half of the talk I will explain some open problems, namely 1) The long-stanting open questions about crystallization at the microscopic scale. 2) The case of interactions corresponding to the more non-local "fat tailed" interaction energies with

s \in [0, d - 2 [

. 3) Some GMT open questions in the case of integer

s

under the constraint that the points belong to a fixed s-rectifiable set.

I will study BV regularity for solutions of variational problems in Optimal Transportation and present some applications. In particular I will focus on

B V

bounds for the Wasserstein projection on the set of measure with density bounded by a prescribed

B V_{l o c}

function

f

. I will also show how to recover BV estimates for solutions of some non-linear parabolic PDE by means of optimal transportation techniques. This is a joint work with A. Meszaros, F. Santambrogio and B. Velichkov.

The idea of compactifying the space of Riemannian manifolds satisfying Ricci curvature lower bounds goes back to Gromov in the ’80ies and was pushed by Cheeger and Colding in the ’90ies who investigated the structure of the spaces arising as Gromov Hausdorff limits of smooth Riemannian manifolds satisfying Ricci curvature lower bounds. A completely new approach via optimal transportation was proposed by Lott-Villani and Sturm almost ten years ago; with this approach one can a give a precise meaning of what means for a non smooth space to have Ricci curvature bounded from below by a constant. This approach has been refined in the last years by a number of authors (let me quote the fundamental work of Ambrosio-Gigli-Savaré among others) and a number of fundamental tools have now been established (for instance the Bochner inequality, the splitting theorem, etc.), permitting to give further insights in the theory. In the seminar I will give an overview of the topic.

A triholomorphic map u between hyperKahler manifolds solves the "quaternion del-bar" equation du=I du i + J du j + K du k. Such a map turns out, under suitable assumptions, to be stationary harmonic. We focus on compactness issues regarding the quantization of the Dirichlet energy and the structure of the blow-up set. We can relax the assumptions on the manifolds, in particular we can take the domain to be merely "quaternion Kahler": this leads to the weaker notion of "almost-stationarity", without however affecting the compacntess results. This is a joint work with G. Tian (Princeton).

We will review some classical problems in Differential Geometry, which lead us to work with manifolds with almost non-negative curvature. In particular, we will explain during the talk why it is natural to wonder weather for these manifolds a topological invariant called

\hat{A}

-genus vanishes (this question was proposed by John Lott in 1997). We will provide a positive answer by investigating sequences of spin manifolds with lower sectional curvature bound, upper diameter bound and the property that the Dirac operator is not invertible. As a key ingredient of the proof we prove a generalization (under weaker curvature assumptions) of the renowned theorem by Gromov about almost flat manifolds. This is joint work with Burkhard Wilking.

I will discuss work on the structure of the branch set of two-valued solutions to the Laplace’s equation and the minimal surface system. Previously, the dimension of the branch set was known; we consider the fine structure of the branch set. In joint work with Neshan Wickramasekera, we show that the branch set is countably

(n - 2)

-rectifiable. Moreover, I have independently shown that the branch set is locally real analytic on a relatively open dense subset of the branch set. Essential ingredients for both results include the monotonicity formula for frequency functions due to F. J. Almgren and a blow-up method, which was originally applied by Leon Simon to multiplicity one classes of minimal submanifolds. The analyticity result requires applying the blow-up method in a new setting where tangent maps are not necessarily translation invariant along

(n - 2)

-dimensional subspaces.

We introduce a new class of estimates for the Ricci flow, and use them both to characterize solutions of the Ricci flow and to provide a notion of weak solutions of the Ricci flow in the nonsmooth setting. Given a family (M,g_t) of Riemannian manifolds, we consider the path space of its space time. Our first characterization says that (M,g_t) evolves by Ricci flow if and only if a sharp infinite dimensional gradient estimate holds for all functions on path space. We prove additional characterizations in terms of the regularity of martingales on path space, as well as characterizations in terms of log-Sobolev and spectral gap inequalities for a family of Ornstein-Uhlenbeck type operators. Our estimates are infinite dimensional generalizations of much more elementary estimates for the linear heat equation on (M,g_t), which themselves generalize the Bakry-Emery-Ledoux estimates for spaces with lower Ricci curvature bounds. Based on our characterizations we can define a notion of weak solutions for the Ricci flow. This is joint work with Aaron Naber.

In this talk, I will discuss a solution to the classical problem of Plateau in the setting of proper metric spaces. Precisely, I will show that among all disc-type surfaces with prescribed Jordan boundary in a proper metric space there exists an area minimizing disc which moreover has a quasi-conformal parametrization. If the space supports a local quadratic isoperimetric inequality for curves then I will show that such a solution is locally Hoelder continuous in the interior and continuous up to the boundary. These results generalize corresponding results of Douglas and Morrey from the setting of Euclidean space and Riemannian manifolds to that of proper metric spaces. This is joint work with Alexander Lytchak.

The harmonic map flow is a gradient flow of the harmonic map energy, introduced by Eells and Sampson in 1964. I will describe an alternative gradient flow that tries to find minimal immersions rather than harmonic maps, and give an overview of the theory so far, including forthcoming work. Joint work with Melanie Rupflin.

KTH, Royal Institute of Technology, Stockholm

University of Cambridge

Goethe-Universität Frankfurt

University of Augsburg

Weierstrass Institute, Berlin

Carnegie Mellon University

Scuola Normale Superiore, Pisa

Écoles Normales Supérieures Lyon

University of Zurich, Zürich

The University of Texas at Austin

University of Firenze

University of Warwick, Coventry

University of Prague

Universität Leipzig

University of Cambridge

MPI for Mathematics in the Sciences, Leipzig

Universität Leipzig

University of Texas at Austin

Max-Planck-Institut für Gravitationsphysik, Golm

University of Texas at Austin

Laboratoire J.-L. Lions, UPMC Paris 6

Max Planck Institute for Mathematics in the Sciences/Wuhan University, Leipzig

Friedrich-Alexander Universität Erlangen-Nürnberg

Universität Leipzig

Hausdorff institute, Bonn

Universitat Duisburg-Essen

Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig

University of Zurich

University of Zurich, Zürich

Oxford University/ Peking University

University of Warwick

Ludwigs-Maximilians-Universität München

Universität Düsseldorf

University of Fribourg

Universität Leipzig

Max-Planck-Institut für Mathematik in den Naturwissenschaften

10:00 - 11:00	Alessio Figalli Local and non-local minimal surfaces Nonlocal minimal surfaces naturally appear when studying the structure of interphases that arise in classical phase field models with very long space correlations. These surfaces are boundaries of sets whose characteristic functions minimize a fractional Sobolev norm, and they generalize the classical notion of minimal surfaces in geometric measure theory. In this talk we’ll explain and compare the general regularity theory for both local and non-local minimal surfaces, and discuss several recent developments and open problems.
11:00 - 11:30	Coffee Break
11:30 - 12:30	Arshak Petrosyan An epiperimetric inequality approach to the regularity of the free boundary in the thin and fractional obstacle problems We will discuss generalizations of Weiss's homogeneity improvement approach to the thin and fractional obstacle problems. The main ingredients are an epiperimetric inequality and a monotonicity formula, which give a powerful combination in the analysis of free boundaries and establish the $C^{1, α}$ regularity of the regular set. The advantage of this method is that it is purely energy based and allows generalization to the case of thin obstacles living on codimension one $C^{1, 1}$ manifolds, or more generally, the thin obstacle problem for the divergence form operators with Lipschitz coefficients. The method can also be used in the study of the obstacle problem for the fractional Laplacian with drift, when the fractional order is greater that one half. Based on joint works with Nicola Garofalo, Camelia Pop, and Mariana Smit Vega Garcia.
12:30 - 14:30	Lunch Break
14:30 - 15:30	Georg Weiss Equilibrium points of a singular cooperative system with free boundary We study maps minimising the energy $\int_{D} (\| \nabla u \|^{2} + 2 \| u \|) d x,$ which, due to Lipschitz character of the integrand, gives rise to the singular Euler equations $Δ u = \frac{u}{\| u \|} χ_{{\| u \| > 0}}, u = (u_{1}, \dots, u_{m}) .$ Our results here concern regularity of the solution as well as that of the free boundary. The main ingredient consists in an epiperimetric inequality.(The result is a joint work with J. Andersson, H. Shahgholian and N. Uraltseva, accepted for publication in Advances in Mathematics)
15:30 - 16:30	Matteo Focardi Endpoint regularity of 2d Mumford-Shah minimizers We discuss an epsilon-regularity result at the endpoint of connected arcs for 2-dimensional Mumford-Shah minimizers obtained in a joint work with C. De Lellis (U. Zuerich). As an outcome of our analysis, if in a ball $B_{r} (x)$ the jump set of a given Mumford-Shah minimizer is sufficiently close in the Hausdorff distance to a radius of $B_{r} (x)$ , then in a smaller ball the jump set is a connected arc terminating at some interior point and $C^{1, α}$ up to the tip.
16:30 - 17:00	Coffee Break
17:00 - 18:00	Stanislav Hencl Sobolev homeomorphism that cannot be approximated by diffeomorphisms in W^{1,1} We show that it is possible to construct a homeomorphism f in the Sobolev space W^{1,1}([0,1]^4,R^4) such that there are no smooth (or piecewise affine) homeomorphisms f_k that converge to f in W^{1,1} norm. This is joint result with Benjamin Vejnar.

09:00 - 10:00	Francesco Maggi Local and nonlocal almost-constant mean curvature hypersurfaces Alexandrov's theorem asserts that a (bounded, embedded) constant mean curvature (cmc) hypersurface must be a sphere. It is well-known that if this condition is relaxed and the mean curvature is just assumed to be close to a constant, then the corresponding hypersurfaces does not need to be close to a sphere. Indeed any family of nearby spheres with equal radii connected by short catenoidal necks can be slightly perturbed to obtain examples of almost-cmc hypersurfaces (Kapouleas, Butscher, Mazzeo). We show that these examples actually capture the only possible behavior of almost-cmc hypersurfaces, by proving various quantitative bounds on the distance between an almost-cmc hypersurface and a collection of tangent spheres of equal radii in terms of their mean curvature oscillation. This is a joint work with G. Ciraolo (U Palermo). We next discuss these issues for the nonlocal mean curvature introduced by Caffarelli and Souganidis, showing in particular a remarkable rigidity property of the nonlocal problem which prevents bubbling phenomena, in other words, every nonlocal almost-cmc hypersurface must be close to a single sphere. This is a joint work with G. Ciraolo, A. Figalli (UT Austin) and M. Novaga (U Pisa).
10:00 - 11:00	Ulrich Menne Weakly differentiable functions on varifolds In this talk a theory of weakly differentiable functions on rectifiable varifolds with locally bounded first variation will be presented. The concept proposed here is defined by means of integration by parts identities for certain compositions with smooth functions. Results include a variety of Sobolev Poincaré type embeddings, embeddings into spaces of continuous and sometimes Hölder continuous functions, pointwise differentiability results both of approximate and integral type as well as coarea formulae. As applications the finiteness of the geodesic distance associated to varifolds with suitable summability of the mean curvature and a characterisation of curvature varifolds are obtained.
11:00 - 11:30	Coffee Break
11:30 - 12:30	Mircea Petrache Asymptotics of particles minimizing power-law interactions: results and questions I will describe a result obtained with Sylvia Serfaty about asymptotic limits of particle systems in $R^{d}$ with pairwise interactions modeled by Riesz kernels $\| x \|^{- s}$ for $s \in [max {0, d - 2}, d [$ . Motivations for such choices of $s$ arise in several fields, including Coulomb gases, eigenvalues for some random matrix ensembles, Fekete sets and spherical designs from approximation theory and the physics of seminconductors immersed in strong magnetic fields. I will briefly recall how the first order term in the asymptotic expansion of the equilibrium energy (the mean field limit) can be obtained. Then I will show how to study and control a next order "renormalized energy" that governs microscopic patterns of points and is an energy of infinite configurations of points. In the second half of the talk I will explain some open problems, namely 1) The long-stanting open questions about crystallization at the microscopic scale. 2) The case of interactions corresponding to the more non-local "fat tailed" interaction energies with $s \in [0, d - 2 [$ . 3) Some GMT open questions in the case of integer $s$ under the constraint that the points belong to a fixed s-rectifiable set.
12:30 - 14:30	Lunch Break
14:30 - 15:30	Guido De Philippis BV estimates in optimal transportation and applications I will study BV regularity for solutions of variational problems in Optimal Transportation and present some applications. In particular I will focus on $B V$ bounds for the Wasserstein projection on the set of measure with density bounded by a prescribed $B V_{l o c}$ function $f$ . I will also show how to recover BV estimates for solutions of some non-linear parabolic PDE by means of optimal transportation techniques. This is a joint work with A. Meszaros, F. Santambrogio and B. Velichkov.
15:30 - 16:30	Andrea Mondino Some properties of non-smooth spaces with Ricci curvature lower bounds The idea of compactifying the space of Riemannian manifolds satisfying Ricci curvature lower bounds goes back to Gromov in the ’80ies and was pushed by Cheeger and Colding in the ’90ies who investigated the structure of the spaces arising as Gromov Hausdorff limits of smooth Riemannian manifolds satisfying Ricci curvature lower bounds. A completely new approach via optimal transportation was proposed by Lott-Villani and Sturm almost ten years ago; with this approach one can a give a precise meaning of what means for a non smooth space to have Ricci curvature bounded from below by a constant. This approach has been refined in the last years by a number of authors (let me quote the fundamental work of Ambrosio-Gigli-Savaré among others) and a number of fundamental tools have now been established (for instance the Bochner inequality, the splitting theorem, etc.), permitting to give further insights in the theory. In the seminar I will give an overview of the topic.
16:30 - 17:00	Coffee Break
17:00 - 18:00	Costante Bellettini Compactness questions for triholomorphic maps A triholomorphic map u between hyperKahler manifolds solves the "quaternion del-bar" equation du=I du i + J du j + K du k. Such a map turns out, under suitable assumptions, to be stationary harmonic. We focus on compactness issues regarding the quantization of the Dirichlet energy and the structure of the blow-up set. We can relax the assumptions on the manifolds, in particular we can take the domain to be merely "quaternion Kahler": this leads to the weaker notion of "almost-stationarity", without however affecting the compacntess results. This is a joint work with G. Tian (Princeton).
19:00 -	Dinner (individual payment)

09:00 - 10:00	Esther Cabezas-Rivas Almost non-negative curvature, what's new? We will review some classical problems in Differential Geometry, which lead us to work with manifolds with almost non-negative curvature. In particular, we will explain during the talk why it is natural to wonder weather for these manifolds a topological invariant called $\hat{A}$ -genus vanishes (this question was proposed by John Lott in 1997). We will provide a positive answer by investigating sequences of spin manifolds with lower sectional curvature bound, upper diameter bound and the property that the Dirac operator is not invertible. As a key ingredient of the proof we prove a generalization (under weaker curvature assumptions) of the renowned theorem by Gromov about almost flat manifolds. This is joint work with Burkhard Wilking.
10:00 - 11:00	Brian Krummel Fine structure and higher regularity of the branch sets I will discuss work on the structure of the branch set of two-valued solutions to the Laplace’s equation and the minimal surface system. Previously, the dimension of the branch set was known; we consider the fine structure of the branch set. In joint work with Neshan Wickramasekera, we show that the branch set is countably $(n - 2)$ -rectifiable. Moreover, I have independently shown that the branch set is locally real analytic on a relatively open dense subset of the branch set. Essential ingredients for both results include the monotonicity formula for frequency functions due to F. J. Almgren and a blow-up method, which was originally applied by Leon Simon to multiplicity one classes of minimal submanifolds. The analyticity result requires applying the blow-up method in a new setting where tangent maps are not necessarily translation invariant along $(n - 2)$ -dimensional subspaces.
11:00 - 11:30	Coffee Break
11:30 - 12:30	Robert Haslhofer Weak solutions for the Ricci flow We introduce a new class of estimates for the Ricci flow, and use them both to characterize solutions of the Ricci flow and to provide a notion of weak solutions of the Ricci flow in the nonsmooth setting. Given a family (M,g_t) of Riemannian manifolds, we consider the path space of its space time. Our first characterization says that (M,g_t) evolves by Ricci flow if and only if a sharp infinite dimensional gradient estimate holds for all functions on path space. We prove additional characterizations in terms of the regularity of martingales on path space, as well as characterizations in terms of log-Sobolev and spectral gap inequalities for a family of Ornstein-Uhlenbeck type operators. Our estimates are infinite dimensional generalizations of much more elementary estimates for the linear heat equation on (M,g_t), which themselves generalize the Bakry-Emery-Ledoux estimates for spaces with lower Ricci curvature bounds. Based on our characterizations we can define a notion of weak solutions for the Ricci flow. This is joint work with Aaron Naber.
12:30 - 14:30	Lunch Break
14:30 - 15:30	Stefan Wenger Area minimizing discs in metric spaces In this talk, I will discuss a solution to the classical problem of Plateau in the setting of proper metric spaces. Precisely, I will show that among all disc-type surfaces with prescribed Jordan boundary in a proper metric space there exists an area minimizing disc which moreover has a quasi-conformal parametrization. If the space supports a local quadratic isoperimetric inequality for curves then I will show that such a solution is locally Hoelder continuous in the interior and continuous up to the boundary. These results generalize corresponding results of Douglas and Morrey from the setting of Euclidean space and Riemannian manifolds to that of proper metric spaces. This is joint work with Alexander Lytchak.
15:30 - 16:00	Coffee Break
16:00 - 17:00	Peter Topping The harmonic map flow revisited The harmonic map flow is a gradient flow of the harmonic map energy, introduced by Eells and Sampson in 1964. I will describe an alternative gradient flow that tries to find minimal immersions rather than harmonic maps, and give an overview of the theory so far, including forthcoming work. Joint work with Melanie Rupflin.

Geometric Analysis, Free Boundary Problems and Measure Theory

Links

Speakers

Costante Bellettini

Esther Cabezas-Rivas

Guido De Philippis

Alessio Figalli

Matteo Focardi

Robert Haslhofer

Stanislav Hencl

Brian Krummel

Francesco Maggi

Ulrich Menne

Andrea Mondino

Mircea Petrache

Arshak Petrosyan

Peter Topping

Georg Weiss

Stefan Wenger

Program

Participants

Gohar Aleksanyan

Peter Bella

Costante Bellettini

Katarina Bellova

Esther Cabezas-Rivas

Judith Campos Cordero

Laszlo Chipot

Eleonora Cinti

Samuel Cohn

Maria Colombo

Guido De Philippis

Antonio De Rosa

Alessio Figalli

Matteo Focardi

Francesco Ghiraldin

Richard Gratwick

Robert Haslhofer

Stanislav Hencl

Mariusz Hynek

Bernd Kirchheim

Brian Krummel

Susanne Kürsten

Tim Laux

Philippe Logaritsch

Stephan Luckhaus

Francesco Maggi

Andrea Marchese

Ulrich Menne

Cornelia Mihaila

Stefano Modena

Andrea Mondino

Mircea Petrache

Arshak Petrosyan

Jacobus Portegies

Hongbing Qiu

Emanuela Radici

Melanie Rupflin

Wenhui Shi

Mariana Smit Vega Garcia

Emanuele Spadaro

Luca Spolaor

Salvatore Stuvard

Xuxu Sun

Peter Topping

Maik Urban

Georg Weiss

Stefan Wenger

Organizers

Bernd Kirchheim

Stephan Luckhaus

Emanuele Spadaro

Administrative Contact

Katja Heid