# Jyväskylä Analysis Seminar

**Fall 2017**

**Fall 2017**### 18.10.2017 at 14:15 MaD380 Panu Lahti

Title: BV functions and fine potential theory for p=1 in metric spaces

Abstract: We consider functions of bounded variation and some topics in fine potential theory in the case p=1, such as a Cartan property and questions related to the fine topology. We do this in the setting of a metric space that is equipped with a doubling measure and supports a Poincaré inequality.

### 25.10.2017 at 14:15 MaD380 Thomas Singer (Aalto)

Title: TBA

### 1.11.2017 at 14:15 MaD380 Luca Rondi (Trieste)

Title: TBA

### 8.11.2017 at 14:15 MaD380 Clifford Gilmore (Helsinki)

Title: TBA

### 15.11.2017at 14:15 MaD380 Tuomas Orponen (Helsinki)

Title: TBA

### 22.11.2017 at 14:15 MaD380 Tapio Rajala

Title: TBA

### 29.11.2017 at 14:15 MaD380 Aleksis Koski

Title: TBA

### 11.10.2017 at 14:15 MaD380 Angel Arroy

Title: Mean value properties in metric measure spaces

### 4.10.2017 at 14:15 MaD380 Daniel Campbell (Charles University)

Title: Sobolev homeomorphisms and monotone maps and their approximation

### 27.9. at 14:15 MaD380 Lauri Hitruhin (Helsinki)

Title: Stretching multifractal spectra of mappings with integrable distortion

Abstract: We present sharp bounds for the stretching multifractal spectra of planar mappings with p-integrable distortion. That is, we find the maximal size of sets in which a mapping with p-integrable distortion can satisfy some specific stretching conditions. We will also mention how finding the sharp multifractal spectra gives sharp area contraction results.

### 20.9. at 14:15 MaD380 Ville Tengvall

Title: Mappings of finite distortion: size of the branch set

### 23.8. at 14:15 MaD380 Eemeli Blåsten (HKUST)

Title: The planar inverse boundary value problem for L^p potentials with p>4/3

Abstract: I will talk about recent work with Leo Tzou and Jenn-Nan Wang. Based on the method of Bukhgeim, we show that Schrödinger operators with two different L^p-potential with p>4/3 always produce different Cauchy data at the boundary. This is an improvement over the previous result of L^2 potentials. What made this possible comes from an earlier interesting result by Sun-Uhlmann, Päivärinta-Serov: the difference of two potentials with the same Cauchy data is actually smoother than either potential is a-priori.

### 6.9. at 14:15 MaD380 Nicola Fusco (University of Naples)

Title: Evolution of material voids by surface diffusion

Abstract: We consider the evolution by surface diffusion of material voids in a linearly elastic solid. We prove short time existence and asymptotic stability when the initial configuration is close to a stable critical point for the energy. Similar results are also obtained for the evolution by surface diffusion of epitaxially thin films.

### 13.9. at 14:15 MaD380 Joonas Ilmavirta

Title: Spectral rigidity and tensor tomography

Abstract: Can you reconstruct a Riemannian manifold up to isometry from the knowledge of the spectrum of the Laplace-Beltrami operator? This question is wide open. An easier version of this problem is spectral rigidity: Is an isospectral deformation necessarily trivial? We will discuss this problem and its connection to geodesic X-ray tomography of tensor fields.

### **Spring 2017**

**Spring 2017**

### Thue 23.5. at 14:15 MaD380 Mikko Stenlund

Title: Stein's method for dynamical system

### 18.5. Alessio Porretta

Title: PDE systems in mean field games theory

10.5 Shirsho Mukherjee

Title : Existence of minimizers for spectral optimization.

Abstract : The main topics of this talk will be about establishing the second eigenvalue of the p-laplacian on quasi open sets, proof of gamma-semicontinuity and a general existence theorem of minimizing domains.

### 3.5. Matthew Romney

Title: Quasiconformal parametrization of metric surfaces.

*26.4.2017* Matti Vihola

Title: Unbiased estimators and multilevel Monte Carlo

Abstract: Multilevel Monte Carlo (MLMC) is a commonly applied method, for instance, in numerical approximation of expectations with respect to diffusions. The unbiased estimators recently proposed by McLeish (Monte Carlo Methods Appl., 2011) and Rhee and Glynn (Oper. Res., 2015) are closely related to MLMC. This talk gives an elementary introduction to these general-purpose estimators. The connection between the estimators is elaborated in terms of a new general class of unbiased estimators, which admits previous debiasing schemes as special cases, but allows straightforward implementation of lower variance modifications. Under general conditions, essentially when the MLMC admits the canonical square root Monte Carlo error rate, the proposed new schemes are asymptotically as efficient as MLMC, both in terms of variance and cost.

### 19.4. Mario Bonk (UCLA)

Title: Sobolev spaces and hyperbolic fillings

Abstract: Boudon and Pajot showed how to identify a certain Besov space on a metric space with the l^p-cohomology of degree 1 of its hyperbolic filling. In joint work with Eero Saksman we used a variant of this idea to give a similar interpetation of certain Sobolev spaces. In my talk I will discuss this and some recent developments.

### 12.4. Yi Zhang

Title: Recent results on planar Sobolev extension domains.

Abstract: I will introduce my recent (new) work with P. Koskela and T. Rajala on the geometric characterizations of planar Sobolev extension domains, and some new understanding of these characterizations.

*5.4.2017* Angkana Rüland (Oxford)

Title: Regularity for the Thin Obstacle Problem

Abstract: Thin obstacle problems arise in the modelling of many chemical, physical and ﬁnancial problems. In this talk I will present a new, robust strategy of proving optimal regularity for low regularity coeﬃcients. Here the central new tool is a Carleman inequality which replaces an Almgren type monotonicity formula. With the optimal regularity at hand, I will discuss quantitative higher regularity results for the thin obstacle problem in the presence of Hölder coeﬃcients. Crucial ingredients in this context are a Hodograph-Legendre transform, the analysis of a fully nonlinear degenerate elliptic equation and the introduction of suitable function spaces. This is joint work with H.Koch and W. Shi.

*29.3.2017* Andrea Mondino

Title: TBA

*22.3.2017* Augusto Gerolin

Title: An overview on Multi-marginal Optimal Transport with Coulomb costs: DFT, Numerics and Entropy.

### 15*.3.2017* Robert Van Leeuwen

Title: The density to potential mapping in Schrödinger dynamics

Abstract: In this presentation we discuss some of the open questions regarding the question whether the particle density uniquely determines the externally applied potential in the Schrödinger dynamics of many-particles [1]. It raises some mathematical issues regarding the properties of the time-dependent Schrödinger equation with explicitly time-dependent potentials, in particular how regularity properties of the external field determine regularity properties of the particle density

[1] M. Ruggenthaler, M.Penz and R. van Leeuwen, Existence, uniqueness and construction of the density-potential mapping in time-dependent density-functional theory, J. Phys: Cond. Matt. 27, 203-202 (2015)

### *8.3.2017* Antti Räbinä

Title: Mappings of finite distortion: decay of the Jacobian

Abstract: Mappings of finite distortion are generalizations of quasiconformal mappings, and they appear e.g. in the theory of nonlinear elasticity. I will talk about new results on the integrability of the reciprocal of the Jacobian.

*1.3.2017* Christopher Hopper (Aalto University)

Title: Partial Regularity for Holonomic Minimisers of Quasiconvex Functionals

**Abstract: **We prove partial regularity for local minimisers of certain strictly quasiconvex integral functionals, over a class of Sobolev mappings into a compact Riemannian manifold, to which such mappings are said to be holonomically constrained. Several applications to variational problems in condensed matter physics with broken symmetries are also discussed, in particular those concerning the superfluidity of liquid helium-3 and nematic liquid crystals.

*8.2.2017* Rami Luisto

Title: Stoilow's theorem revisited

Abstract: Stoilow’s theorem from 1928 states that a continuous, light, and open mapping between surfaces is a discrete map with a discrete branch set. This result implies that such mappings between orientable surfaces are locally modelled by complex power mappings and admit a holomorphic factorization. The purpose of the talk is to give a proof of this classical theorem having the readers interested in discrete and open mappings in mind. The talk is based on a preprint (https://arxiv.org/abs/1701.05726), joint with Pekka Pankka.

*1.2.2017 *Tony Liimatainen

Title: The Calderon problem for the conformal Laplacian

**Abstract**: We consider a conformally invariant version of the Calderoon problem, where the objective is to determine the con- formal class of a Riemannian manifold with boundary from the Dirichlet-to-Neumann map for the conformal Laplacian. The main result states that a locally conformally real-analytic manifold in dimensions ≥ 3 can be determined in this way, giving a positive answer to an earlier conjecture by Lassas and Uhlmann (2001). The proof proceeds as in the standard Calderoon problem on a real-analytic Riemannian manifold, but new features appear due to the con- formal structure. In particular, we introduce a new coordinate system that replaces harmonic coordinates when determining the conformal class in a neighborhood of the boundary.

### *25.1.2017 *Amal Attouchi

**Title:** Recent advances in diffusive Hamilton-Jacobi equations.

**Abstract**: Diffusive Hamilton-Jacobi equations appear in optimal control theory and in physical models describing the growth and roughening of surfaces. In this talk I will present some old and new results concerning well-posedness, regularity and homogenization.

**Fall 2016**

### 14.12.2016 Eero Ruosteenoja: Local regularity for the normalized p-Poisson problem

**Abstract (preliminary):** The normalized p-Poisson problem arises for example from stochastic games and has applications in image processing. I will prove H\"{o}lder continuity for gradients of viscosity solutions of this problem.

### ***

### 7.12.2016 Stefan Wenger: Canonical parametrizations of metric discs

**Abstract:** I will show how to use recent results on the existence and regularity of area and energy minimizing discs in metric spaces to obtain canonical parametrizations of metric surfaces. In particular, we obtain a new and conceptually simple proof of a well-known theorem of Bonk and Kleiner on the existence of quasisymmetric parametrizations of linearly locally connected, Ahlfors 2-regular metric 2-spheres. If time permits, I will furthermore describe some generalizations and applications to the geometry of such surfaces as well. Joint work with Alexander Lytchak.

### 30.11.2016 Espen Jakobsen (NTNU)

### 23.11.2016 Simone Di Marino (Paris, Dauphine):"BV functions and finite perimeter sets in metric measure spaces: functional and geometric approaches"

**Abstract: **In the last years there has been a huge development in the analysis in nonsmooth spaces, starting from the seminal work of Cheeger. Miranda introduced the notion of BV function in metric measure spaces, and so also the notion of finite perimeter sets with a functional approach. We will relate this functional notion with a more geometrical one, namely the relaxation of the Minkowski outer content, showing that they are indeed equivalent.

### 16.11.2016 Sylvester Eriksson-Bique (Courant Institute, New York University): "Poincare inequalities via quantitative connectivity, synthetic Ricci curvature, and differentiability in metric measure spaces"

**Abstract:** The talk will outline a new existence result for Poincare inequalities via a notion of quantitatively connecting pairs of points. This result is applied to a variety of spaces to exhibit Poincare inequalities, such as MCP-spaces with a very weak Ricci bound defined by Ohta, weighted metric measure spaces with Muckenhoupt-type weights and certain classes of Lipschitz differentiability spaces. New types of self-improvement results in the spirit of Keith-Zhong are presented. Further a rectifiability result in terms of spaces admitting Poincare inequalities is outlined, and we introduce techniques to make spaces more connected by gluing curves.

### 9.11.2016 Aleksis Koski (Univ. of Helsinki) : Nonlinear and autonomous Beltrami equations: Improved regularity

**Abstract**

### 2.11.2016: Yi Zhang: A quantitative form of the Faber-Krahn inequality. (Joint work with Nicola Fusco

### 26.10.2016: Lashi Bandara (Goteborg): Riesz continuity of the Atiyah-Singer Dirac operator under perturbations of the metric.

* Abstract: *We prove that the Atiyah-Singer Dirac operator $\slashed{D}_g$ in $L^2$ depends Riesz continuously on $L^\infty$

perturbations of a complete metric $g$. The Lipschitz bound for the map $g \to \slashed{D}_g/\sqrt{1 + \slashed{D}_g^2}$ depends on curvature bounds and a lower bound on injectivity radius. Our proof uses harmonic analysis techniques related to Calder\'on's first commutator and the Kato square root problem. We also show perturbation results for more general functions of general Dirac-type operators on vector bundles.

This is joint work with Alan McIntosh (Australian National University)

and Andreas Ros\'en (Gothenburg University).

### 19.10.2016: Jani Onninen:How did I meet Ball & Evans?

**Abstract:**

### 12.10.2016: Tomas Soto:Traces of function spaces on metric measure spaces

* Abstract:* "We study the traces of Besov, Triebel-Lizorkin and Sobolev functions in the setting of metric measure spaces using a hyperbolic filling of the underlying space. Joint work with Eero Saksman."

### 5.10.2016: Lasse Leskelä (Aalto):Diclique clustering in directed random intersection graph

* Abstract: *I will discuss a notion of clustering in directed graphs, which describes how likely two followers of a node are to follow a common target. The associated network motifs, called dicliques or bi-fans, have been found to be key structural components in various real-world networks. A two-mode statistical network model consisting of actors and auxiliary attributes is introduced, where an actor i decides to follow an actor j whenever i demands an attribute supplied by j. This directed random graph model admits nontrivial clustering properties of the aforementioned type, as well as power-law indegree and outdegree distributions. The talk is based on joint work with Mindaugas Bloznelis (U Vilnius), available at arXiv:1607.02278.

### 28.9.2016: Antti Käenmäki:* *"Equilibrium states for singular value function and dimension drop on self-affine sets"

* Abstract*: "With Ian Morris, we managed to completely characterize the equilibrium states for the singular value function in dimension three. Concerning self-affine sets in R^3, our result gives a positive answer to a folklore open question whether removing one of the defining affine maps results in a strict reduction of the Hausdorff dimension. In R^2, this follows from my earlier result with De-Jun Feng. The question remains open in higher dimensions."

### 21.9.2016: Elefterios Soultanis:* *Homotopy and existence of energy minimizers in metric spaces

* Abstract: *I discuss the problem of minimizing the p-energy of Newtonian maps in a given homotopy class. For the metric space setting a new proof strategy is required, so far only allowing target spaces with hyperbolic fundamental group. I will sketch the proof and point out where this assumption arises.

### 10.8.2016: Leo Tzou (University of Sydney),

15:15-16:15 (NOTE: unusual time!) MaD 380

Title: **Detecting a magnetic gauge by wave scattering**

**Abstract: **On a Riemann surface with Euclidean ends we recover the magnetic Schr\"odinger operator, up to gauge equivalence, by wave scattering. This problem turns out to be intimately related to the topology of the surface and we see how classical index theorems such as Riemann-Roch can play a significant role in understanding this relationship.** **

1.8.2016 Mourad Sini (Radon Institute, Linz)

Title: **The acoustic diffraction by multiple small inhomogeneities and applications to metamaterials and imaging**

**Spring 2016**

### 1.6.2016: Ian Morris (University of Surrey)

Title: **"Ergodic properties of matrix equilibrium states"**

**Abstract: **"The classical formula of Hutchinson for the Hausdorff dimension of a self-similar set is proved by constructing a self-similar measure on the set which has Hausdorff dimension equal to the expected value. In the more general context of self-affine rather than self-similar sets, the analogue of this dimension-maximising measure is defined via a variational problem on a space of symbolic sequences. I will describe some recent results on these measures in the simplest case, where the expected value of the Hausdorff dimension is lower than 1."

### 25.5.2016: Shirsho Mukherjee

Title: **Regularity of p-Laplace equations in the Heisenberg Group**

**Abstract: **There has been several publications in the past few decades that deals with

the regularity theory of p-Laplace equations in the Heisenberg Group. The Holder and Sobolev regularity of weak solutions is well established, but there has been only partial results for higher regularity, so far. In this talk, I would like to discuss the proof of local Holder continuity of the horizontal gradient of weak solutions for p-Laplace equations, for all finite p > 1. This is a joint work with X. Zhong.

### 18.5.2016: Filippo Cagnetti (University of Sussex)

Title: **The rigidity problem for symmetrization inequalities**

**Abstract: **We will discuss several symmetrizations (Steiner, Ehrhard, and spherical symmetrization) that are known not to increase the perimeter. We will show how it is possible to characterize those sets whose perimeter remains unchanged under symmetrization. We will also characterize rigidity of equality cases. By rigidity, we mean the situation when those sets whose perimeter remains unchanged under symmetrization, are trivially obtained through a rigid motion of the (Steiner, Ehrhard or spherical) symmetral. We will achieve this through the introduction of a suitable measure-theoretic notion of connectedness, and through a fine analysis of the barycenter function for a special class of sets. These results are obtained together with several collaborators (Maria Colombo, Guido De Philippis, Francesco Maggi, Matteo Perugini, Dominik Stoger).

### 11.5.2016: Changlin Xiang

Title: **On a Liouville type problem in a quarter plane**

**Abstract:** Xavier Cabre and Jinggang Tan proposed a Liouville type problem in a quarter plane in 2010 in Adv. Math.. In the same paper they gave a partial solution to this problem via the method of moving planes and a Hamiltonian identity. In this talk, I will give a complete solution to their problem via a different idea which is inspired by a work of Yanyan Li and Chang-Shou Lin.

### 4.5.2016: Jan Maas (IST Austria)

Title: **Discrete Ricci curvature and functional inequalities for**** interacting particle systems**

**Abstract:** In the past decade there has been a lot of progress in the

analysis on metric measure spaces based on ideas from optimal transport.

We discuss how some of these ideas can be applied to discrete spaces,

using a discrete analogue of the Kantorovich-Wasserstein metric. In

particular we present a discrete notion of Ricci curvature based on

geodesic convexity of the entropy, which allows us to obtain discrete

functional inequalities, such as spectral gap and logarithmic Sobolev

inequalities. We also discuss recent applications to interacting

particle systems.

This is based on joint works with Matthias Erbar (Bonn), Prasad Tetali

(Georgia Tech), and Max Fathi (Berkeley).

### 27.4 Juhana Siljander

Title: **Gradient estimates for the porous medium equation**

**Abstract: **The sharp regularity of weak solutions to the porous medium equation is a long-standing open problem. The fundamental solution of the problem, the so called Barenblatt solution, is known to be C^{1/(m-1)}-HÃ¶lder continuous, and it is a natural question whether this is the worst possible behavior. A counter-example by Aronson and Graveleau, however, suggests that this is not the case - leaving the question wide open. Known results by Gilding and Ivanov provide a tight link between the HÃ¶lder-regularity and gradient estimates for the solutions. In this talk, I will present some new results concerning the topic. The talk is based on a joint work with professor Ugo Gianazza.

### 20.4.2016: Augusto Gerolin (University of Bath):

Title:** Optimal Transport and Density Functional Theory**

**Abstract: **My intention is to present some recent results related to Multi-marginal Optimal Transport Theory for Coulomb and repulsive harmonic costs. In particular, questions related to the existence of Monge-type solutions in this multi-marginal setting. During the talk, I intend to give a brief survey on the general theory of multi-marginal optimal transport and discuss some of its main issues. Most of the results I will present are contained in a joint work with S. Di Marino and L. Nenna.

### 14.4.2016 (Thursday): Bastian von Harrach (Goethe Universität Frankfurt) :

Title: **Monotonicity-based methods for inverse coefficient problems**

**Abstract:** Some newly emerging imaging methods lead to the inverse problem of determining one or several coefficient function(s) in an elliptic partial differential equation from (partial) knowledge of its solutions. In particular let us mention electrical impedance tomography (EIT), where electrical currents are driven through a patient to image its interior. The mathematical challenges behind such inverse coefficient problems reach from theoretical uniqueness questions to the construction of convergent numerical algorithms and stability issues. In this talk, we will describe recent advances on these problems that are based on monotonicity relations with respect to operator definiteness and the concept of localized potentials.

### 7.4.2016: Daniel Campbell (Charles University Prague):

Title: **Structure of homeomorphic solutions to the Beltrami equation**

**Abstract: **Let $\Omega\subseteq\mathbb R^2$ be a domain, let $\Phi$ be a $\Delta_2$ Young function and let $f\in W^{1,\Phi}(\Omega,\mathbb R^2)$ be a homeomorphism between $\Omega$ and $f(\Omega)$. Then there exists a sequence of diffeomorphisms $f_k$ converging to $f$ in the Sobolev-Orlicz space $W^{1,\Phi}(\Omega,\mathbb R^2)$.

On the contrary, under the assumption that $n\geq 4$ and $p<[\tfrac{n}{2}]$, where $[x]$ denotes the integer part of $x$, we prove that a similar approximation theorem cannot hold. That is we construct a homeomorphism in $W^{1,p}(\mathbb R^n,\mathbb R^n)$ which cannot be approximated in $W^{1,p}$ by diffeomorphisms or piece-wise affine homeomorphisms.

### 6.4.2016: Tuomo Kuusi (Aalto University):

Title: **The additive structure of elliptic homogenization**

**Abstract: **One of the principal difficulties in stochastic homogenization is transferring quantitative ergodic information from the coefficients to the solutions, since the latter are nonlocal functions of the former. In this talk, I will address this problem in a new way, in the context of linear elliptic equations in divergence form, by showing that certain quantities associated to the energy density of solutions are essentially additive. As a result, we are able to prove quantitative estimates on the first-order correctors which are optimal in both scaling and stochastic integrability. The proof of the additivity is a bootstrap argument: using the regularity theory recently developed for stochastic homogenization, we accelerate the weak convergence of the energy density, flux and gradient of the solutions as we pass to larger and larger length scales, until it saturates at the CLT scaling. This is a joint work with S. Armstrong and J.-C. Mourrat.

### 30.3.2016: Jarmo Jääskeläinen:

Title: **Structure of homeomorphic solutions to the Beltrami equation**

**Abstract:** In the context of G-compactness of Beltrami operators it was recently proved that there is one-to-one correspondence between two-dimensional vector spaces of quasiconformal maps and linear Beltrami equations.

In this talk we tackle similar questions in the nonlinear setting. We discuss the structure of homeomorphic solutions to (nonlinear) Beltrami equations, in particular, we talk about the manifold structure and the uniqueness properties of normalised solutions.

The talk is based on joint works with Kari Astala, Albert Clop, Daniel Faraco, Aleksis Koski and Laszlo Szekelyhidi, Jr.

### 16.3.2016: Tapio Rajala:

Title:** L^\infty estimates in optimal mass transportation**

**Abstract:** L^p-transportation distances between probability measures have been used in many areas of analysis and geometry. For 1 < p < \infty these distances have many nice properties which are not valid at the limiting cases p=1 and p=\infty. In particular, the case p=\infty is quite cumbersome. Nevertheless, also the L^\infty-transportation distance arises in many applications. In this talk I will discuss how L^\infty-transportation distance is related to the L^p-transportation distances (with 1 < p < \infty) and in particular I will discuss the existence of quantitative lower bounds of L^p-distance with the L^\infty-distance. This is joint work with Heikki Jylhä.

### 2.3.2016: Vesa Julin:

Title: **Stability of the Gaussian isoperimetric inequality**

### 24.2.2016: Katrin Fässler:

Title: **Curve packing and modulus estimates**

**Abstract:** The modulus of curve families is an ubiquitous tool in geometric mapping theory. Motivated by a question about quasiregular maps, we study the modulus of families that contain an isometric copy of every curve of unit length in the plane. We show that such families have positive p-modulus for p>4. This strengthens an earlier result by J. Marstrand, who proved that the curves in such a family must cover positive area.

The talk is based on joint work with T. Orponen.

### 17.2.2016: David Herron (University of Cincinatti):

Title: **Universal Convexity for Classes of QuasiHyperbolic Type Metrics**

*Abstract: *The problem of finding a best possible route, seen in applied and pure mathematics, is both old and hard. In metric geometry, this is the "geodesic problem", to find a shortest path between two points. Due to its difficulty, it is worthwhile to know information about the location of geodesics.

For example: Suppose we conformally deform a "nice" space. Do the geodesics in the deformed space stay inside balls in the original space?

We answer this question for the case of certain conformal metrics defined on domains in Euclidean n-space (or in the n-sphere).

In particular, for these classes of conformal metrics (on regions in the sphere) we characterize the open sets which are geodesically convex in any containing domain.

### 10.2.2016: Antti Käenmäki

Title: **Ledrappier-Young formula and exact dimensionality of self-affine measures**

*Abstract: *We solve the long standing open problem on exact dimensionality of self-affine measures. We show that every self-affine measure on the plane is exact dimensional regardless of the choice of the defining iterated function system. The measures also satisfy the Ledrappier-Young formula.

### 3.2: Prof. Hui-Chun Zhang (Sun Yat-Sen University)

Title: **Lipschitz continuity of harmonic maps from Alexandrov spaces to NPC spaces**

### 27.1.2016: Aleksis Koski (Helsinki University)

Title: **Nonlinear Beltrami equations, positivity of the Jacobian**

### 20.1.2016: Prof. Gaofeng Zheng (Central China Normal University):

Title:** Some results on Maxwell's equations**

**Fall 2015**

### 16.12.2015: Ville Tengvall (Charles University in Prague):

**A Sobolev homeomorphism that cannot be approximated by diffeomorphisms in $W^{1,p}$**

*Abstract: *The problem of approximating homeomorphisms by diffeomorphisms has proven to be both very challenging and of great interest in a variety of contexts. In the case of $L^{\infty}$-norm these questions can be traced back to works by Rado, Moise, Bing, Connell, Bing, Kirby-Siebenmann-Wall, and Donaldson-Sullivan.

In the case of Sobolev $W^{1,p}$-homeomorphism with $1 \le p < \infty$ the celebrated breakthrough in the planar case was given by Iwaniec, Kovalev and Onninen, where they found diffeomorphic approximations to any planar homeomorphism $f\in W^{1,p}$, for any $1<p<\infty$ in the $W^{1,p}$-norm. The remaining missing case $p=1$ was solved by Hencl and Pratelli, and these results were further generalized in the Orlicz-Sobolev setting by Campbell.

However, very recently Hencl and Vejnar proved that the approximation might fail in the higher dimensional case $n \ge 4$. We will now study further this problem by constructing for a given $p \in [1,[n/2])$ a Sobolev homeomorphism $f \in W^{1,p}(Q^{n},Q^{n})$ of a unit cube $Q^{n}:=(-1,1)^{n}$ with $n \ge 4$ which cannot be approximated by diffeomorphism in $W^{1,p}$. Here $[a]$ stands for the integer part of real number $a$.

This is joint work with D. Campbell, P. Hajlasz and S. Hencl.

### 25.11.2015: Eino Rossi:

**Local structure of fractal sets**

**Abstract:****One common definition of a fractal is that it consists of smaller copies of itself. This indicates that when zooming into a fractal set one could expect to see the same set over and over again. In some sense this is true, but even for self-similar sets this is not the whole truth. In this talk, I try to give a more detailed study of what can we actually see when looking deeper and deeper into a fractal set. The main focus is on the geometric limit objects, such as tangent sets and micro sets, of this zooming in process. The talk is based on my thesis.**

### 18.11.2015: Anders Björn (Linnköping University):

**Boundary regularity, barriers and Petrovskii's condition for $p$-parbolic equations**

**Abstract:****Consider the Dirichlet (boundary value) problem for the Laplacian, i.e. take a domain $G$ in ${\bf R}^n$ and a continuous function $f$ on the boundary $\partial G$ and find the harmonic function $u_f$ which has $f$ as boundary values. For general domains this problem can't always be solved, and for existence of a solution we have to allow the boundary values to be taken in some weak sense (e.g. using Perron solutions). A boundary point $x \in \partial G$ is regular if $$ \lim_{G \ni y \to x} u_f(y)=f(x) \quad \text{for all } f \in**

> C(G).$$ One can similarly define regularity for other equations, e.g. the (elliptic nonlinear) $p$-harmonic equation $\Delta_p u := {\rm div}(|\nabla u|^{p-2} \nabla u)=0$, the (parabolic linear) heat equation $\partial_t u = \Delta u$, and the (parabolic nonlinear) $p$-parabolic equation $\partial_t u = \Delta_p u$.

In the first three cases, boundary regularity can be characterized by the existence of a barrier, whereas in the last case one can use the existence of a barrier family. An open problem for 20 years has been whether the existence of a single barrier can be used to characterize regularity for the $p$-parabolic equation. I will show that this is not possible.

Petrovskii (1935) showed that the origin is regular for the heat equation with respect to \[

\{(x,t): |x|<A\sqrt{-t}\sqrt{\log |{\log(-t)}|} \text{ and } -1\le t<0\}, \] if $A=2$, while it is irregular if $A >2$. I will discuss nonlinear $p$-parabolic analogs of this result.

This talk is based on joint work with Jana Björn and Ugo Gianazza.

### 11.11.2015: Daniel Seco:

**Cyclicity vs. orthogonal polynomials**.

**Abstract:**We say that a function f in a Hilbert space of analytic functions Y is cyclic if the space X, of polynomials multiplied by f, is dense in Y. In a previous work, the authors introduced an optimizational point of view, characterizing cyclicity in terms of some polynomials we called optimal approximants. In the work we present, we find new properties of such approximants and establish a correspondence between these and some families of orthogonal polynomials on the space X.

### 28.10.2015: Hannes Luiro:

**New results on quasihyperbolic geometry.**

**Abstract:**Unless the quasihyperbolic metric is a standard tool for example in the theory of quasiconformal mappings, the quasihyperbolic geometry itself is not very well known. Many elementary problems have remained open, like the following problems/conjectures:

1) Uniqueness of geodesics in simply connected planar domains.

2) Uniqueness of geodesics in arbitrary planar domains if d(x,y)<\pi, d is quasihyperbolic distance.

3) Existence of universal constant c>0 (independent of dimension, as well)

such that quasihyperbolic balls with radius less than c are convex.

Recently, I managed to show the validity of these conjectures. This is the topic of the talk.

### 28.10.2015: Vyron Vellis:

**Quasiconformal non-parametrizability of punctured smooth spheres.**

**Abstract:***In 1996, Semmes constructed a metric on the unit sphere S^3 that resembles the Euclidean metric geometrically (linearly locally contractible) and measure theoretically (3-regular) but is not quasiconformal to the Euclidean metric. Semmes's surprising result was extended by Heinonen and Wu in every dimension n>3. Semmes' metric is Riemannian outside of a Cantor set on S^3 while the metric of Heinonen and Wu is Riemannian outside of a co-dimension 3 subset of S^n. In this talk, we improve these results by constructing a metric on S^n, for each n>2, that is linearly locally contractible, n-regular and Riemannian outside of a single point, but is not quasiconformal to the Euclidean metric. This is a joint work with Pekka Pankka.*

### 21.10.2015: Amal Attouchi:

**The maximum principle, a key tool in the study of elliptic and parabolic PDE.**

**Abstract:***The maximum principle is a basic but very efficient tool in the study of PDE.*

In this talk we will revisit some of its application going from the moving planes argument of Alexandrov to the Bernstein and Friedman- Mcleod techniques to deal with some some recent results about the diffusive Hamilton-Jacobi equation.

### 14.10.2015: Renjin Jiang:

**On the equivalence of quantitative regularity of harmonic functions and heat kernels and its applications**

**Abstract:**In this talk, I will first review some recent developments on harmonic functions and heat kernels on metric measure spaces, then I will present an equivalence between quantitative regularity of harmonic functions and heat kernels.

Let $(X,d,\mu)$ be a complete metric measure space, with $\mu$ being a doubling measure, that supports an $L^2$-Poincar\'e inequality. We will show that, for a fixed $p\in (2,\infty]$, the $L^p$-boundedness of the gradient heat semigroup is equivalent to the $L^p$-reverse H\"older inequality for the gradient of harmonic functions.

Applications to the Riesz transform, isoperimetric inequalities and Sobolev inequalities will be discussed. Analogies of the

theories on $(X,d,\mu)$, with $\mu$ being a local doubling measure that supports a local $L^2$-Poincar\'e inequality, will also be presented.

This is based on a joint work with Pekka Koskela and Dachun Yang.

### 7.10.2015: Palina Salanevich (Jacobs University)

Title: **Problem of phaseless recovery: from structured measurements to random and back**

**Abstract:**

In many areas of imaging science, such as diffraction imaging, astronomical imaging, microscopy, etc., optical detectors can often only record the squared modulus of the Fraunhofer or Fresnel diffraction pattern of the radiation that is scattered from an object. In such setting, it is not possible to measure the phase of the optical wave reaching the detector. So, it is needed to reconstruct a signal from intensity measurements only. This problem is called phase retrieval. The measurements obtained in this way are of the form of pointwise squared modulus of masked Fourier transform of the object at hand. But this type of measurements is too specific to build up a theory around it. Thus in most known reconstruction algorithms, such as PhaseLift, the measurement frame considered to be randomly generated. This grants a lot of benefits for the theoretical analysis but makes the algorithms unimplementable in practice. As such, the question of constructing an efficient and stable recovery algorithm working with structured measurements still remains an important open problem. We are going to consider the case when the measurement frame is a Gabor frame, that is, the case of time-frequency structured measurements. The main motivation is that in this case, the frame coefficients are of the form of masked Fourier coefficients, where the masks are time shifts of the Gabor window. This makes measurements meaningful for applications, but at the same time preserves the flexibility of the frame-theoretic approach.

### 23.9.2015: Ludek Kleprlik (Jyväskylä Yliopisto)** **

Title: **Composition operators on $W^{1}X$ and quasiconformal mappings**

**Abstract:**

Let $\Omega\subset\R^n$ be an open set and $X(\Omega)$ be any rearrangement invariant function space close to $L^q(\Omega)$. We show that each homeomorphisms $f$ which introduces composition operator $u\rightarrow u\circ f$ from $W^1 X$ to $W^1 X$ is necessarily a $q$-quasiconformal mapping.

### 16.9.2015: Yi Zhang (Jyväskylä Yliopisto)

Title: **A density problem for Sobolev spaces on planar domains**

*Abstract:*

In 1987 J. L. Lewis proved that global smooth functions are dense in $W^{1,\,p}(\Omega)$ for $1<p<\infty$, provided that $\Omega$ is a bounded planar Jordan domain. Moreover in 2007, A. Giacomini and P. Trebeschi shown that $W^{1,\,2}(\Omega)$ is dense in $W^{1,\,p}(\Omega)$ for $1 \le p<2$ when $\Omega$ is a bounded simply connected planar domain. In the talk I would like to present recent work (arXiv:1508.01400) with Pekka Koskela which improves these two results.

### 26.8.2015: Carlos Perez

Title: **New results for commutators of Singular Integrals with BMO functions**

**Abstract:**

Commutators of Singular Integrals with BMO functions were introduced in the seventies by Coifman-Rochberg and Weiss. These operators are very interesting for many reasons and has many connections with different branches of Analysis and PDE. From the theoretical point of view one they are very interesting since they are "more singular" than the classical Singular Integral Operators. In this lecture we plan to give several reasons showing the "bad" behavior of these operators.

### 19.8.2015 Jose Manuel Enriquez de Salamanca Garcia

Title: **The corona theorem for a domain bounded by a $C^{1+\alpha}$ curve**

**Abstract:**

Let f be a quasiconformal mapping on the plane with complex dilatation $\mu$. If $\mu$ satisfies a certain Carleson measure condition, then one can transfer $H^\infty$ on the upper half plane onto the corresponding space in the complement of the quasicircle f(R). This condition characterizes $C^{1+\alpha}$ curves and allows us to prove the corona theorem for domains whose boundary lies in one of these curves.