# Analysis seminar

### Next Seminars

**21.11.2018 Zheng Zhu**Title: The product of Sobolev extension domains

Abstract: If u and V are W^{1,p}-extension domain, then U\times V is also a W^{1,p}-extension domain. Conversely, if U\times V is a W^{1,p}-extension domain, then both U and V must be W^{1,p}-extension domains. This is a joint work with Prof. Pekka Koskela.

**28.11.2018 Yi-Hsuan Lin **Title: On fractional inverse problems

Abstract: We demonstrate recent progress in the fractional inverse problems, especially for the famous Calderón type problem, where one tries to determine an unknown coefficient in an anisotropic Schrödinger equation from exterior measurements of solutions. This equation enjoys remarkable uniqueness and approximation properties, which turn out to yield strong results in related inverse problems.

**5.12.2018 Darya Apushkinskaya (Peoples' Friendship University of Russia, Moscow) **Title: When Hopf's lemma remains valid?

Abstract: The Hopf lemma, known also as the boundary point principle, is one of the important tools in qualitative analysis of partial differential equations. This lemma states that a supersolution of a partial differential equation with a minimum value at a boundary point, must increase linearly away from its boundary minimum provided the boundary is smooth enough. For general operators of non-divergence type with bounded measurable coefficients this result was established in elliptic case independently by E. Hopf and O. Oleinik (1952) and in parabolic case by L. Nirenberg (1953). The first result for elliptic equations with divergence structure was proved by R. Finn and D. Gilbarg (1957). Later the efforts of many mathematicians were aimed at the extension of the classes of admissible opeartors and at the reduction of the boundary smoothness. We present several versions of the Hopf lemma for general elliptic and parabolic equations in divergence and non-divergence forms under the sharp requirements on the coefficients of equations and on the boundary of a domain. Also we provide a new sharp counterexample.

### Fall 2018

**19.9.2018 Anna Kausamo**Title: On the Monge problem in Multi-Marginal Optimal Mass Transportation

**26.9.2018 Rakesh **(University of Delaware)Title: Inverse problems for the wave equation

Abstract: We describe some problems and results for formally determined (not over-determined) inverse (coefficient determination) problems for the perturbed wave equation. The problems have little geometry since the wave speed is constant. The difficulties are in the analysis associated with proving the injectivity of non-linear or even linear maps. Even some basic problems remain unsolved.

**3.10.2018 Christian Webb (Aalto University)**Title: On the statistical behavior of the Riemann zeta function

Abstract: A notoriously difficult problem of analytic number theory is to describe the behavior of the Riemann zeta function on its so-called critical line. After reviewing some basic facts about thezeta function, I will discuss what can be said if the problem is relaxed slightly, namely if one is only interested in the behavior of the zeta function in the vicinity of a typical point on the critical line. Time permitting, I will also discuss how this problem is related to various models of probability theory and mathematical physics. The talk is based on joint work with Eero Saksman.

**10.10. 2018 Atte LohvansuuTitle: **Duality of moduli in regular metric spaces

**17.10.2018 Ivan Yaroslavtsev (Technical University of Delft, Netherlands.)**Title: The Hilbert transform and orthogonal martingales in Banach spaces

Abstract: Due to a classical argument of Doob it is well-known that the periodic Hilbert transform has a representation in terms of stochastic integrals with respect to a 2-dimensional Brownian motion. These stochastic integrals happen to be orthogonal martingales, so any estimates for orthogonal martingales lead to the same estimates for the periodic Hilbert transform. The goal of this talk is to show the converse dependence. Namely, we show that for any Banach space $X$ and any convex continuous functions $\Phi, \Psi:X \to \mathbb R_+$ one has that for any orthogonal martingales $M$ and $N$ such that $N$ is weakly differentially subordinate to $M$

\[

\mathbb E \Psi(N_t) \leq C_{\Phi, \Psi} \mathbb E \Phi(M_t),\;\;\; t\geq 0,

\]

where the sharp constant C_{\Phi, \Psi} coincides with the $\Phi, \Psi$-norm of the periodic Hilbert transform.

This estimate has a lot of applications. In particular, it will allow us to show that the $L^p$-norms of the periodic Hilbert transform and the discrete Hilbert transform coincide for all $1<p<\infty$ and for any Banach space $X$. This extends the result of Bañuelos and Kwaśnicki, who showed in 2017 that the $L^p$-norms of the periodic Hilbert transform and the discrete Hilbert transform are equal in the real-valued setting, which had been an open problem for 90 years. This talk is based on joint work with Adam Osękowski (University of Warsaw).

**24.10.2018 Marco Caroccia (University of Lisbon)**

Title: Perimeter with densities: An energy for epitaxial growth

Abstract: In this talk I will introduce a free energy, first studied by Ratz and Voigt, defined as the integration over the reduced boundary of a crystal E of a suitable psi(u(x)) where psi is a convex function and u denotes the adatom density (the density of atoms not attached to the crystal and free to move on the surface). This energy leads the surface diffusion evolution in absence of the elastic stress and for small adatom densities. We treat the variational point of view, by analyzing the semicontinuity issues and the relaxad functional.

**31.10.2018 Rami Luisto**Title: Regularity of the inverse of BV homeomorphisms in dimension three

Abstract: By the classical inverse function theorem the inverse of a C^1 homeomorphism is again a C^1 homeomorphism when the Jacobian is strictly positive. It is natural to ask if similar results hold under milder regularity assumptions; natural candidates being Sobolev- and BV-homeomorphisms which are commonly used in the study of nonlinear elasticity. The problem of the weak regularity of the inverse has attracted much attention in the past decade, starting from the planar results of Hencl, Koskela and Onninen (2006, 2007) and extending to higher dimensions with the work of Csörnyei, Hencl and Maly (2010). These results show that for a Sobolev (1,n-1) homeomorphism f the inverse is a BV-mapping, and belongs to W^(1,n-1) when the mapping f is of finite distortion. In this talk we discuss our recent result in [arxiv:1804.03449], joint with Stanislav Hencl and Aapo Kauranen, which states that in dimension three the inverse of a BV-homeomorphism is BV if and only if the distributional adjoint of the mapping is a finite Radon measure. The distributional adjoint defined here is a new concept motivated both by our characterization and the success of the distributional Jacobian in the study of nonlinear elasticity.

**7.11.2018 Olli Tapiola**Title: Carleson measures, rectifiability and epsilon-approximability of harmonic functions in L^p

Abstract: The study of uniform rectifiability started as a hunt for optimal geometric conditions for various aspects of Calderón-Zygmund type harmonic analysis but more recently, the emphasis has been on exploring its connections to partial differential equations and other types of analysis. In this talk, we discuss some recent trends in uniform rectifiability, related topics and an L^p version of \epsilon-approximability of Hytönen and Rosén that generalizes the L^\infty type approximability property of Varopoulos. The talk is partially based on my previous and on-going work with S. Hofmann and S. Bortz.

**14.11.2018 Aleksis Koski**Title: Sobolev homeomorphic extensions

Abstract: In the mathematical theory of nonlinear elasticity one typically represents elastic bodies as domains in Euclidean space, and the main object of study are deformations (mappings) between two such bodies. The class of acceptable deformations one considers usually consists of Sobolev homeomorphisms between the respective domains, for example, with some given boundary values. It is hence a fundamental question in this theory to ask whether a given boundary map admits a homeomorphic extension in the Sobolev class or not. We share some recent developements on this subject, including sharp existence results and counterexamples.

_______________________________________________________________________

### Spring 2018

**23.5.2018 Tomas Soto**Title: Banach envelopes

**16.5.2018 Juan J. Manfredi (University of Pittsburgh)**Title: Some random walks in the Heisenberg group

**9.5.2018 Leyter Potenciano**

Title: An inverse problem value problem for the magnetic Schrödinger operator with local data

Abstract: The main goal of this talk is to discuss an Inverse Boundary Value Problem associated with a magnetic Schrödinger operator on a bounded domain. It is quite well-known that one can recover the coefficients within the domain related to the Schrödinger operator (such coefficients represent the magnetic and electric potentials) by making current and voltage measurements on the whole boundary of the domain.

We will show that this result still remains true when the measurements are taken on subsets of the boundary.

**25.4.2018 Sun-Sig Buyn (Seoul National University)**

Title: Regularity estimates for nonlinear parabolic problems in nonsmooth domains

**18.4.2018 Jere Lehtonen**Title: Geodesic X-ray transform on non-compact Riemannian manifolds

Abstract: I will discuss the following question: Suppose we have an unknown function on a non-compact Riemannian manifold. If we know its integrals over maximal geodesics then can we determine the function itself?

The talk is based on a joint work with M. Salo and J. Railo.

**11.4.2018 Cristobal Meronno (Universidad Autonoma de Madrid)**Title: Recovery of the singularities of a potential from backscattering data.

Abstract: I will introduce one of the main Inverse scattering problems for the Schrödinger equation with a potential. The objective is to discuss some recent results showing that it is possible to recover the singularities of the potential from the scattering data, in every dimension. We will also see that there are precise limitations on how much regularity information can be obtained from the data depending on the a priori regularity of the potential.

**4.4.2018 Domenico La Manna (Naples)**Title: An Isoperimetric problem with a Coulombic repulsion and attractive term

**28.3.2018 Mikko Salo**Title: The fractional Calderón problem

Abstract: We show global uniqueness in an inverse problem for the fractional Schrödinger equation: an unknown potential in a bounded domain is uniquely determined by exterior measurements of solutions. We also show global uniqueness in the partial data problem where measurements are taken in an arbitrary open, possibly disjoint, subsets of the exterior..

**21.3.2018 Dimitrios Ntalampekos (UCLA)**Title: (NON)-REMOVABILITY OF THE SIERPINSKI GASKET

Abstract: Removability of sets for quasiconformal maps and Sobolev functions has applications in Complex Dynamics, in Conformal Welding, and in other problems that require “gluing” of functions to obtain a new function of the same class. We, therefore, seek geometric conditions on sets which guarantee their removability. In this talk, I will discuss some very recent results on the (non)-removability of the Sierpinski gasket.

A first result is that the Sierpinski gasket is removable for continuous functions of the class W^{1,p} for p > 2. The method used applies to more general fractals that resemble the Sierpinski gasket, such as Apollonian gaskets and generalized Sierpinski gasket Julia sets. Then, I will sketch a proof that the Sierpinski gasket is non-removable for quasiconformal maps and thus for W^{1,p} functions, for 1 ≤ p ≤ 2. The argument involves the construction of a non-Euclidean sphere, and then the use of the Bonk-Kleiner theorem to embed it quasisymmetrically to the plane.

**14.3.2018 at 14:15 MaD302! Debanjan Nandi**Title: Density results in Sobolev spaces. Some new results on density of functions with bounded derivatives in Sobolev spaces in (Gromov) hyperbolic type domains.

**28.2.2018 at 14:15 MaD380 Joonas Heino**Title: A characterization of solutions for parabolic p(x,t)-Laplace type equations

Abstract: In this talk, we formulate a two-player zero-sum stochastic differential game in continuous time that represents the unique viscosity solution to a terminal value problem for a parabolic partial differential equation involving the normalized p(x,t)-Laplacian. The game is formulated in a way that covers the full range 1<p(x,t)<\infty.

**21.2.2018 No seminar on the conference week.**

**14.2.2018 at 14:15 MaD380 Alex Karrila (Aalto)**Title: Branches in a uniform spanning tree and conformal invariance

Abstract: It is predicted by physicists that continuum limits of critical random models on planar lattices should be described by conformally invariant quantum field theories. The aim of this talk is to characterize the limit of a certain collection of random interfaces, on increasingly dense lattices, in terms of conformally invariant random geometry. In more detail, let $\Lambda$ be a bounded and simply connected planar domain and $\Lambda^\delta$ its natural approximation by the square grid $\delta \mathbb{Z}^2$. We consider a uniform random spanning tree of the graph $\Lambda^\delta$, and condition it on the existence of certain boundary-to-boundary branches. The weak limit of the corresponding random interfaces, as $\delta \to 0$, is a conformally invariant family of random curves called the local multiple $SLE(2)$. Partly based on joint work with Kalle Kytölä (Aalto) and Eveliina Peltola (Geneva).

**7.2.2018 at 14:15 MaD380 Laurent Dufloux (Oulu)**Title: (TBA)

31.1.2018 at 14:15 MaD380 Timo Schultz

Title: Optimal transport maps in metric measure spaces with curvature bounded below

24.1.2018 at 14:15 MaD380 Michael Seidl (Dept. of Theor. Chemistry, VU Amsterdam.)

Title: Strictly correlated electrons in quantum mechanics: Density functional theory (DFT) meets optimal transport theory (OT)

Abstract: The quantum-mechanical limit of infinitely strong repulsion between electrons ("strictly correlated electrons": SCE) provides important information for DFT (density functional theory). Nowadays, DFT is the method of choice for a wide class of electronic structure calculations (e.g.: quantum chemistry, solid state physics). Interestingly, the SCE concept has turned out to provide solutions to certain optimal transport (OT) problems in mathematics. This talk intends to highlight this connection between quantum physics (DFT) and mathematics (OT theory).

**31.1.2018 Timo Schulz**

Title: Optimal transport maps in metric measure spaces with curvature bounded below

**24.1.2018 Michael Seidl (Dept. of Theor. Chemistry, VU Amsterdam.)**

Title: Strictly correlated electrons in quantum mechanics: Density functional theory (DFT) meets optimal transport theory (OT)

Abstract: The quantum-mechanical limit of infinitely strong repulsion between electrons ("strictly correlated electrons": SCE)

provides important information for DFT (density functional theory). Nowadays, DFT is the method of choice for a wide class

of electronic structure calculations (e.g.: quantum chemistry, solid state physics). Interestingly, the SCE concept has turned out

to provide solutions to certain optimal transport (OT) problems in mathematics. This talk intends to highlight this connection

between quantum physics (DFT) and mathematics (OT theory).

**Fall 2017**

**29.11.2017 at 14:15 MaD380 Aleksis Koski**

Title: Radial symmetry of p-harmonic minimizers

Abstract: Motivated by models in elasticity theory, we study the minimization of p-harmonic energy among Sobolev homeomorphisms between planar doubly connected domains. The main problem in this setting is that there is no guarantee that an energy-minimal homeomorphism exists, as the Sobolev weak limits of homeomorphisms need not be injective themselves (nor continuous when p < 2). Hence our first points of discussion will be

1) The correct notion of a minimizer

2) The regularity and properties of such a minimizer

The main topic of this talk is the radially symmetric minimization problem between planar annuli for p < 2. This talk is based on joint work with Jani Onninen (__https://arxiv.org/abs/1710.01067__).

**22.11.2017 at 14:15 MaD380 Tapio Rajala**Title: Quasiconvex domains Abstract: A domain is quasiconvex if any two of its points can be connected by a curve inside the domain that has length comparable to the distance between the points. In this talk, we will study closed sets in the Euclidean space with quasiconvex complements. In particular, we will look at metrically removable sets. This is joint work with Sergei Kalmykov and Leonid Kovalev.

**15.11.2017 at 14:15 MaD380 David Bate (Helsinki)**Title: Characterising rectifiable metric spaces using typical Lipschitz functions.

**15.11.2017 at 15:15 MaD380 Tuomas Orponen (Helsinki)**

Title: Sharpening Marstrand‘s projection theorem

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

Title: Growth rates of frequently hypercyclic harmonic functions

Abstract:The notion of frequent hypercyclicity stems from ergodic theory and wasintroduced by Bayart and Grivaux (2004). Many natural continuous linear operators are frequently hypercyclic, for instance the differentiation operator on the space of entire functions. We consider the partial differential operator acting on the space of harmonic functions on R^n and we identify sharp growth rates, in terms of the L^2-norm on spheres, of its frequently hypercyclic vectors. This answers a question posed by Blasco, Bonilla and Grosse-Erdmann (2010). This is joint work with Eero Saksman and Hans-Olav Tylli.

**1.11.2017 at 14:15 MaD380 Luca Rondi (Trieste)**Title: Stability for the direct electromagnetic scattering problem

Abstract:I discuss the scattering problem for time-harmonic electromagnetic waves, due to the presence of scatterers and of inhomogeneities in the medium. I show a stability result for the solution to the corresponding exterior boundary value problem, with respect to variations of the scatterer and of the inhomogeneity, under minimal regularity assumptions for both of them. For example, both obstacles and screen-type scatterers are allowed at the same time. As a consequence, one can obtain bounds on solutions to these scattering problems which are uniform with respect to extremely general classes of admissible scatterers and inhomogeneities. In order to prove such a stability result, two key ingredients were developed: the first one is Mosco convergence for H(curl) spaces; the second one is a higher integrability property of solutions to Maxwell equations in nonsmooth domains. This is a joint work with Hongyu Liu and Jingni Xiao (Hong Kong Baptist University)

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

**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.

**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.