## Geometric function and mapping theory

### dr hab. Tomasz Adamowicz

Thursday 14.15-15.45, room 106**20-Oct: **Tomasz Adamowicz (IM PAN)

**Title**: "Prime ends and mappings"

**Abstract**: The studies of prime ends have long history involving various approaches, for example due to Caratheodory, Nakki, Vaisala and Zorich. In the talk we present some of those theories, including recent developments in metric measure spaces. Moreover, we discuss problems of continuous and homeomorphic extensions of mappings to the topological and prime end boundaries in the Euclidean setting and the setting of metric measure spaces for a class of mappings generalizing quasiconformal mappings.

**27-Oct:** Tomasz Adamowicz (IM PAN)

**Title**: "Prime ends and mappings, p. II"

**3-Nov: **Ben Warhurst (MIM UW)

**Title**: "Sub-Laplacians and Mean values"

**Abstract**: The talk will examine the relationship between the property of a function on a Carnot group being harmonic with respect to a sub-Laplacian, and the property of a function satisfying a mean value property.

**10-Nov: **Michał Miśkiewicz (MIM UW)

**Title**: "Rectifiable Reifenberg-type theorems"

**Abstract**: Reifenberg's topological disk theorem gives a criterion (the so-called Reifenberg flatness condition) on a closed subset S of R^n implying that S is bi-Holder with R^k. As shown by elementary examples, Reifenberg flat sets can have infinite k-dimensional measure and even dimension greater than k. There are several generalizations with stronger assumptions that yield an upper bound on the k-dimensional measure of S (T. Toro, G. David). In this talk I will discuss a recent result due to A. Naber and D. Valtorta, which has relatively weak assumptions and allows some interesting generalizations.

**17-Nov: **Antoni Kijowski (IM PAN)

**Title**: "Connections between harmonicity and mean value property"

**Abstract**: One of many properties of harmonic function *u* is that the mean value of u over any ball *B(x,r)* is equal to the *u(x)* (mean value property). The converse is also true, meaning that if a continous function u has mean value property on every ball in its domain, then u is harmonic. During my talk we will see how can we weaken the assumptions, to still obtain harmonicity, particularly considering lower regularity of function or smaller number of radii on which the mean value property holds (e.g. one radius and two radius theorems). A number of examples will help us understand which assumptions are essential.

**24-Nov: **Martyna Patera (IM PAN)

**Title**: "The Mean Value Property for Harmonic Functions on Trees"

**Abstract**: One of the most important properties of harmonic functions is the mean value property, which says that the average value of a harmonic function over any admissible ball is equal to the value of the function at the center of the ball. Conversely, functions having the mean value property are harmonic (under some assumptions).

The main part of my talk will be dedicated to one radius theorems in the discrete setting. For a function satisfying the mean value property for one radius (depending on the point), we will see what conditions on the radius function are needed for the function to be harmonic. We will consider harmonic functions in R^n and on trees. We will define them on trees in two ways: using the mean value property and using the discrete Laplacian. We will also investigate connections between these two definitions.

**1-Dec: **Jarosław Mederski (IM PAN & UMK)

**Title**: "Linear and semilinear curl-curl problems"

**Abstract**: We present recent results concerning linear and semilinear equations involving the curl-curl operator. Our problems are motivated by nonlinear materials and solutions to the curl-curl problems lead to time-harmonic electromagnetic waves propagating in the materials. We discuss different nonlinear materials e.g. with the optical Kerr effect.

**8-Dec: **No seminar: *Winter school in PDEs,* Będlewo (Simons Semester)

**15-Dec: **Jacek Gałęski (MIM UW)

**Title**: "Besicovitch-Federer projection theorem for mappings having constant rank of the Jacobian matrix"

**Abstract**: For an m-unrectifiable set S of finite Hausdorff measure and a mapping f from R^n to R^n having constant, equal to m, rank of the Jacobian matrix, we prove that in any open neighbourhood of f in C^1 topology there exist a mapping g such that Hausdorff measure of the image g(S) is zero. Local (restriction to a small set in the domain) version of the above theorem is an easy consequence of the Besicovitch-Federer projection theorem. The main body of the talk is the construction of one mapping from parts (mapping defined on small open domain) that do not agree with other parts on intersections of their individual domains.

**22-Dec, 29-Dec, 5-Jan: **No seminars, Christmas Holidays

**12-Jan: **Tomasz Kostrzewa (MiNI PW)

**Title**: "Differential operators on LCA groups"

**Abstract**: In my talk I will introduce the notion of an invariant differential operator on locally compact abelian group and study its properties and relations to the Bruhat-Schwartz class. I will also talk about my attempts on using this notion in the theory of Sobolev spaces on LCA groups.

**19-Jan: **Marta Szumańska (IM PAN & MIM UW)

**Title**: "Gol'dshtein and Ukhlov results on traces of functions of Dirichlet spaces defined on simply connected domains"

**Abstract**:The Dirichlet space is a space of locally integrable functions whose weak gradient is square integrable on a given domain. Such functions are defined up to the set of measure zero, thus their values on the boundary of the domain can be considered only in terms of trace. Gol'dshtein and Ukhlov proved that if the Dirichlet space is defined on a simply connected domain in the plane, connected at each boundary point, then the unique quasi-continuous representative of a function in the Dirichlet space can be extended to quasi-continuous function on the closure of the domain; the extended function is defined on the boundary up to the set of capacity zero.

In case of broader class of domains, i.e. simply connected domains, one can construct the extension to the Caratheodory boundary (up to the set of capacity zero).

During the talk I am planning to show main ideas of the proofs of the abovementioned theorems and introduce necessary tools and notions (properties of the Dirichlet space and capacities, notions of quasi-continuous functions, the Caratheodory boundary and the capacitary boundary).

The talk is based on the paper ,,Traces of functions of L_2^1 Dirichlet spaces on the Caratheodory boundary" by V. Gol'dshtein A. Ukhlov; Studia Mathematica 235 (2016), 209-224.

**26-Jan: **Stanislav Hencl (Charles University, Prague), room **321**

**Title**: "Jacobians of Sobolev homeomorphisms"

**Abstract**: We show that if $n$ is 2 or 3, $f$ is a homeomorphism of a domain $\Omega$ in $R^n$ and $f$ is in the Sobolev space $W^{1,1}_{\loc}(\Omega,R^n)$, then the Jacobian determinan $J_f$ is either non-negative or non-positive a.e. in $\Omega$. This answers an open problem by P. Hajlasz. It is a joint work with J. Maly.

**2-Feb: **Teri Soultanis (IM PAN), room 403

**Title**: "Energy minimization of maps between metric spaces, and hyperbolicity in the target"

**Abstract**: I present the class of Newtonian maps between metric spaces and discuss the notion of homotopy of such maps. The role of nonpositive curvature in the target for proving existence of energy minimizers in homotopy classes is well known but in this generality the proof seems to require an extra hyperbolicity assumption. I will explain the main ideas in the proof and illustrate where the hyperbolicity comes in.

**2-Mar: **Yevhen Sevostianov (Zhytomyr State University), r. 403

**Title**: "Geometric Approach in the Theory of Spatial Mappings"

**Abstract**: The talk is devoted to the investigation of the space mappings with non-bounded characteristics of quasiconformality. In particular, we mean here the so-called mappings with finite distortion which are intensively investigated by leading mathematicians in the last decade. The series of properties of the so-called ring Q–mappings are obtained. The above mappings include the mappings with bounded distortion by Reshetnyak (quasiregular mappings). In particular, the differentiability and the analogues of the theorems of Sokhotski–Weierstrass, Liouville, Picard, Iversen, Montel etc. are obtained for the above mappings.

**9-Mar: **Yevhen Sevostianov (Zhytomyr State University), r. 403

**Title**: "On boundary behavior of mappings in terms of prime ends"

**Abstract**: A boundary behavior of mappings, which are closely related with the Sobolev and Orlicz--Sobolev classes in the plane and in the space, is investigated. There are obtained theorems on the boundary behavior of classes mentioned above.

**16-Mar: **No seminar. "Workshop on Nonstandard Growth Analysis and its Applications 2017"

http://crossfields.impan.pl/growth.html

**23-Mar: **Michał Miśkiewicz (MIM UW), r. 403

**Title**: "Fractional differentiability of p-harmonic functions"

**Abstract**: It is well known that the gradient of any p-harmonic function is locally Holder continuous (with some small exponent α dependent on p and the dimension) - this can be viewed as existence of derivatives of order α in terms of Besov spaces. My talk is concerned with existence of higher order derivatives: for p>2 of order 2/p (G. Mingione) and for 2

**30-Mar: **Michał Gaczkowski (IM PAN & MiNI PW), r. 403

**Title**: "Concentration compactness in variable exponent spaces"

**Abstract**: When dealing with PDEs, compact embeddings are one of the most powerful tools. Unfortunately, they are no always valid. One can try to deal with this by seeking convergence in some other sense. In my talk I would like to introduce results of Fernandez and Silva about concentration compactness of P.L. Lions for variable exponent in $R^n$. I will also discuss how such kind of results could be possibly use to improve compact embeddings.

**6-Apr: **Sławomir Kolasiński (MIM UW), r. 403

**Title**: ''Solution of an anisotropic inhomogeneous Plateau problem''

**Abstract**: The Plateau problem is about finding a "surface" which minimizes "area" amongst competitors which "span" a given boundary. I shall briefly describe various formulations of the problem and shortcomings of the solutions. Next, I will present a sketch of the existence proof for an abstractly formulated problem destiled from Almgren's 1968 paper.

This talk will be based on my joint work with Yangqin Fang from the Max-Planck Institute in Potsdam.

**13-Apr:** No seminar (Good Thursday)

**20-Apr: **Martyna Patera (IM PAN), r. **408**

"Doubling measures on regular trees"

**Abstract**:

A measure on a metric space is a doubling measure if the measure of any ball

with radius 2r is bounded (with some multiplicative constant) by the measure of

a ball with radius r. A metric space equipped with a doubling measure holds

many useful properties.

The main part of my talk will be dedicated to studying a doubling condition on a

tree. On a regular rooted tree we can definie a metric and a measure to

obtain a metric measure space. We will investigate what properties a

radial measure, i.e. dependent only on the distance from the root, must

posses to be a doubling measure. In particular, we will look at a

uniformizing metric and the weighted measure associated with it.

**27-Apr: **Olli Toivanen (IM PAN), r. 403

**Title**: Regularity in generalized Orlicz spaces

**Abstract**: A generalized Orlicz space is a generalization of, for example, Orlicz spaces and Lebesgue spaces, both classical and of variable exponent. To put it in the terms of the integrability condition for a function t and a location x, with some exponent (function) p:

Lebesgue = t^p

Orlicz = \phi(t)

variable Lebesgue = t^{p(x)}

generalized Orlicz = \phi(x,t)

Recent research has began to look for the generalization of the various regularity theories of these cases.

I will discuss recent results by Harjulehto, Hästö, Klen, Cruz-Uribe, myself and others, starting from "Hölder regularity of quasiminimizers under generalized growth conditions" (Harjulehto, Hästö, T.; Calc. Var. PDEs, 56 (2), 2017.)

**4-May: **Antoni Kijowski (IM PAN), r. 106

**Title**: Regularity of weakly and strongly harmonic functions

**Abstract**: A strongly harmonic function is a function defined on metric measure space, having the mean value property for all balls. Weakly harmonic function are defined in a similar way, i.e. it has the mean value property at every point for at least one radius of a ball. During my talk I will discuss some consequences of endowing a distance function and a measure with additional properties on such type of harmonicity, i.e. the Hölder and Lipschitz continuity. I will present some examples, among them strongly harmonic functions which are spherically symmetric.

The talk is based on paper "Harmonic functions on metric measure spaces", T. Adamowicz, M. Gaczkowski, P. Górka, arXiv:1601.03919 and a work with T. Adamowicz.

**11-May: **Iwona Skrzypczak (IMPAN)

**Title**: Renormalized solutions to nonlinear elliptic partial differential equations

**Abstract**: The talk will concern existence of renormalized solutions to general nonlinear elliptic equation in Musielak-Orlicz space. The growth of

the leading part of the operator $A$ is controlled by a generalized nonhomogeneous and anisotropic $N$-function $M$. The approach does

not require any particular type of growth condition of $M$ or its conjugate $M^*$ (neither $\Delta_2$, nor $\nabla_2$ conditions).

The condition we impose is log-H\"older continuity of $M$, which results in good approximation properties of the space. The proof of the main results

uses truncation ideas, the Young measures methods and monotonicity arguments.

The results are the based on the joint work with Piotr Gwiazda and Anna Zatorska-Goldstein.

**18-May: **Ben Warhurst (MIM UW)

**Title**: Geometric function theory on Jets

**Abstract**: Jet spaces can be viewed as a generalisation of the Heisenberg group, particularly from a sub-Riemannian perspective. The talk will introduce jets as sub-Riemannian spaces and discuss some exemplary results.

**25-May: **Grzegorz Łysik (IM PAN & UJK)

**Title**: Twierdzenie o jednoznaczności dla funkcji $\rho$-analitycznych

**Abstract**: Zacznę od podania warunków koniecznych i dostatecznych na zachodzenie równości w nierówności trójkąta dla metryki niezmienniczej $\rho$ na $R^n$. Następnie zdefiniuję funkcje $\rho$-analityczne i pokażę, że posiadają one własność jednoznaczności, tzn. jeśli funkcja znika na zbiorze otwartym, to znika wszędzie.

**1-June: **Ben Warhurst (MIM UW)

**Title**: Geometric function theory on Jets, p.II

**Abstract**: Jet spaces can be viewed as a generalisation of the Heisenberg group, particularly from a sub-Riemannian perspective. The talk will introduce jets as sub-Riemannian spaces and discuss some exemplary results.