## Geometric function and mapping theory

### dr hab. Tomasz Adamowicz

Thursday 14.15-15.45, room 106Academic year 2017/2018

**5-Oct: **Sebastiano Golo (University of Jyvaskyla, Finland)

**Title**: Homogeneous distances and shape of spheres

**Abstract**: A graded groups are a generalization of Carnot groups introduced already by Folland and Stein in the Eighties. On graded groups it is possible to define canonical anisotropic dilations. The distances that are both left-invariant and 1-homogeneous with respect to these dilations are called "homogeneous distances". I will present some results on the regularity of spheres of homogeneous distances on graded group. As an application, I will state a theorem about the Besicovitch Covering Property in the Heisenberg group.

**12-Oct: **Sebastiano Golo (University of Jyvaskyla, Finland)

**Title**: Minimal surfaces in the Heisenberg group

**Abstract:** The Heisenberg group has a well studied sub-Riemannian structure. Geometric measure theory in the sub-Riemannian setting is still in development and several fundamental questions are still open. One reason is that sets of finite sub-Riemannian perimeter may have fractal behaviours. I will present the state of art (in my knowledge) of the study of minimal surfaces in this setting, with some recent results about contact variations of the area functional.

**19-Oct: **Michael Eastwood (University of Adelaide)

**Title**: Puncture repair in conformal geometry

**Abstract: **If you remove a point from a compact Riemann surface and ask for a conformal compactification of the result, then the only option is to replace the point. Roughly speaking, there is no difference between a punctured Riemann surface and a marked Riemann surface. The same result holds for conformal geometry in higher dimensions, where the proof should be easier but all the proofs I know are more difficult! There are plenty of ways to attempt to prove these results. I'll explain what I know. Sadly, after one successful argument, this will mostly be an explanation of what goes wrong or things I cannot prove. In other words, mostly sketches of various failed attempts. Needless to say, help from the audience would be much appreciated!

**26-Oct: **Michał Gaczkowski (IM PAN)

**Title**: Improved compactness in Hajłasz spaces

**Abstract: **On the Riemannian manifolds a compact Sobolev embedding can be improved in the presence of symmetries. We would like to investigate if such kind of results holds for Sobolev spaces defined on metric spaces. In this case most methods are very difficult to use. However, it turns out that using concentration compactness of P.L. Lions one can obtain better compact embeddings (unfortunately, not as good as expected). In my talk I would like to show how one can obtain such improvement in Hajłasz spaces.

**2-Nov: **Katrin Fässler (University of Fribourg, Switzerland)

**Title**: Singular integrals, intrinsic graphs and Lipschitz harmonic functions in the Heisenberg group

**Abstract: **The Heisenberg group H is a Lie group whose Lie algebra is given by the bracket condition [X,Y]=T. The differential operator X^2+Y^2 (sub-Laplacian) admits a fundamental solution G, which is closely related to a homogeneous norm on H. The resulting metric space (H,d) exhibits an interesting geometry. In this talk I will discuss the L^2 boundedness of certain 3-dimensional singular integral operators T with respect to 3-regular measures in (H,d). I will show that proving uniform L^2 boundedness of T on vertical planes -- which is essentially a Euclidean problem -- suffices to prove L^2 boundedness of T on intrinsic graphs of compactly supported intrinsic C^{1,\alpha} functions over vertical planes. This result applies to the operator T whose kernel is the gradient of G. As a corollary we infer that the intrinsic graphs mentioned above are non-removable for Lipschitz continuous solutions to the sub-Laplace equation. This is joint work with V. Chousionis and T. Orponen.

**9-Nov: **Antoni Kijowski (IMPAN)

**Title**: Characterization of strongly harmonic functions

**Abstract: **We say that a locally integrable function on a metric measure space is strongly harmonic if it attains the mean value property. In case of an open subset of Euclidean space with induced metric and measure it is equivalent, by the converse to the Gauss theorem, to solving the Laplace equation. However, in a general metric measure space X the class H of all such functions is not well characterized.

During my talk I will try to give the audience an insight into the structure and properties of H by investigating strongly harmonic functions on the Euclidean space equipped with a distance function and a weighted Lebesgue measure. I will show characterization of the class H in two cases: (1) when the distance is arbitrary and the measure is Lebesgue, and (2) with an arbitrary weight and the Euclidean distance. Finally, I will describe the dimension of space H.

**16-Nov: **No seminar. The conference "International Conference on Symmetry and Geometric Structures" at IM PAN

**23-Nov: **Iwona Skrzypczak (IMPAN)

**Title**: Lavrentiev's phenomenon in the Musielak-Orlicz spaces

**Abstract: ** Lavrentiev's phenomenon originally meant the situation when the infimum of the variational problem over the smooth functions is strictly greater than infimum taken over the set of all functions satisfying the same boundary conditions. The notion of the Lavrentiev phenomenon became naturally generalised to describe the situation, where functions from certain spaces cannot be approximated by regular ones (e.g. smooth).

Typical examples of the Musielak-Orlicz spaces are the variable exponent spaces, the Orlicz-Sobolev spaces, and the double-phase space.

It is well known that the smooth functions are dense in the variable exponent spaces only if the exponent is regular enough. Similar fact holds for the double-phase space. In the Orlicz-Sobolev spaces the result corresponding to the Meyers-Serrin theorem is Gossez's approximation theorem (Studia Math. '82), who infers that weak derivatives are strong derivatives with respect to the modular topology.

The talk will describe unification of these approaches provided in the manuscript arXiv:1711.06145: results on modular density of smooth functions. The precision of the method is confirmed by sharp results in all above mentioned cases. The possible applications to PDEs will be mentioned.

**30-Nov: **No seminar

**7-Dec: **Ben Warhurst (MIMUW)

**Title**: Puncture repair in metric measure spaces

**Abstract: **The talk will discuss a geometric measure space approach to the puncture repair problem in conformal Riemannian geometry discussed in the seminar earlier this semester by Mike Eastwood.

**14-Dec: **Maria J. Gonzalez (Universidad de Cádiz, Spain)

**Title**: Geometry of quasicircles and Beurling type operators

**Abstract: **The general problem is the following: Given a quasiconformal mapping ρ on the plane, what conditions on its dilatation µ guarantee certain geometric properties of the quasicircle Γ =ρ(R)? In this talk we will mainly consider the case where Γ is a chord-arc curve. Denoting by S the Beurling operator, we will show that the invertivility of the operator (I−μS) on a particular weighted L^2 space characterizes such curves. (Joint work with K. Astala)

Academic year 2016/2017

**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.