Conference on the occasion of the 75th birthday of
Olli Martio
December 15.-17. 2016
University of Jyväskylä
Jyväskylä, Finland
Abstracts
Serban Costea:
Sobolev-Lorentz spaces in the Euclidean setting and counterexamples
This paper studies the inclusions between different Sobolev-Lorentz spaces
W 1,(p,q) (Ω) defined on open sets Ω ⊂ Rn , where n ≥ 1 is an integer, 1 <
p < ∞ and 1 ≤ q ≤ ∞. We prove that if 1 ≤ q < r ≤ ∞, then W 1,(p,q) (Ω) is
strictly included in W 1,(p,r) (Ω).
We also extend the Morrey embedding theorem to the Sobolev-Lorentz
1,(p,q)
spaces H0
(Ω) when n < p < ∞. Namely, we prove that the Sobolev1,(p,q)
Lorentz spaces H0
(Ω) embed into the space of Hölder continuous functions on Ω with exponent 1 − np whenever Ω ⊂ Rn is open, 1 ≤ n < p < ∞
and 1 ≤ q ≤ ∞.
David Drasin:
Old and middle aged complex analysis
The talk has two separate themes. (I) Olli Martio, whose anniversary is
being honored today. I focus on the beginnings of the study of quasi regular
mappings in Finland, especially the work of JOS, starting about 50 years
ago. While the original tools used depending on the contemporary theory
of two-dimensional quasiconformal mappings, in time the power of PDE
approach, initiated in the former USSR became recognized and has been an
important link to the widest possible audience in analysis.
(II) Finnish mathematics became famous almost a century ago with Rolf
Nevanlinna’s value-distribution theory. In the last 50 years, as characteristic of classical fields in general, the big open problems take decades to be
resolved. I will discuss a very special case of one problem, which considers a refinement of Picard’s theorem - we consider only entire functions of
oder between one and two; it is not hard to discuss more general situations,
but the main difficulties show up here. The issue is how far the sum of
(Nevanlinna) deficiencies must be from two if f has order ρ ∈ (1, 2).
We give a concrete conjecture, which suggests that unless the function
f (z) is very close topologically to the presumed extremal functions, its deficiency sum cannot reach even 3/2, to say nothing of 2.
Parisa Hariri:
Hyperbolic type metrics in Geometric function theory
My talk is based on four papers [CHKV, HVW, HVZ, DHV], where metrics
are central theme. As well-known, Geometric Function Theory is a field
where metrics are recurrent: some examples are Euclidean, chordal and hyperbolic metrics. Here we study several other metrics defined on subdomains
of Rn , n ≥ 2. Examples include
• the triangular ratio metric
• the visual angle metric
• the distance ratio metric and its modification.
The first two of these metrics have been introduced and studied recently.
Some of the basic questions are:
• How are these metrics related to other metrics such as hyperbolic or
quasihyperbolic metric?
• How are these metrics transformed under well-known classes of mappings such as Möbius transformations, quasiconformal maps, Lipschitz
maps?
The answers will depend on the domains studied.
References
[CHKV]J. Chen, P. Hariri, R. Klén, and M. Vuorinen: Lipschitz
conditions, triangular ratio metric, and quasiconformal maps. Ann. Acad.
Sci. Fenn. 40, 2015, 683–709, doi:10.5186/aasfm.2015.4039, arXiv:1403.6582
[math.CA].
[DHV]O. Dovgoshey, P. Hariri, M. Vuorinen: Comparison theorems for hyperbolic type metrics. Complex Var. Elliptic Equ. 61 (2016),
no. 11, 1464–1480.
[HVW] P. Hariri, M. Vuorinen, G. Wang: Some remarks on the
visual angle metric. Comput. Methods Funct. Theory 16 (2016), no. 2,
187–201.
[HVZ] P. Hariri, M. Vuorinen, and X. Zhang: Inequalities and
bilipschitz conditions for triangular ratio metric.-Rocky Mountain J. Math.
(To appear) arXiv: 1411.2747 [math.MG] 21pp.
Petteri Harjulehto:
Minimizers under generalized growth conditions
We study the Dirichlet φ-energy integral with Sobolev boundary values.
The function φ has generalized Orlicz growth. Special cases include variable
exponent and double phase growths. We prove Harnack’s inequality for local
quasiminimizers. As a consequence, we obtain the local Hölder continuity
of local quasiminimizers. We study regular boundary points for minimisers
and show that a boundary point is regular if the complement is fat in the
φ-capacity sense.
Ritva Hurri-Syrjänen:
On the John-Nirenberg 2nd result and beyond
The goal of my talk is to address some inequalities which Fritz John and
Louis Nirenberg proved to be valid for certain functions defined in a cube.
I will discuss the validity of similar inequalities for functions defined in an
arbitrary bounded domain. My talk is based on joint work with Niko Marola
and Antti V. Vähäkangas.
Peter Hästö:
Harmonic analysis in generalized Orlicz spaces
In this talk, I present recent results on the boundedness of the maximal operator and other operators in generalized Orlicz spaces. Both necessity and
sufficiency of the conditions are considered. I also motivate the research with
applications from PDE. Further work on PDE with non-standard growth is
given in the talk by Petteri Harjulehto.
Tero Kilpeläinen:
Reflections
Martio’s reflection principle states that a p-harmonic function can be reflected over a piece of an hyperplane if it vanishes there. This result might
be referred to in my talk.
Juha Kinnunen:
Parabolic weighted norm inequalities
We discuss parabolic Muckenhoupt weights and functions of bounded mean
oscillation (BMO) related to a doubly nonlinear parabolic partial differential equation. In the natural geometry for the doubly nonlinear equation the
time variable scales as the modulus of the space variable raised to a power.
Consequently the Euclidean balls and cubes have to be replaced by parabolic
rectangles respecting this scaling in all estimates. An extra challenge is given
by the time lag appearing in the estimates. The main result gives a full characterization of weak and strong type weighted norm inequalities for parabolic
forward in time maximal operators. In addition, we give a Jones type factorization result for the parabolic Muckenhoupt weights and a CoifmanRochberg type characterization of the parabolic BMO through parabolic
Muckenhoupt weights and maximal functions. We also discuss connections
and applications of the results to regularity of nonlinear parabolic partial
differential equations.
Riikka Korte:
How to define the Dirichlet problem when p = 1?
Existence, uniqueness, continuity and stability of solutions to the Dirichlet
problem for p-harmonic functions in metric measure space setting is now
reasonably well understood when 1 < p < ∞. The corresponding problem
for p = 1 is to find a BV function of least gradient in the given domain,
with prescribed trace on the boundary. In this case, several problems arise
already in Euclidean spaces.
We will discuss two different notions of the Dirichlet problem in the case
p = 1. The first one is based on minimizing the BV -energy in the closure of
the domain. In the second approach, the energy being minimized includes
the integral of the jump of the inner trace measured with respect to the
interior perimeter of the domain. As an application, we will consider the
stability of the solutions to p-Dirichlet problems when p & 1.
This is based on joint work with Panu Lahti, Xining Li and Nageswari
Shanmugalingam.
Pekka Koskela:
Lipα -extension domains revisited
The class of Lipα -extension domains was introduced by Fred Gehring and
Olli Martio in 1985 towards the study of uniform Hölder continuity of quasiconformal mappings. I will review this work and discuss another setting
where this class has turned up.
Mika Koskenoja:
Reflection principle and the Monge-Ampère equation
We study the reflection principle for the Monge-Ampère equation in real
and complex spaces. First we consider the reflection principle for viscosity
solutions of the homogeneous real Monge-Ampère equation. Before stating
our main result in this setting we consider the reflection of convex functions. Then in the complex space setting we give reflection principles for
pluriharmonic and plurisubharmonic functions. Finally, we study the reflection principle for solutions of the homogeneous generalized complex MongeAmpère equation. In particular, we state a reflection principle for maximal
plurisubharmonic functions.
Tuomo Kuusi:
Vectorial nonlinear potential theory
We discuss new pointwise potential estimates obtained for vectorial p-Laplacian
involving measure data. The estimates allow to give sharp descriptions of
fine properties of solutions which are the exact analog of the ones in classical linear potential theory. For instance, sharp characterizations of Lebesgue
points and optimal regularity criteria for solutions are provided exclusively
in terms of potentials. This is a joint work with G. Mingione.
John Lewis:
What comes around goes around
After a few comments on the US election and Olli, I will talk on several
problems arising from my very early work on the p Laplace equation which
coauthors and I have recently made progress on.
Jan Malý:
A version of the Stokes theorem
Consider a parametrized surface f with values in a manifold Ω. How to
recognize whether f induces a boundary (in the sense of currents) of an
integer valued BV function on Ω? We give a characterization of this situation
in terms of modulus of a path family. This is a joint work with Olli Martio
and Vendula Honzlová Exnerová.
Kai Rajala:
Quasiregular maps with small dilatation
We discuss Martio’s long-standing KI < 2 -conjecture as well as other open
problems on stability and invertibility of quasiregular maps in space.
Heli Tuominen:
Removable singularities for div v = f in weighted Lebesgue spaces
We will discuss about removable singularities for the divergence equation.
A compact set S ⊂ Rn is Lp -removable for the equation div v = 0, if for all
v ∈ Lp (Rn , Rn ), the property div v = f in Rn \ S implies that div v = f in
Rn (divergences in the weak sense).
We will first recall the characterizations of Lp -removable sets and then
give the corresponding results in weighted Lp -spaces.
Harri Varpanen:
Null recurrence and transience in random juggling
Random juggling models are random walks on countable digraphs having
various combinatorial structure. We prove a conjecture by Bouttier et al.
(2015), where a phase transition in the juggling process is obtained by a
parameter change in the throw distribution.
Matti Vuorinen:
Metrics and quasiconformal maps
This talk gives an overview of my recent research interests, connected with
the theory of quasiconformal (qc) and quasiregular (qr) mappings in the
Euclidean space Rn , n ≥ 2. These mappings generalize conformal maps and
analytic functions to the higher dimensional case. Their theory started fifty
years ago due to the work of F. W. Gehring, J. Väisälä, Yu. G. Reshetnyak,
O. Martio, S. Rickman and has been further developed by many authors.
When the important parameter K , the maximal dilatation of a mapping,
tends to unity, we get these classical maps, conformal maps and analytic
functions as the limiting case K = 1 . The talk will discuss the distortion
theory of these mappings, i.e. how qc and qr maps transform distances
between points. Some novel metrics are used in this research. The talk is
based on joint work with several coauthors, mostly with my former students.
In particular, the three latest coauthors are G. Wang, X. Zhang, and P.
Hariri.
Xiao Zhong:
Liouville theorems for steady flows in the plane
I will talk about Leray’s problem on the steady flows of incompressible fluids
past a body in the plane.