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Tomasz Adamowicz

Publications and source records attributed to Tomasz Adamowicz.

28 records · Page 2Linked to original sources
arXiv2017

Mean value property and harmonicity on Carnot-Carathéodory groups

We study strongly harmonic functions in Carnot-Carathéodory groups defined via the mean value property with respect to the Lebesgue measure. For such functions we show their Sobolev regularity and smoothness. Moreover, we prove that strongly harmonic functions satisfy the sub-Laplace equation for the appropriate gauge norm and that the inclusion is sharp. We observe that spherical harmonic polynomials in $\mathbb{H}_1$ are both strongly harmonic and satisfy the sub-Laplace equation. Our presentation is illustrated by examples.

math.AP↗
arXiv2016

Prime ends in metric spaces and boundary extensions of mappings

By using the inner diameter distance condition we define and investigate new, in such a generality, class $\mathcal{F}$ of homeomorphisms between domains in metric spaces and show that, under additional assumptions on domains, $\mathcal{F}$ contains (quasi)conformal, bi-Lipschitz and quasisymmetric mappings as illustrated by examples. Moreover, we employ a prime ends theory in metric spaces and provide conditions allowing continuous and homeomorphic extensions of mappings in $\mathcal{F}$ to topological closures of domains, as well as homeomorphic extensions to the prime end boundary. Domains satisfying the bounded turning condition, locally and finitely connected at the boundary and the structure of prime end boundaries for such domains play a crucial role in our investigations. We apply our results to show the Koebe theorem on arcwise limits for mappings in $\mathcal{F}$. Furthermore, relations between the Royden boundary and the prime end boundary are presented. Our work generalizes results due to Carathéodory, Näkki, Väisälä and Zorich.

math.MG↗
arXiv2016

Harmonic functions on metric measure spaces

We introduce and study strongly and weakly harmonic functions on metric measure spaces defined via the mean value property holding for all and, respectively, for some radii of balls at every point of the underlying domain. Among properties of such functions we investigate various types of Harnack estimates on balls and compact sets, weak and strong maximum principles, comparison principles, the Hölder and the Lipshitz estimates and some differentiability properties. The latter one is based on the notion of a weak upper gradient. The Dirichlet problem for functions satisfying the mean value property is studied via the dynamical programming method related to stochastic games. We employ the Perron method to construct a harmonic function with continuous boundary data. Finally, we discuss and prove the Liouville type theorems. Our results are obtained for various types of measures: continuous with respect to a metric, doubling, uniform, measures satisfying the annular decay condition. Relations between such measures are presented as well. The presentation is illustrated by examples.

math.MG↗
arXiv2015

Prime ends in the Heisenberg group $\mathbb{H}_{1}$ and the boundary behavior of quasiconformal mappings

We investigate prime ends in the Heisenberg group $\mathbb{H}_{1}$ extending Näkki's construction for collared domains in Euclidean spaces. The corresponding class of domains is defined via uniform domains and the Loewner property. Using prime ends we show the counterpart of Caratheodory's extension theorem for quasiconformal mappings, the Koebe theorem on arcwise limits, the Lindelöf theorem for principal points and the Tsuji theorem.

math.MG↗
arXiv2015

Hölder continuity of quasiminimizers with nonstandard growth

We show the Hölder continuity of quasiminimizers of the energy functionals $\int f(x,u,\nabla u)\,dx$ with nonstandard growth under the general structure conditions $$ |z|^{p(x)} - b(x)|y|^{r(x)}-g(x) \leq f(x,y,z) \leq μ|z|^{p(x)} + b(x)|y|^{r(x)} + g(x). $$ The result is illustrated by showing that weak solutions to a class of $(A,B)$-harmonic equations $$ -{\rm div} A(x,u,\nabla u) = B(x,u,\nabla u), $$ are quasiminimizers of the variational integral of the above type and, thus, are Hölder continuous. Our results extend works by Chiadò Piat-Coscia, Fan-Zhao and Giusti-Giaquinta.

math.AP↗
arXiv2014

The boundary Harnack inequality for variable exponent $p$-Laplacian, Carleson estimates, barrier functions and $p(\cdot)$-harmonic measures

We investigate various boundary decay estimates for $p(\cdot)$-harmonic functions. For domains in $\mathbb{R}^n, n\geq 2$ satisfying the ball condition ($C^{1,1}$-domains) we show the boundary Harnack inequality for $p(\cdot)$-harmonic functions under the assumption that the variable exponent $p$ is a bounded Lipschitz function. The proof involves barrier functions and chaining arguments. Moreover, we prove a Carleson type estimate for $p(\cdot)$-harmonic functions in NTA domains in $\mathbb{R}^n$ and provide lower- and upper- growth estimates and a doubling property for a $p(\cdot)$-harmonic measure.

math.AP↗
arXiv2013

The geometry of planar $p$-harmonic mappings: convexity, level curves and the isoperimetric inequality

We discuss various representations of planar $p$-harmonic systems of equations and their solutions. For coordinate functions of $p$-harmonic maps we analyze signs of their Hessians, the Gauss curvature of $p$-harmonic surfaces, the length of level curves as well as we discuss curves of steepest descent. The isoperimetric inequality for the level curves of coordinate functions of planar $p$-harmonic maps is proven. Our main techniques involve relations between quasiregular maps and planar PDEs. We generalize some results due to P. Lindqvist, G. Alessandrini, G. Talenti and P. Laurence.

math.AP↗
arXiv2013

Regularity of $p(\cdot)$-superharmonic functions, the Kellogg property and semiregular boundary points

We study various boundary and inner regularity questions for $p(\cdot)$-(super)harmonic functions in Euclidean domains. In particular, we prove the Kellogg property and introduce a classification of boundary points for $p(\cdot)$-harmonic functions into three disjoint classes: regular, semiregular and strongly irregular points. Regular and especially semiregular points are characterized in many ways. The discussion is illustrated by examples. Along the way, we present a removability result for bounded $p(\cdot)$-harmonic functions and give some new characterizations of $W^{1, p(\cdot)}_0$ spaces. We also show that $p(\cdot)$-superharmonic functions are lower semicontinuously regularized, and characterize them in terms of lower semicontinuously regularized supersolutions.

math.AP↗
arXiv2012

Prime ends for domains in metric spaces

In this paper we propose a new definition of prime ends for domains in metric spaces under rather general assumptions. We compare our prime ends to those of Carathéodory and Näkki. Modulus ends and prime ends, defined by means of the \p-modulus of curve families, are also discussed and related to the prime ends. We provide characterizations of singleton prime ends and relate them to the notion of accessibility of boundary points, and introduce a topology on the prime end boundary. We also study relations between the prime end boundary and the Mazurkiewicz boundary. Generalizing the notion of John domains, we introduce almost John domains, and we investigate prime ends in the settings of John domains, almost John domains and domains which are finitely connected at the boundary.

math.MG↗