University of Jyväskylä

Dissertation: 17.12 Applications of chaining, Poincaré and pointwise decay of measures (Karak)

Start date: Dec 17, 2014 12:00 AM

End date: Dec 17, 2014 03:00 PM

Location: Mattilanniemi, Agora Beeta

Nijjwal Karak. Kuvaaja: Shirsho MukherjeeM.Sc. Nijjwal Karak defends his doctoral dissertation in Mathematics. Opponent Assistant Professor Jonas Azzam (Universitat Autònoma de Barcelona) and custos Academy Professor Pekka Koskela (University of Jyväskylä). The event is held in English.

The thesis makes contribution to the field of non-smooth analysis. It produces interesting and new results using the tools of Analysis on metric spaces. We have used chaining of balls, Poincar´e inequality and pointwise decay of measures to investigate the fine behaviour of non-smooth functions on Euclidean space and on met-ric spaces. The major result of the thesis regards the size of the set of non-Lebesgue points of a priori non-smooth function defined on a metric space. We have also found some relation between capacity and Hausdorff measure of a set in a metric space, which is valuable as both of them are used in modern mathematics to measure the size of small sets. These results have had important applications to partial differ-ential equations and applied mathematics. Roughly speaking a priori non-smooth functions arise naturally in mathematical modelling and the associated partial differential equations. Another result is about removability of Orlicz-Sobolev functions in Euclidean space of all di-mensions using porous sets.

The dissertation is published in University of Jyväskylä, Department of Mathematics and Statistics, number 145, 76p., Jyväskylä 2014, ISSN: 1457-8905 , ISBN: 978-951-39-5966-1. It is available at the University Librarys Publications Unit, tel. +358 40 805 3825, myynti@library.jyu.fi.

  • Further information:

Nijjwal Karak, +358 46 656 4424, nijjwal.n.karak@jyu.fi

Communications intern Birgitta Kemppainen, +358 40 805 4483, tiedotus@jyu.fi

 

Abstract:

The area of the thesis is Geometric analysis on Euclidean space and on metric spaces. We use the tools from Potential analysis to solve the problems in Geometric analysis.

The thesis consists of three manuscript. In the first manuscript, we study removable sets for the Orlicz-Sobolev space W1,Ψ, for functions of the form Ψ(t) = tp logλ(e+t). We show that (p,λ)-porous sets lying in a hyperplane of Rn are removable and this result is sharp. In the second, we show that in a Q-doubling metric space, that supports a Q-Poincar´e inequality and satisfies a chain condition, sets of Q-capacity zero have generalized Hausdorff h-measure zero for h(t) = log1−Q−(1/t). In the third, we show that in a Q-doubling metric space, which satisfies a chain condition, if we have a Q-Poincar´e inequality for a of function (u,g) where g ∈ LQ, then u has Lebesgue points Hh-a.e. for h(t) = log1−Q−(1/t).