Niko Marola

On the problem of unique continuation for the p-Laplace equation

Abstract We study if two different solutions of the p -Laplace equation ∇ ⋅ ( | ∇ u | p − 2 ∇ u ) = 0 , where 1 p ∞ , can coincide in an open subset of their common domain of definition. We obtain some partial results on this interesting problem.

Local Behavior of p-harmonic Green’s Functions in Metric Spaces

We describe the behavior of p-harmonic Green’s functions near a singularity in metric measure spaces equipped with a doubling measure and supporting a Poincaré inequality.

Aspects of local-to-global results

We establish local-to-global results for a function space which is larger than the well-known bounded mean oscillation space, and was also introduced by John and Nirenberg

Boundary measures, generalized Gauss-Green formulas, and mean value property in metric measure spaces

We study mean value properties of harmonic functions in metric measure spaces. The metric measure spaces we consider have a doubling measure and support a (1, 1)-Poincare inequality. The notion of harmonicity is based on the Dirichlet form defined in terms of a Cheeger differentiable structure. By studying fine properties of the Green function on balls, we characterize harmonic functions in terms of a mean value property. As a consequence, we obtain a detailed description of Poisson kernels. We shall also obtain a Gauss–Green type formula for sets of finite perimeter which posses a Minkowski content characterization of the perimeter. For the Gauss–Green formula we introduce a suitable notion of the interior normal trace of a regular ball.

On a frequency function approach to the unique continuation principle

Abstract In this survey we discuss the frequency function method so as to study the problem of unique continuation for elliptic partial differential equations. The methods used in the note were mainly introduced by Garofalo and Lin.

Mental disorders and the use of primary health care services among homeless shelter users in the Helsinki metropolitan area, Finland

BackgroundHomelessness is associated with increased morbidity, mortality and health care use. The aim of this study was to examine the role of mental disorders in relation to the use of 1) daytime primary health care services and 2) after hours primary health care emergency room (PHER) services among homeless shelter users in the Helsinki Metropolitan Area, Finland.MethodsThe study cohort consists of all 158 homeless persons using the four shelters operating in the study area during two selected nights. The health records were analyzed over a period of 3 years prior to the sample nights and data on morbidity and primary health care visits were gathered. We used negative binomial regression to estimate the association between mental disorders and daytime visits to primary health care and after hours visits to PHERs.ResultsDuring the 3 years the 158 homeless persons in the cohort made 1410 visits to a physician in primary health care. The cohort exhibited high rates of mental disorders, including substance use disorders (SUDs); i.e. 141 persons (89%) had a mental disorder. We found dual diagnosis, defined as SUD concurring with other mental disorder, to be strongly associated with daytime primary health care utilization (IRR 11.0, 95% CI 5.9–20.6) when compared with those without any mental disorder diagnosis. The association was somewhat weaker for those with only SUDs (IRR 4.9, 95% CI 2.5–9.9) or with only other mental disorders (IRR 5.0, 95% CI 2.4–10.8). When focusing upon the after hours visits to PHERs we observed that both dual diagnosis (IRR 14.1, 95% CI 6.3–31.2) and SUDs (11.5, 95% CI 5.7–23.3) were strongly associated with utilization of PHERs compared to those without any mental disorder. In spite of a high numbers of visits, we found undertreatment of chronic conditions such as hypertension and diabetes.ConclusionsDual diagnosis is particularly strongly associated with primary health care daytime visits among homeless persons staying in shelters, while after hours visits to primary health care level emergency rooms are strongly associated with both dual diagnosis and SUDs. Active treatment for SUDs could reduce the amount of emergency visits made by homeless shelter users.

Characterizations of Sets of Finite Perimeter Using Heat Kernels in Metric Spaces

The overarching goal of this paper is to link the notion of sets of finite perimeter (a concept associated with N1,1-spaces) and the theory of heat semigroups (a concept related to N1,2-spaces) in the setting of metric measure spaces whose measure is doubling and supports a 1-Poincaré inequality. We prove a characterization of sets of finite perimeter in terms of a short time behavior of the heat semigroup in such metric spaces. We also give a new characterization of BV functions in terms of a near-diagonal energy in this general setting.

A characterization of {{\mathrm{BMO}}} self-maps of a metric measure space

This paper studies functions of bounded mean oscillation (

Phragmén--Lindelöf theorem for infinity harmonic functions

{{\mathrm{BMO}}}