Hardy Type Inequalities for Δλ\Delta_\lambda-Laplacians

We derive Hardy type inequalities for a large class of sub-elliptic operators that belong to the class of $\Delta_\lambda$-Laplacians and find explicit values for the constants involved. Our results generalize previous inequalities obtained for Grushin type operators $$\Delta_{x}+ |x|^{2\alpha}\Delta_{y},\qquad\ (x,y)\in\mathbb{R}^{N_1}\times\mathbb{R}^{N_2},\ \alpha\geq 0,$$ which were proved to be sharp

Wiener-Landis criterion for Kolmogorov-type operators

We establish a necessary and sufficient condition for a boundary point to be regular for the Dirichlet problem related to a class of Kolmogorov-type equations. Our criterion is inspired by two classical criteria for the heat equation: the Evans-Gariepy's Wiener test, and a criterion by Landis expressed in terms of a series of caloric potentials

Instantaneous blowup and singular potentials on Heisenberg groups

In this paper we generalize the instantaneous blowup result from [3] and [15] to the heat equation perturbed by singular potentials on the Heisenberg group

The role of fundamental solution in Potential and Regularity Theory for subelliptic PDE

In this survey we consider a general Hormander type operator, represented as a sum of squares of vector fields plus a drift and we outline the central role of the fundamental solution in developing Potential and Regularity Theory for solutions of related PDEs. After recalling the Gaussian behavior at infinity of the kernel, we show some mean value formulas on the level sets of the fundamental solution, which are the starting point to obtain a comprehensive parallel of the classical Potential Theory. Then we show that a precise knowledge of the fundamental solution leads to global regularity results, namely estimates at the boundary or on the whole space. Finally in the problem of regularity of non linear differential equations we need an ad hoc modification of the parametrix method, based on the properties of the fundamental solution of an approximating problem

Potential theory results for a class of PDOs admitting a global fundamental solution

We outline several results of Potential Theory for a class of linear par-tial differential operators L of the second order in divergence form. Under essentially the sole assumption of hypoellipticity, we present a non-invariant homogeneous Harnack inequality for L; under different geometrical assumptions on L (mainly, under global doubling/Poincar\ue9 assumptions), it is described how to obtainan invariant, non-homogeneous Harnack inequality. When L is equipped with a global fundamental solution \u393, further Potential Theory results are available (such as the Strong Maximum Principle). We present some assumptions on L ensuring that such a \u393 exists

Home care in Europe: a systematic literature review

<p>Abstract</p> <p>Background</p> <p>Health and social services provided at home are becoming increasingly important. Hence, there is a need for information on home care in Europe. The objective of this literature review was to respond to this need by systematically describing what has been reported on home care in Europe in the scientific literature over the past decade.</p> <p>Methods</p> <p>A systematic literature search was performed for papers on home care published in English, using the following data bases: Cinahl, the Cochrane Library, Embase, Medline, PsycINFO, Sociological Abstracts, Social Services Abstracts, and Social Care Online. Studies were only included if they complied with the definition of home care, were published between January 1998 and October 2009, and dealt with at least one of the 31 specified countries. Clinical interventions, instrument developments, local projects and reviews were excluded. The data extracted included: the characteristics of the study and aspects of home care 'policy & regulation', 'financing', 'organisation & service delivery', and 'clients & informal carers'.</p> <p>Results</p> <p>Seventy-four out of 5,133 potentially relevant studies met the inclusion criteria, providing information on 18 countries. Many focused on the characteristics of home care recipients and on the organisation of home care. Geographical inequalities, market forces, quality and integration of services were also among the issues frequently discussed.</p> <p>Conclusions</p> <p>Home care systems appeared to differ both between and within countries. The papers included, however, provided only a limited picture of home care. Many studies only focused on one aspect of the home care system and international comparative studies were rare. Furthermore, little information emerged on home care financing and on home care in general in Eastern Europe. This review clearly shows the need for more scientific publications on home care, especially studies comparing countries. A comprehensive and more complete insight into the state of home care in Europe requires the gathering of information using a uniform framework and methodology.</p