Partial Regularity for Stationary Solutions to Liouville-Type Equation in dimension 3

In dimension $n=3$, we prove that the singular set of any stationary solution to the Liouville equation $-\Delta u=e^u$, which belongs to $W^{1,2}$, has Hausdorff dimension at most 1.Comment: 20 page

Some Remarks on Pohozaev-Type Identities

The aim of this note is to discuss in more detail the Pohozaev-type identities that have been recently obtained by the author, Paul Laurain and Tristan Rivi\`ere in the framework of half-harmonic maps defined either on $R$ or on the sphere $S^1$ with values into a closed manifold $N^n\subset R^m$. Weak half-harmonic maps are critical points of the following nonlocal energy \int_{R}|(-\Delta)^{1/4}u|^2 dx~~\mbox{or}~~\int_{S^1}|(-\Delta)^{1/4}u|^2\ d\theta. If $u$ is a sufficiently smooth critical point of the above energy then it satisfies the following equation of stationarity \frac{du}{dx}\cdot (-\Delta)^{1/2} u=0~~\mbox{a.e in $R$}~~\mbox{or}~~\frac{\partial u}{\partial \theta}\cdot (-\Delta)^{1/2} u=0~~\mbox{a.e in $S^1$.} By using the invariance of the equation of stationarity in $S^1$ with respect to the trace of the M\"obius transformations of the $2$ dimensional disk we derive a countable family of relations involving the Fourier coefficients of weak half-harmonic maps $u\colon S^1\to N^n.$ In the same spirit we also provide as many Pohozaev-type identities in $2$-D for stationary harmonic maps as conformal vector fields in $R^2$ generated by holomorphic functions

The jeely nursery, letting the children lead: final report to the robertson trust

This is the final report written at the conclusion of a three year Robertson Trust funded project at the Jeely Nursery in Castlemilk, Glasgow, 2007 to 2010. The project purpose was to meet the particular needs of children vulnerable to highly adverse social and economic circumstances, including those living with parental addiction. The aim was to develop a collaborative strategy which would, by involving children, nursery staff and parents together, help to build enduring resources for the emotional resilience needed by children to overcome adversity and improve their chances of achieving educational success. The well validated premise underpinning the child-led pedagogy, Special Playtime, is that early negative attachment experiences can be transformed through direct positive experience with trained staff. The report examines the project using a three dimensional conceptual framework located in the literature on attachment, resilience and child-led pedagogy and focuses on the manner in which the several and differing relationships within the project interacted with and sustained each other

Letting the children lead: the jeely nursery - a first interim report to the robertson trust

This is the first interim report undertaken for the Robertson Trust (RT) of an ongoing project developed by the Jeely Nursery (JN) in Castlemilk Glasgow 2007 – 2010. The Robertson Trust funding has enabled the JN to initiate a radical multi faceted programme of development focussed on children living in highly adverse socioeconomic conditions who may in addition be subject to the negative effects of living in families coping with substance abuse. They represent one of the most vulnerable groups of children in contemporary society. The task ahead for the nursery staff is complex, challenging and long term, requiring a high degree of consistency and commitment to both professional and personal growth. There is no question that the challenge is understood and accepted at all levels; that commitment to the project is well established and that a high level of motivation is sustained in spite of a number of difficult circumstances occurring over the year, unrelated to the project

Remarks on Neumann boundary problems involving Jacobians

In this short note we explore the validity of Wente-type estimates for Neumann boundary problems involving Jacobians. We show in particular that such estimates do not in general hold under the same hypotheses on the data for Dirichlet boundary problems