Existence and Spectral Theory for Weak Solutions of Neumann and Dirichlet Problems for Linear Degenerate Elliptic Operators with Rough Coefficients

In this paper we study existence and spectral properties for weak solutions of Neumann and Dirichlet problems associated to second order linear degenerate elliptic partial differential operators $X$, with rough coefficients of the form $$X=-\text{div}(P\nabla )+{\bf HR}+{\bf S^\prime G} +F$$ in a geometric homogeneous space setting where the $n\times n$ matrix function $P=P(x)$ is allowed to degenerate. We give a maximum principle for weak solutions of $Xu\leq 0$ and follow this with a result describing a relationship between compact projection of the degenerate Sobolev space $QH^{1,p}$ into $L^q$ and a Poincar\'e inequality with gain adapted to $Q$

Harnack's inequality and H\"older continuity for weak solutions of degenerate quasilinear equations with rough coefficients

We continue to study regularity results for weak solutions of the large class of second order degenerate quasilinear equations of the form \begin{eqnarray} \text{div}\big(A(x,u,\nabla u)\big) = B(x,u,\nabla u)\text{ for }x\in\Omega\nonumber \end{eqnarray} as considered in our previous paper giving local boundedness of weak solutions. Here we derive a version of Harnack's inequality as well as local H\"older continuity for weak solutions. The possible degeneracy of an equation in the class is expressed in terms of a nonnegative definite quadratic form associated with its principal part. No smoothness is required of either the quadratic form or the coefficients of the equation. Our results extend ones obtained by J. Serrin and N. Trudinger for quasilinear equations, as well as ones for subelliptic linear equations obtained by Sawyer and Wheeden in their 2006 AMS memoir article.Comment: 39 page

Erythrocytes in multiple sclerosis: forgotten contributors to the pathophysiology?

Multiple sclerosis (MS) is an autoimmune disease characterised by lymphocytic infiltration of the central nervous system and subsequent destruction of myelin and axons. On the background of a genetic predisposition to autoimmunity, environmental triggers are assumed to initiate the disease. The majority of MS research has focused on the pathological involvement of lymphocytes and other immune cells, yet a paucity of attention has been given to erythrocytes, which may play an important role in MS pathology. The following review briefly summarises how erythrocytes may contribute to MS pathology through impaired antioxidant capacity and altered haemorheological features. The effect of disease-modifying therapies on erythrocytes is also reviewed. It may be important to further investigate erythrocytes in MS, as this could broaden the understanding of the pathological mechanisms of the disease, as well as potentially lead to the discovery of novel and innovative targets for future therapies

Institutional memory: we need a more dynamic understanding of the way institutions remember

Institutional memory is central to the task of governing. But existing understandings of how institutional memory works are too limiting and rooted in an ontological falsehood, argue Jack Corbett, Dennis C. Grube, Heather Lovell, and Rodney Scott. They explain why a more dynamic approach is needed