We study a new formulation for the eikonal equation |∇u| = 1 on a bounded subset of R. Instead of a vector field ∇u, we consider a field P of orthogonal projections on 1-dimensional subspaces, with div P ∈ L. We prove that solutions of this equation propagate direction as in the classical eikonal equation. We also show that solutions exist if and only if the domain is a tubular neighbourhood of...

OBJECTIVE To investigate potential mechanisms mediating the neuroprotective effect of thymoquinone (TQ) on dopaminergic neurons. METHODS This study was conducted in the Chemistry and Biochemistry Institute, University of Veterinary Medicine, Vienna, Austria between June and August 2013. Primary cultures were prepared from embryonic mouse mesencephala (OFI/SPF) at gestation day 14. Four sets o...

In this paper, we present several results concerning vector potentials and scalar potentials with data in Sobolev spaces with negative exponents, in a not necessarily simply-connected, three-dimensional domain. We then apply these results to Poincaré’s theorem and to Korn’s inequality. 1 Weak versions of a classical theorem of Poincaré In this work, (the results of which were announced in [2]),...

A Bernstein-Bézier basis is developed for H(div)-conforming finite elements that gives a clear separation between the curls of the Bernstein basis for the polynomial discretisation of the space H1, and the non-curls that characterize the specific H(div) finite element space (Raviart-Thomas in our case). The resulting basis has two distinct components reflecting this separation with the basis fu...

Global error estimates in L2(Q), L°°(Q), and H~S(Q), Q in R2 or R3, are derived for a mixed finite element method for the Dirichlet problem for the elliptic operator Lp = -div(a grad p + bp) + cp based on the Raviart-Thomas-Nedelec space V^ X Wh c H(div; Í2) X L2(ü). Optimal order estimates are obtained for the approximation of p and the associated velocity field u = -(a grad p + bp) in L2(fl) ...

We give an existence result of a renormalized solution for a class of nonlinear parabolic equations ∂b(x, u) ∂t − div (a(x, u,∇u)+ (u)) = f , where the righthand side belongs to L((0, T )× ) and where b(x, u) is an unbounded function of u and where −div(a(t, x, u,∇u) + (u)) is a Leray–Lions type operator with growth |∇u|p−1 in ∇u, but without any growth assumption on u. AMS subject classificati...

We show that given a nonvanishing particular solution of the equation (div p grad+q)u = 0, (1) the corresponding differential operator can be factorized into a product of two first order operators. The factorization allows us to reduce the equation (1) to a first order equation which in a two-dimensional case is the Vekua equation of a special form. Under quite general conditions on the coeffic...

PURPOSE The aim of this study was to investigate the potential effects of mild hypoxia in the mature and immature brain. METHODS We prepared organotypic slice cultures of the hippocampus and used hippocampal tissue cultures at 7 and 14 days in vitro (DIV) to represent the immature and mature brain, respectively. Tissue cultures were exposed to 10% oxygen for 60 minutes. Twenty-four hours afte...

Certain anesthetics exhibit neurotoxicity in the brains of immature but not mature animals. Gamma-aminobutyric acid (GABA), the primary inhibitory neurotransmitter in the adult brain, is excitatory on immature neurons via its action at the GABAA receptor, due to a reversed transmembrane chloride gradient. GABAA receptor activation in immature neurons is sufficient to open L-type voltage-gated c...