A Strong Maximum Principle for Weak Solutions of Quasi-Linear Elliptic Equations with Applications to Lorentzian and Riemannian Geometry

The strong maximum principle is proved to hold for weak (in the sense of support functions) sub- and super-solutions to a class of quasi-linear elliptic equations that includes the mean curvature equation for $C^0$ spacelike hypersurfaces in a Lorentzian manifold. As one application a Lorentzian warped product splitting theorem is given.Comment: 37 pages, 1 figure, ams-latex using eepi

Implications of liberalisation for methods of setting retail gas prices in Belgium

The liberalisation of the gas and electricity markets entailed the establishment of new pricing mechanisms, including those concerning consumers in the residential sector. The purpose of the article is to analyse how Belgian gas suppliers have adopted those mechanisms. After a brief description of the wholesale and retail gas markets in Belgium, the second section focuses on examining retail prices for residential consumers. The analysis is based on the tariff data of the five main suppliers active in that segment. It shows that the method of setting the retail gas prices used in variable price contracts is based on very similar principles, using specific indexation formulas freely determined by each operator. The formulas generate selling price indexation based on parameters which pass on changes in purchase costs to the final consumer according to a price risk transfer principle. That indexation is convenient for all operators and does not entail any additional consumer information costs. The disclosure of the automatic indexation mechanisms has the advantage of being relatively simple and transparent in regard to fundamental movements in parameters and their influence on prices. However, for the average consumer, the calculation of indexed prices appears complex and the information supplied seems incomplete. Discretionary adjustments to the said indexations are not very clear even though they are reflected in an increase in the portion of the price which is not linked to changes in the energy parameters. The situation of gas suppliers active on the retail market in Belgium is then assessed in comparison with that in neighbouring European countries, which apply officially regulated prices, price approval procedures, price caps and prices free of any regulation. However, those prices are still at least subject to “close” supervision owing to the authorities’ concern that the retail prices charged should reflect the true cost of a product which is subject to a public service obligation, which is an item of essential household expenditure, and for which the price must be determined by the market. Since consumer prices of gas are more volatile in Belgium than in other countries, with the ensuing second-round effects on inflation, measures capable of reducing that volatility could be attractive. However, the effects of any such measures require careful appraisal. Two measures might be envisaged, in line with developments seen in other countries : the introduction of “supervised” freedom to set tariffs, with effective checks on the justification for tariff adjustments, and information on tariff formulas which is easier for consumers to understand.consumer price index, Belgium, gas price, retail gas market