We study if two different solutions of the p-Laplace equation ∇ · (|∇u| p−2 ∇u) = 0, where 1 < p < ∞, can coincide in an open subset of their common domain of definition. We obtain some partial results on this interesting problem.

The Perron-Wiener-Brelot (PWB) method is applied to an important nonlinear situation. Unbounded subsolutions, their approximation and a counterpart of the harmonic measure are considered.Introduction. The Perron-Wiener-Brelot (PWB-) method as introduced by O. Perron [P] and refined by several mathematicians is wellknown in Potential Theory and it is mainly used in the theory of harmonic functions although it has a wider scope of applications [CC]. The PWBmethod was generalized by E. Beckenbach and L. Jackson [BJ, J] to the non-linear situation. Their approach used the strong maximum principle for the difference of two solutions [BJ, Postulate 2]. The purpose of this note is to show that the PWB-method can be employed without this assumption in certain important non-linear cases. We are also able to deal with unbounded subsolutions.We consider weak solutions, called F-extremals, of an Euler equation

For wireless mobile Internet users the length of the battery life is one of the most important performance factors. The energy efficiency of the data transmission over radio is a key component affecting the battery lifetime. This paper investigates WLAN energy consumption in network communication on a Mobile handset. We introduce an energy model that allows analysis and simulation of the energy efficiency of the Internet protocols on a Wireless Network Interface, and have extended the NS-2 simulation platform to allow investigating the energy consumption of the Radio Modem and the Power Amplifier in WLAN 802.11g network interface of a mobile device. We have also validated our model against measurements on real wireless hardware, and show that the simulation results closely match the real world behavior. We claim to present more detailed and accurate model of the WLAN energy consumption than what is done by the past work that allows designing and optimizing future Internet protocols towards more energy efficient behavior.

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Abstract.Let /: G -» R" be quasiregular and / = / F(x,Vu) dm a conformally invariant variational integral. Holder-continuity, Harnack's inequality and principle are proved for the extremals of /. Obstacle problems and their connection to subextremals are studied. If « is an extremal or a subextremal of /, then u ° / is again an extremal or a subextremal if an appropriate change in F is made.