Lipschitz continuity of state functions in some optimal shaping

We prove local Lipschitz continuity of the solution to the state equation in two kinds of shape optimization problems with constraint on the volume: the minimal shaping for the Dirichlet energy, with no sign condition on the state function, and the minimal shaping for the first eigenvalue of the Laplacian. This is a main first step for proving regularity of the optimal shapes themselves.Received: 11 November 2003, Accepted: 10 May 2004, Published online: 8 February 2005Mathematics Subject Classification (2000): 49Q10, 35R35, 49K20, 35J20     

Résistivité électrique due aux macles dans les métaux monovalents

Crystals off. c. c. sodium twinned in the direction of (111) planes at low temperatures, are studied in the approximation of nearly free electrons. The potential of the lattice is treated as a perturbation. The scattering of electrons by the twinning plane occurs in the direction of two particular wave vectors characterized by their coefficients of reflexion. The electrical resistivity arising from this scattering is calculated with the variational method. The use of this method is justified for certain experimental conditions where the resistivity due to the twinning is a small part of the total one. The numerical calculation shows that the resistivity, given in tensor form, is consistent with the experimental results on the resistivity of stacking faults of other monovalent metals.     

Discrete degenerate representations of non-compact unitary groups

Two degenerate principal series of irreducible unitary representations of an arbitrary non-compact unitary groupU(p,1) are derived. These series are determined by the eigenvalues of the first and second-order invariant operators, which are shown to possess a discrete spectrum. The explicit form of the corresponding harmonic functions is derived and the properties of the discrete representations are discussed in detail. Moreover, in the Appendix, we derive the properties of the corresponding degenerate representations of an arbitrary compactU(p) group.On leave of absence from Institute of Nuclear Research, Warsaw, Poland.On leave of absence from Institute of Physics of the Czechoslovak Academy of Sciences, Prague, Czechoslovakia.     

Conductivite electrique des magnetites finement divisees substituees a l'aluminium et au chrome

The magnetites substituted for aluminium and chromium show a high conductivity decrease as well as a change in the conduction type under vacuum when passing from the inverse to the normal spinel during substitution. This type of conduction is confirmed by the study of oxygen chemisorption at temperatures close to room temperature. At higher temperatures, volume oxidation is accompanied by a sharp change in conductivity, which differs depending on whether the spinel is inverse or normal.     

Random time change and recurrent boundary

The purpose of this paper is to study some properties of random time changes in recurrent potential theory. In particular we show that the Martin recurrent boundary is not invariant under a random time change. We then obtain a characterization of random time change destroying a boundary point. We also give some complement about the recurrent boundary connected with “special additive functionals”. We have for example a representation at the boundary of solutions of the Poisson's equation $\text{?(I-U}{}_1\text{)=-U}{}_1\text{(x,·)}$ by using local time at x.