Gábor Francsics

Generators of a Picard modular group in two complex dimensions.

The goal of the article is to prove that four explicitly given transformations, two Heisenberg translations, a rotation and an involution generate the Picard modular group with Gaussian integers acting on the two dimensional complex hyperbolic space. The result answers positively a question raised by A. Kleinschmidt and D. Persson.

Analytic singularities of the Bergman kernel for tubes

Let ⊂ R be a bounded, convex, and open set with real analytic boundary. Let T ⊂ C be the tube with base , and let B be the Bergman kernel of T . If is strongly convex, then B is analytic away from the boundary diagonal. In the weakly convex case this is no longer true. In this situation we relate the off-diagonal points where analyticity fails to the characteristic lines. These lines are contained in the boundary of T , and they are projections of the Treves curves. These curves are symplectic invariants that are determined by the CR (Cauchy-Riemann) structure of the boundary of T . Note that Treves curves exist only when has at least one weakly convex boundary point.

On the porous medium equations with lower order singular nonlinear terms

where rn_~l, 2>0, p>0 , b>0, c > 0 are constants. The function uo(x) is supposed to be bounded, continuous and nonnegative, satisfying uo(x) >0 on land uo(x) =-0 in R \ I for a bounded interval I = (al, a2)c R. Equation (1) is parabolic when u>0 and degenerates when u=0. In the case b = c = 0 (1) is called the equation of nonlinear heat conductivity or filtration equation. The terms cu p and b(uX)~ in (1) correspond to the absorption and to the transport of heat or matter, respectively. The Cauchy problem (1), (2) usually has no classical solution having continuous derivatives which appear in (1) even in the case b = c = 0 (see Zeldovich and Komponeets [7]). Oleinik, Kalastmikov and YuMin [6] proved that the Cauchy problem (1), (2) has a unique generalized solution in the case b = c = 0 .