Volume 2006, Issue 596

In the present paper we continue our study of free boundary regularity problems for harmonic measure and Poisson kernels, below the continuous threshold. We focus our attention on 2-phase problems. We show that under the appropriate hypothesis for the free boundary data, if each phase is geometrically non-degenerate then the free boundary is regular. The main new point of this work is that no flatness condition on the free boundary is assumed a priori.

Consider the equations of Navier-Stokes in the exterior of a rotating domain. It is shown that, after rewriting the problem on a fixed domain Ω, the solution of the corresponding Stokes equation is governed by a C 0 -semigroup on L σ p (Ω), 1 < p < ∞, with generator . Moreover, for and initial data u 0 ∈ L σ p (Ω), we prove the existence of a unique local mild solution to the Navier-Stokes problem.

We prove the Kirillov-Reshetikhin conjecture for all untwisted quantum affine algebras: we prove that the characters of Kirillov-Reshetikhin modules solve the Q -system and we give an explicit formula for the character of their tensor products. In the proof we show that Kirillov-Reshetikhin modules are special in the sense of monomials and that their q -characters solve the T -system (functional relations appearing in the study of solvable lattice models). Moreover we prove that the T -system can be written in the form of an exact sequence. For simply-laced cases, these results were proved by Nakajima in [ H. Nakajima , Quiver Varieties and t -Analogs of q -Characters of Quantum Affine Algebras, Ann. Math. 160 (2004), 1057–1097.], [ H. Nakajima , t -analogs of q -characters of Kirillov-Reshetikhin modules of quantum affine algebras, Represent. Th. 7 (2003), 259–274.] with geometric arguments (main result of [ H. Nakajima , Quiver Varieties and t -Analogs of q -Characters of Quantum Affine Algebras, Ann. Math. 160 (2004), 1057–1097.]) which are not available in general. The proof we use is different and purely algebraic, and so can be extended uniformly to non simply-laced cases.

We show that a nodal hypersurface X in ℙ 3 of degree d with a sufficiently large number l of nodes, , is algebraically quasi-hyperbolic, i.e. X can only have finitely many rational and elliptic curves. Our results use the theory of symmetric differentials and algebraic foliations and give a very striking example of the jumping of the number of symmetric differentials in families.

In a previous article ( Orbites unipotentes et pôles d'ordre maximal de la fonction μ de Harish-Chandra , to appear in Canad. J. Math.), we have assumed the existence of the local Langlands correspondence for supercuspidal representations and deduced from this a local Langlands correspondence for discrete series representations and beyond (without going into the structure of the L -packets). The aim of the present article is to show that this extension of the local Langlands correspondence for supercuspidal representations (and some of the assumptions in the article above) is compatible with the theory of L -functions due to Langlands-Shahidi.

Let p be an odd prime number and let S be a finite set of prime numbers congruent to 1 modulo p . We prove that the group G S (ℚ)( p ) has cohomological dimension 2 if the linking diagram attached to S and p satisfies a certain technical condition, and we show that G S (ℚ)( p ) is a duality group in these cases. Furthermore, we investigate the decomposition behaviour of primes in the extension ℚ S ( p )/(ℚ) and we relate the cohomology of G S (ℚ)( p ) to the étale cohomology of the scheme Spec(ℤ) – S . Finally, we calculate the dualizing module.

This paper considers the one-variable p -adic L -function which interpolates the L -values of anticyclotomic Hecke characters of an imaginary quadratic field K in which p splits. Fixing such a character with root number +1, the Iwasawa μ-invariant of the associated branch of the p -adic L -function is computed as a sum of simple local contributions at the inert primes of K . The proof uses a result of Tonghai Yang to relate the p -adic L -function to a measure constructed out of the special values of theta functions with complex multiplication by K .

In this paper we introduce a new class of finitely presented pro- p -groups G of cohomological dimension 2 called mild groups. If d ( G ), r ( G ) are respectively the minimal number of generators and relations of G , we give an infinite family of mild groups G with r ( G ) ≧ d ( G ) and d ( G ) ≧ 2 arbitrary. These groups can be constructed with G /[ G , G ] finite, answering a question of Kuzmin. If G = G S ( p ) is the Galois group of the maximal p -extension of ℚ unramified outside a finite set of primes S and p ≠ 2, we show that G is mild for a co-final class of sets S , even in the case p ∉ S .

The solutions of the (nonlinear) Painlevé VI differential equation having icosahedral linear monodromy group will be classified up to equivalence under Okamoto's affine F 4 Weyl group action and many properties of the solutions will be given. There are 52 classes, the first ten of which correspond directly to the ten icosahedral entries on Schwarz's list of algebraic solutions of the hypergeometric equation. The next nine solutions are simple deformations of known P VI solutions (and have less than five branches) and five of the larger solutions are already known, due to work of Dubrovin and Mazzocco and Kitaev. Of the remaining 28 solutions we will find 20 explicitly using the method of [ P. P. Boalch , From Klein to Painlevé via Fourier, Laplace and Jimbo, Proc. London Math. Soc. 90 (2005), no. 3, 167–208.] (via Jimbo's asymptotic formula). Amongst those constructed there is one solution that is “generic” in that its parameters lie on none of the affine F 4 hyperplanes, one that is equivalent to the Dubrovin-Mazzocco elliptic solution and three elliptic solutions that are related to the Valentiner three-dimensional complex reflection group, the largest having 24 branches.

Given a generically surjective holomorphic vector bundle morphism f : E → Q , E and Q Hermitian bundles, we construct a current R f with values in Hom( Q , H ), where H is a certain derived bundle, and with support on the set Z where f is not surjective. The main property is that if ϕ is a holomorphic section of Q , and R f ϕ = 0, then locally f ψ = ϕ has a holomorphic solution ψ. In the generic case also the converse holds. This gives a generalization of the corresponding theorem for a complete intersection, due to Dickenstein-Sessa and Passare. We also present results for polynomial mappings, related to M. Noether's theorem and the effective Nullstellensatz. The construction of the current is based on a generalization of the Koszul complex. By means of this complex one can also obtain new global estimates of solutions to f ψ = ϕ, and as an example we give new results related to the H p -corona problem.