Solutions of fully nonlinear elliptic equations with patches of zero gradient: Existence, regularity and convexity of level curves

In this paper, we first construct ``viscosity' solutions (in the Crandall-Lions sense) of fully nonlinear elliptic equations of the form

In fact, viscosity solutions are surprisingly weak. Since candidates for solutions are just continuous, we only require that the ``test' polynomials (those tangent from above or below to the graph of at a point ) satisfy the correct inequality only if . That is, we simply disregard those test polynomials for which .

Nevertheless, this is enough, by an appropriate use of the Alexandroff-Bakelman technique, to prove existence, regularity and, in two dimensions, for , (0$">) and constant boundary conditions on a convex domain, to prove that there is only one convex patch.

     

Extensions for finite Chevalley groups II

Let be a semisimple simply connected algebraic group defined and split over the field with elements, let be the finite Chevalley group consisting of the -rational points of where , and let be the th Frobenius kernel. The purpose of this paper is to relate extensions between modules in and with extensions between modules in . Among the results obtained are the following: for 2$"> and , the -extensions between two simple -modules are isomorphic to the -extensions between two simple -restricted -modules with suitably ``twisted" highest weights. For , we provide a complete characterization of where and is -restricted. Furthermore, for , necessary and sufficient bounds on the size of the highest weight of a -module are given to insure that the restriction map is an isomorphism. Finally, it is shown that the extensions between two simple -restricted -modules coincide in all three categories provided the highest weights are ``close" together.

     

Block representation type of reduced enveloping algebras

Let be an algebraically closed field of characteristic , a connected, reductive -group, , and the reduced enveloping algebra of associated with . Assume that is simply-connected, is good for and has a non-degenerate -invariant bilinear form. All blocks of having finite and tame representation type are determined.

     

Automorphisms of finite order on Gorenstein del Pezzo surfaces

In this paper we shall determine all actions of groups of prime order with on Gorenstein del Pezzo (singular) surfaces of Picard number 1. We show that every order- element in ( , being the minimal resolution of ) is lifted from a projective transformation of . We also determine when is finite in terms of , and the number of singular members in . In particular, we show that either for some , or for every prime , there is at least one element of order in (hence is infinite).

     

Sets of uniqueness for spherically convergent multiple trigonometric series

A subset of the -dimensional torus is called a set of uniqueness, or -set, if every multiple trigonometric series spherically converging to outside vanishes identically. We show that all countable sets are -sets and also that sets are -sets for every . In particular, , where is the Cantor set, is an set and hence a -set. We will say that is a -set if every multiple trigonometric series spherically Abel summable to outside and having certain growth restrictions on its coefficients vanishes identically. The above-mentioned results hold also for sets. In addition, every -set has measure , and a countable union of closed -sets is a -set.

     

Topological dynamics on moduli spaces II

Let be an orientable genus 0$"> surface with boundary . Let be the mapping class group of fixing . The group acts on the space of -gauge equivalence classes of flat -connections on with fixed holonomy on . We study the topological dynamics of the -action and give conditions for the individual -orbits to be dense in .

     

An application of the Littlewood restriction formula to the Kostant-Rallis Theorem

Consider a symmetric pair of linear algebraic groups with , where and are defined as the +1 and -1 eigenspaces of the involution defining . We view the ring of polynomial functions on as a representation of . Moreover, set , where is the space of homogeneous polynomial functions on of degree . This decomposition provides a graded -module structure on . A decomposition of is provided for some classical families when is within a certain stable range.

The stable range is defined so that the spaces are within the hypothesis of the classical Littlewood restriction formula. The Littlewood restriction formula provides a branching rule from the general linear group to the standard embedding of the symplectic or orthogonal subgroup. Inside the stable range the decomposition of is interpreted as a -analog of the Kostant-Rallis theorem.

     

Character degrees and nilpotence class of finite groups

Let be a finite set of powers of containing 1. It is known that for some choices of , if is a finite -group whose set of character degrees is , then the nilpotence class of is bounded by some integer that depends on , while for some other choices of such an integer does not exist. The sets of the first type are called class bounding sets. The problem of determining the class bounding sets has been studied in several papers whose results made it tempting to conjecture that a set is class bounding if and only if . In this article we provide a new approach to this problem. Our main result shows the relevance of certain -adic space groups in this problem. With its help, we are able to prove some results that provide new class bounding sets. We also show that there exist non-class-bounding sets such that .

     

Sharp Sobolev-Poincaré inequalities on compact Riemannian manifolds

Given a smooth compact Riemannian -manifold, , we return in this article to the study of the sharp Sobolev-Poincaré type inequality

where is the critical Sobolev exponent, and is the sharp Euclidean Sobolev constant. Druet, Hebey and Vaugon proved that is true if , that is true if and the sectional curvature of is a nonpositive constant, or the Cartan-Hadamard conjecture in dimension is true and the sectional curvature of is nonpositive, but that is false if and the scalar curvature of is positive somewhere. When is true, we define as the smallest in . The saturated form of reads as

We assume in this article that , and complete the study by Druet, Hebey and Vaugon of the sharp Sobolev-Poincaré inequality . We prove that is true, and that possesses extremal functions when the scalar curvature of is negative. A fairly complete answer to the question of the validity of under the assumption that the scalar curvature is not necessarily negative, but only nonpositive, is also given.

     

Braid pictures for Artin groups

We define the braid groups of a two-dimensional orbifold and introduce conventions for drawing braid pictures. We use these to realize the Artin groups associated to the spherical Coxeter diagrams , and and the affine diagrams , , and as subgroups of the braid groups of various simple orbifolds. The cases , , and are new. In each case the Artin group is a normal subgroup with abelian quotient; in all cases except the quotient is finite. We also illustrate the value of our braid calculus by giving a picture-proof of the basic properties of the Garside element of an Artin group of type .

     

On Fano manifolds with Nef tangent bundles admitting 1-dimensional varieties of minimal rational tangents

Let be a Fano manifold of Picard number 1 with numerically effective tangent bundle. According to the principal case of a conjecture of Campana-Peternell's, should be biholomorphic to a rational homogeneous manifold , where is a simple Lie group, and is a maximal parabolic subgroup.

In our opinion there is no overriding evidence for the Campana-Peternell Conjecture for the case of Picard number 1 to be valid in its full generality. As part of a general programme that the author has undertaken with Jun-Muk Hwang to study uniruled projective manifolds via their varieties of minimal rational tangents, a new geometric approach is adopted in the current article in a special case, consisting of (a) recovering the generic variety of minimal rational tangents , and (b) recovering the structure of a rational homogeneous manifold from . The author proves that, when and the generic variety of minimal rational tangents is 1-dimensional, is biholomorphic to the projective plane , the 3-dimensional hyperquadric , or the 5-dimensional Fano homogeneous contact manifold of type , to be denoted by .

The principal difficulty is part (a) of the scheme. We prove that is a rational curve of degrees , and show that resp. 2 resp. 3 corresponds precisely to the cases of resp. resp. . Let be the normalization of a choice of a Chow component of minimal rational curves on . Nefness of the tangent bundle implies that is smooth. Furthermore, it implies that at any point , the normalization of the corresponding Chow space of minimal rational curves marked at is smooth. After proving that is a rational curve, our principal object of study is the universal family of , giving a double fibration , which gives -bundles. There is a rank-2 holomorphic vector bundle on whose projectivization is isomorphic to . We prove that is stable, and deduce the inequality from the inequality resulting from stability and the existence theorem on Hermitian-Einstein metrics. The case of is ruled out by studying the structure of the curvature tensor of the Hermitian-Einstein metric on in the special case where .

     

An inverse problem for scattering by a doubly periodic structure

Consider scattering of electromagnetic waves by a doubly periodic structure with for integers , . Above the structure, the medium is assumed to be homogeneous with a constant dielectric coefficient. The medium is a perfect conductor below the structure. An inverse problem arises and may be described as follows. For a given incident plane wave, the tangential electric field is measured away from the structure, say at for some large . To what extent can one determine the location of the periodic structure that separates the dielectric medium from the conductor? In this paper, results on uniqueness and stability are established for the inverse problem. A crucial step in our proof is to obtain a lower bound for the first eigenvalue of the following problem in a convex domain :

     

Rational -equivariant homotopy theory

We give an algebraicization of rational -equivariant homotopy theory. There is an algebraic category of `` -systems' which is equivalent to the homotopy category of rational -simply connected -spaces. There is also a theory of ``minimal models' for -systems, analogous to Sullivan's minimal algebras. Each -space has an associated minimal -system which encodes all of its rational homotopy information, including its rational equivariant cohomology and Postnikov decomposition.

     

Detection of renewal system factors via the Conley index

Let be an isolating neighborhood for a map . If we can decompose into the disjoint union of compact sets and , then we can relate the dynamics on the maximal invariant set to the shift on two symbols by noting which component of each iterate of a point lies in. We examine a method, based on work by Mischaikow, Szymczak, et al., for using the discrete Conley index to detect explicit subshifts of the shift associated to . In essence, we measure the difference between the Conley index of and the sum of the indices of and .

     

Maximal functions with polynomial densities in lacunary directions

Given a real polynomial in one variable such that , we consider the maximal operator in ,

0\,,\,i,j\in \mathbb{Z}}\frac{1... ...t f\big (x_{1}-2^{i}p(t),x_{2}-2^{j}p(t)\big )\big \vert\,dt . \end{displaymath}">

We prove that is bounded on for 1$"> with bounds that only depend on the degree of .

     

Nonradial solvability structure of super-diffusive nonlinear parabolic equations

We study the solvability of the Cauchy problem for the nonlinear parabolic equation

when in , with a given nonnegative function. It is known from earlier works of the authors that the asymptotic radial growth , for the spherical averages of is critical for local solvability, in particular ensuring it if is radially symmetric. We show that if the initial data behaves in polar coordinates like , for large with nonnegative and -periodic, then the following holds: If vanishes on some interval of length 0$">, then there is no local solution of the initial value problem. On the other hand, if such an interval does not exist, then the initial value problem is locally solvable and the time of existence can be estimated explicitly.

     

Thick points for intersections of planar sample paths

Let denote the number of visits to of the simple planar random walk , up to step . Let be another simple planar random walk independent of . We show that for any , there are points for which . This is the discrete counterpart of our main result, that for any , the Hausdorff dimension of the set of thick intersection points for which , is almost surely . Here is the projected intersection local time measure of the disc of radius centered at for two independent planar Brownian motions run until time . The proofs rely on a ``multi-scale refinement' of the second moment method. In addition, we also consider analogous problems where we replace one of the Brownian motions by a transient stable process, or replace the disc of radius centered at by for general sets .

     

Lines tangent to

We show that for there are complex common tangent lines to general spheres in and that there is a choice of spheres with all common tangents real.

     

-estimates for the linearized Monge-Ampère equation

Let be a strictly convex domain and let be a convex function such that    det in . The linearized Monge-Ampère equation is

where det is the matrix of cofactors of . We prove that there exist and depending only on , and such that

for all solutions to the equation .