Morse index and uniqueness for positive solutions of radial $p$-Laplace equations
Amandine
Aftalion;
Filomena
Pacella
4255-4272
Abstract: We study the positive radial solutions of the Dirichlet problem $\Delta_p u+f(u)=0$ in $B$, $u>0$ in $B$, $u=0$ on $\partial B$, where $\Delta_p u=\operatorname{div}(\vert\nabla u\vert^{p-2}\nabla u)$, $p>1$, is the $p$-Laplace operator, $B$ is the unit ball in $\mathbb{R} ^n$ centered at the origin and $f$ is a $C^1$ function. We are able to get results on the spectrum of the linearized operator in a suitable weighted space of radial functions and derive from this information on the Morse index. In particular, we show that positive radial solutions of Mountain Pass type have Morse index 1 in the subspace of radial functions of $W_0^{1,p}(B)$. We use this to prove uniqueness and nondegeneracy of positive radial solutions when $f$ is of the type $u^s+u^q$ and $p\geq 2$.
Localization for a porous medium type equation in high dimensions
Changfeng
Gui;
Xiaosong
Kang
4273-4285
Abstract: We prove the strict localization for a porous medium type equation with a source term, $u_{t}= \nabla(u^ {\sigma} \nabla u)+u^ \beta$, $x \in \mathbf{R}^ N$, $N>1$, $\beta >\sigma +1$, $\sigma>0,$ in the case of arbitrary compactly supported initial functions $u_0$. We also otain an estimate of the size of the localization in terms of the support of the initial data $\operatorname{supp}u_0$ and the blow-up time $T$. Our results extend the well-known one dimensional result of Galaktionov and solve an open question regarding high dimensions.
A class of \boldmath{$C^*$}-algebras generalizing both graph algebras and homeomorphism \boldmath{$C^*$}-algebras I, fundamental results
Takeshi
Katsura
4287-4322
Abstract: We introduce a new class of $C^*$-algebras, which is a generalization of both graph algebras and homeomorphism $C^*$-algebras. This class is very large and also very tractable. We prove the so-called gauge-invariant uniqueness theorem and the Cuntz-Krieger uniqueness theorem, and compute the $K$-groups of our algebras.
Multi-point Taylor expansions of analytic functions
José
L.
López;
Nico
M.
Temme
4323-4342
Abstract: Taylor expansions of analytic functions are considered with respect to several points, allowing confluence of any of them. Cauchy-type formulas are given for coefficients and remainders in the expansions, and the regions of convergence are indicated. It is explained how these expansions can be used in deriving uniform asymptotic expansions of integrals. The method is also used for obtaining Laurent expansions in several points as well as Taylor-Laurent expansions.
Nonexistence of abelian difference sets: Lander's conjecture for prime power orders
Ka
Hin
Leung;
Siu Lun
Ma;
Bernhard
Schmidt
4343-4358
Abstract: In 1963 Ryser conjectured that there are no circulant Hadamard matrices of order $>4$ and no cyclic difference sets whose order is not coprime to the group order. These conjectures are special cases of Lander's conjecture which asserts that there is no abelian group with a cyclic Sylow $p$-subgroup containing a difference set of order divisible by $p$. We verify Lander's conjecture for all difference sets whose order is a power of a prime greater than 3.
On the $L_{p}$-Minkowski problem
Erwin
Lutwak;
Deane
Yang;
Gaoyong
Zhang
4359-4370
Abstract: A volume-normalized formulation of the $L_{p}$-Minkowski problem is presented. This formulation has the advantage that a solution is possible for all $p\ge 1$, including the degenerate case where the index $p$ is equal to the dimension of the ambient space. A new approach to the $L_{p}$-Minkowski problem is presented, which solves the volume-normalized formulation for even data and all $p\ge 1$.
3-manifolds that admit knotted solenoids as attractors
Boju
Jiang;
Yi
Ni;
Shicheng
Wang
4371-4382
Abstract: Motivated by the study in Morse theory and Smale's work in dynamics, the following questions are studied and answered: (1) When does a 3-manifold admit an automorphism having a knotted Smale solenoid as an attractor? (2) When does a 3-manifold admit an automorphism whose non-wandering set consists of Smale solenoids? The result presents some intrinsic symmetries for a class of 3-manifolds.
On the Harnack inequality for a class of hypoelliptic evolution equations
Andrea
Pascucci;
Sergio
Polidoro
4383-4394
Abstract: We give a direct proof of the Harnack inequality for a class of degenerate evolution operators which contains the linearized prototypes of the Kolmogorov and Fokker-Planck operators. We also improve the known results in that we find explicitly the optimal constant of the inequality.
Isolating blocks near the collinear relative equilibria of the three-body problem
Richard
Moeckel
4395-4425
Abstract: The collinear relative equilibrium solutions are among the few explicitly known periodic solutions of the Newtonian three-body problem. When the energy and angular momentum constants are varied slightly, these unstable periodic orbits become normally hyperbolic invariant spheres whose stable and unstable manifolds form separatrices in the integral manifolds. The goal of this paper is to construct simple isolating blocks for these invariant spheres analogous to those introduced by Conley in the restricted three-body problem. This allows continuation of the invariant set and the separatrices to energies and angular momenta far from those of the relative equilibrium.
The loss of tightness of time distributions for homeomorphisms of the circle
Zaqueu
Coelho
4427-4445
Abstract: For a minimal circle homeomorphism $f$ we study convergence in law of rescaled hitting time point process of an interval of length $\varepsilon>0$. Although the point process in the natural time scale never converges in law, we study all possible limits under a subsequence. The new feature is the fact that, for rotation numbers of unbounded type, there is a sequence $\varepsilon_{n}$ going to zero exhibiting coexistence of two non-trivial asymptotic limit point processes depending on the choice of time scales used when rescaling the point process. The phenomenon of loss of tightness of the first hitting time distribution is an indication of this coexistence behaviour. Moreover, tightness occurs if and only if the rotation number is of bounded type. Therefore tightness of time distributions is an intrinsic property of badly approximable irrational rotation numbers.
Oppenheim conjecture for pairs consisting of a linear form and a quadratic form
Alexander
Gorodnik
4447-4463
Abstract: Let $Q$ be a nondegenerate quadratic form and $L$ a nonzero linear form of dimension $d>3$. As a generalization of the Oppenheim conjecture, we prove that the set $\{(Q(x),L(x)):x\in\mathbb{Z} ^d\}$ is dense in $\mathbb{R} ^2$ provided that $Q$ and $L$ satisfy some natural conditions. The proof uses dynamics on homogeneous spaces of Lie groups.
Spécialisation de la $R$-équivalence pour les groupes réductifs
Philippe
Gille
4465-4474
Abstract: Soit $G/k$ un groupe réductif défini sur un corps $k$ de caractéristique distincte de $2$. On montre que le groupes des classes de $R$-équivalence de $G(k)$ne change pas lorsque l'on passe de $k$ au corps des séries de Laurent $k((t))$, c'est-à-dire que l'on a un isomorphisme naturel $G(k)/R \buildrel\sim\over\longrightarrow G\bigl( k((t)) \bigr)/R$. ABSTRACT. Let $G/k$ be a reductive group defined over a field of characteristic $\not =2$. We show that the group of $R$-equivalence for $G(k)$ is invariant by the change of fields $k((t))/k$ given by the Laurent series. In other words, there is a natural isomorphism $G(k)/R \buildrel\sim\over\longrightarrow G\bigl( k((t)) \bigr)/R$.
Radon's inversion formulas
W.
R.
Madych
4475-4491
Abstract: Radon showed the pointwise validity of his celebrated inversion formulas for the Radon transform of a function $f$ of two real variables (formulas (III) and (III$'$) in J. Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Ber. Verh. Sächs. Akad. Wiss. Leipzig, Math.-Nat. kl. 69 (1917), 262-277) under the assumption that $f$ is continuous and satisfies two other technical conditions. In this work, using the methods of modern analysis, we show that these technical conditions can be relaxed. For example, the assumption that $f$ be in $L^p(\mathbb{R} ^2)$for some $p$ satisfying $4/3<p<2$ suffices to guarantee the almost everywhere existence of its Radon transform and the almost everywhere validity of Radon's inversion formulas.
Variation inequalities for the Fejér and Poisson kernels
Roger
L.
Jones;
Gang
Wang
4493-4518
Abstract: In this paper we show that the $\varrho$-th order variation operator, for both the Fejér and Poisson kernels, are bounded from $L^p$ to $L^p$, $1<p<\infty$, when $\varrho >2$. Counterexamples are given if $\varrho =2$.
How to make a triangulation of $S^3$ polytopal
Simon
A.
King
4519-4542
Abstract: We introduce a numerical isomorphism invariant $p(\mathcal{T})$ for any triangulation $\mathcal{T}$ of $S^3$. Although its definition is purely topological (inspired by the bridge number of knots), $p(\mathcal{T})$ reflects the geometric properties of $\mathcal{T}$. Specifically, if $\mathcal{T}$ is polytopal or shellable, then $p(\mathcal{T})$is ``small'' in the sense that we obtain a linear upper bound for $p(\mathcal{T})$ in the number $n=n(\mathcal{T})$ of tetrahedra of $\mathcal{T}$. Conversely, if $p(\mathcal{T})$ is ``small'', then $\mathcal{T}$is ``almost'' polytopal, since we show how to transform $\mathcal{T}$ into a polytopal triangulation by $O((p(\mathcal{T}))^2)$ local subdivisions. The minimal number of local subdivisions needed to transform $\mathcal{T}$ into a polytopal triangulation is at least $\frac{p(\mathcal{T})}{3n}-n-2$. Using our previous results [The size of triangulations supporting a given link, Geometry & Topology 5 (2001), 369-398], we obtain a general upper bound for $p(\mathcal{T})$ exponential in $n^2$. We prove here by explicit constructions that there is no general subexponential upper bound for $p(\mathcal{T})$ in $n$. Thus, we obtain triangulations that are ``very far'' from being polytopal. Our results yield a recognition algorithm for $S^3$ that is conceptually simpler, although somewhat slower, than the famous Rubinstein-Thompson algorithm.
Subgroups of $\operatorname{Diff}^{\infty}_+ (\mathbb S^1)$ acting transitively on $4$-tuples
Julio
C.
Rebelo
4543-4557
Abstract: We consider subgroups of $C^{\infty}$-diffeomorphisms of the circle $\mathbb S^1$which act transitively on $4$-tuples of points. We show, in particular, that these subgroups are dense in the group of homeomorphisms of $\mathbb S^1$. A stronger result concerning $C^{\infty}$-approximations is obtained as well. The techniques employed in this paper rely on Lie algebra ideas and they also provide partial generalizations to the differentiable case of some results previously established in the analytic category.
Value groups, residue fields, and bad places of rational function fields
Franz-Viktor
Kuhlmann
4559-4600
Abstract: We classify all possible extensions of a valuation from a ground field $K$ to a rational function field in one or several variables over $K$. We determine which value groups and residue fields can appear, and we show how to construct extensions having these value groups and residue fields. In particular, we give several constructions of extensions whose corresponding value group and residue field extensions are not finitely generated. In the case of a rational function field $K(x)$ in one variable, we consider the relative algebraic closure of $K$ in the henselization of $K(x)$ with respect to the given extension, and we show that this can be any countably generated separable-algebraic extension of $K$. In the ``tame case'', we show how to determine this relative algebraic closure. Finally, we apply our methods to power series fields and the $p$-adics.
Existence and characterization of regions minimizing perimeter under a volume constraint inside Euclidean cones
Manuel
Ritoré;
César
Rosales
4601-4622
Abstract: We study the problem of existence of regions separating a given amount of volume with the least possible perimeter inside a Euclidean cone. Our main result shows that nonexistence for a given volume implies that the isoperimetric profile of the cone coincides with the one of the half-space. This allows us to give some criteria ensuring existence of isoperimetric regions: for instance, local convexity of the cone at some boundary point. We also characterize which are the stable regions in a convex cone, i.e., second order minima of perimeter under a volume constraint. From this it follows that the isoperimetric regions in a convex cone are the euclidean balls centered at the vertex intersected with the cone.
Real loci of symplectic reductions
R.
F.
Goldin;
T.
S.
Holm
4623-4642
Abstract: Let $M$ be a compact, connected symplectic manifold with a Hamiltonian action of a compact $n$-dimensional torus $T$. Suppose that $M$ is equipped with an anti-symplectic involution $\sigma$ compatible with the $T$-action. The real locus of $M$ is the fixed point set $M^\sigma$ of $\sigma$. Duistermaat introduced real loci, and extended several theorems of symplectic geometry to real loci. In this paper, we extend another classical result of symplectic geometry to real loci: the Kirwan surjectivity theorem. In addition, we compute the kernel of the real Kirwan map. These results are direct consequences of techniques introduced by Tolman and Weitsman. In some examples, these results allow us to show that a symplectic reduction $M/ /T$ has the same ordinary cohomology as its real locus $(M/ /T)^{\sigma_{red}}$, with degrees halved. This extends Duistermaat's original result on real loci to a case in which there is not a natural Hamiltonian torus action.
Viscosity solutions, almost everywhere solutions and explicit formulas
Bernard
Dacorogna;
Paolo
Marcellini
4643-4653
Abstract: Consider the differential inclusion $Du\in E$ in $\mathbb{R} ^{n}$. We exhibit an explicit solution that we call fundamental. It also turns out to be a viscosity solution when properly defining this notion. Finally, we consider a Dirichlet problem associated to the differential inclusion and we give an iterative procedure for finding a solution.
Convolution roots of radial positive definite functions with compact support
Werner
Ehm;
Tilmann
Gneiting;
Donald
Richards
4655-4685
Abstract: A classical theorem of Boas, Kac, and Krein states that a characteristic function $\varphi$ with $\varphi(x) = 0$ for $\vert x\vert \geq \tau$ admits a representation of the form \begin{displaymath}\varphi(x) = \int u(y) \hspace{0.2mm} \overline{u(y+x)} \, \mathrm{d}y, \qquad x \in \mathbb{R}, \end{displaymath} where the convolution root $u \in L^2(\mathbb{R})$ is complex-valued with $u(x) = 0$ for $\vert x\vert \geq \tau/2$. The result can be expressed equivalently as a factorization theorem for entire functions of finite exponential type. This paper examines the Boas-Kac representation under additional constraints: If $\varphi$ is real-valued and even, can the convolution root $u$ be chosen as a real-valued and/or even function? A complete answer in terms of the zeros of the Fourier transform of $\varphi$ is obtained. Furthermore, the analogous problem for radially symmetric functions defined on $\mathbb{R}^d$ is solved. Perhaps surprisingly, there are compactly supported, radial positive definite functions that do not admit a convolution root with half-support. However, under the additional assumption of nonnegativity, radially symmetric convolution roots with half-support exist. Further results in this paper include a characterization of extreme points, pointwise and integral bounds (Turán's problem), and a unified solution to a minimization problem for compactly supported positive definite functions. Specifically, if $f$ is a probability density on $\mathbb{R}^d$ whose characteristic function $\varphi$ vanishes outside the unit ball, then \begin{displaymath}\int \vert x\vert^2 f(x) \, \mathrm{d}x = - \Delta \varphi(0) \geq 4 \, j_{(d-2)/2}^2 \end{displaymath} where $j_\nu$ denotes the first positive zero of the Bessel function $J_\nu$, and the estimate is sharp. Applications to spatial moving average processes, geostatistical simulation, crystallography, optics, and phase retrieval are noted. In particular, a real-valued half-support convolution root of the spherical correlation function in $\mathbb{R}^2$ does not exist.