Domains of Holomorphy with Edges and Lower Dimensional Boundary Singularities

It is shown that a domain in C ^N with piecewise smooth boundary (and also of some more general shape) is a domain of holomorphy, provided the Levi form at every regular point is positively semidefinite and the tangent cone is convex at every point outside a boundary subset of zero Hausdorff (2N-2)-dimensional measure.     

Fatou–Bieberbach Domains as Basins of Attraction of Automorphisms Tangent to the Identity

We prove that there exists an automorphism of ℂ² tangent to the identity with a domain of attraction D to the origin, biholomorphic to ℂ², along a degenerate characteristic direction. Our automorphism of ℂ² is conjugate to a translation in D. We also prove the existence of a curve Γ, a biholomorphic copy of ℂ, entirely contained in the boundary of D. In our construction Γ is tangent to the z-axis in a neighborhood of the origin. The automorphisms we construct also fix the w-axis; therefore we obtain D, a Fatou–Bieberbach domain that does not intersect two biholomorphic copies of ℂ locally transversal at the origin.     

Backward iteration in strongly convex domains

We prove that a backward orbit with bounded Kobayashi step for a hyperbolic, parabolic or strongly elliptic holomorphic self-map of a bounded strongly convex C² domain in Cd necessarily converges to a repelling or parabolic boundary fixed point, generalizing previous results obtained by Poggi-Corradini in the unit disk and by Ostapyuk in the unit ball of Cd.     

The blow-up curve of solutions of mixed problems for semilinear wave equations with exponential nonlinearities in one space dimension, I

In this paper, we consider mixed problems with a spacelike boundary derivative condition for semilinear wave equations with exponential nonlinearities in a quarter plane. Results similar to those obtained earlier by Caffarelli-Friedman for Cauchy problems and power nonlinearities are proved in the present situation, namely we show that solutions either are global or blow up on a spacelike curve. Weaker results are also obtained if the boundary vector field is tangent to the characteristic which leaves the domain in the future. Received January 7, 2000 / Accepted July 17, 2000 /Published online December 8, 2000     

On proper holomorphic self-maps of generalized complex ellipsoids

The purpose of this paper is to prove that every proper holomorphic self-mapping of a Reinhardt domain Ω in C ⁿ which is a generalization of a complex ellipsoid is biholomorphic. The main novelty of our result is that Ω is a domain in C ⁿ such that it is allowed to have a boundary point at which the Levi determinant has infinite order of vanishing.     

On the h-vector of a cohen-macaulay domain in positive characteristic

Here we study the Hilbert function of a Cohen-Macaulay homogeneous domain over an algebraically closed field of positive characteristic. The main tool (and an essential part of the main geometrical results) is the study of the Hilbert function of a general hyperplane section X?P ^r of an integral curve C?P ^r+1 , which is pathological in some sense. In §1, we study the case when Cis a strange curve, i.e., all tangent lines to Cat its simple points pass through a fixed point υ∈P ^r+1 . In §2, we give more refined results under the assumption that the Trisecant Lemma fails for C, i.e., any line spanned by two points of Ccontains one more point of C.     

Uniform Decay properties of a model in structural acoustics

We investigate decay properties for a system of coupled partial differential equations which model the interaction between acoustic waves in a cavity and the walls of the cavity. In this system a wave equation is coupled to a structurally damped plate or beam equation. The underlying semigroup for this system is not uniformly stable, but when the system is appropriately restricted we obtain some uniform stability. We present two results of this type. For the first result, we assume that the initial wave data is zero, and the initial plate or beam data is in the natural energy space; then the corresponding solution to system decays uniformly to zero. For the second result, we assume that the initial condition is in the natural energy space and the control function is L²(0,∞) (in time) into the control space; then the beam displacement and velocity are both L²(0,∞) into a space with two spatial derivatives.     

Construction of Solutions and L^1-error Estimates of Viscous Methods for Scalar Conservation Laws with Boundary

This paper is concerned with an initial boundary value problem for strictly convex conservation laws whose weak entropy solution is in the piecewise smooth solution class consisting of finitely many discontinuities. By the structure of the weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos-Leroux Nedelec, we give a construction method to the weak entropy solution of the initial boundary value problem. Compared with the initial value problem, the weak entropy solution of the initial boundary value problem includes the following new interaction type： an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary. According to the structure and some global estimates of the weak entropy solution, we derive the global L^1-error estimate for viscous methods to this initial boundary value problem by using the matching travelling wave solutions method. If the inviscid solution includes the interaction that an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary, or the inviscid solution includes some shock wave which is tangent to the boundary, then the error of the viscosity solution to the inviscid solution is bounded by O（ε^1/2） in L^1-norm; otherwise, as in the initial value problem, the L^1-error bound is O（ε｜ In ε｜）.     

A link between sets of tensors stable under lamination and quasiconvexity

A link is found between quasiconvexity and the conditions for a set L of conductivity or elasticity tensors to be stable under lamination. These conditions, derived in the companion paper, are shown here to be equivalent to the condition that for every point B on the boundary of the set L an operator T_B dependent on the tangent plane and curvature of the set at B is a quasiconvex translation operator. A separate class of quasiconvex translation operators is obtained which are candidates for proving that L is stable under homogenization. The region stable under homogenization associated with any one of these operators shares a common boundary point and tangent plane with the set L and has curvature at that point not greater than the curvature of L. The conditions under which there exists a representative subclass of these operators such that the associated regions stable under homogenization wrap around L remains unresolved. It is proved that L can be characterized by minimizations of sums and dual energies in much the same way that convex sets can be characterized by their Legendre transforms. © 1994 John Wiley & Sons, Inc.     

The Bézout equation for functions of log‐type growth in convex domains of finite type

The aim of this paper is to solve a division problem for the algebra of functions, which are holomorphic in a domain D ? C ⁿ, n > 1, and grow near the boundary not faster than some power of –log dist(z, bD). The domain D is assumed to be smoothly bounded and convex of finite d'Angelo type (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Pattern formations of 2D Rayleigh–Bénard convection with no‐slip boundary conditions for the velocity at the critical length scales

We study the Rayleigh–Bénard convection in a 2D rectangular domain with no‐slip boundary conditions for the velocity. The main mathematical challenge is due to the no‐slip boundary conditions, because the separation of variables for the linear eigenvalue problem, which works in the free‐slip case, is no longer possible. It is well known that as the Rayleigh number crosses a critical threshold R_c, the system bifurcates to an attractor, which is an (m ? 1)‐dimensional sphere, where m is the number of eigenvalues, which cross zero as R crosses R_c. The main objective of this article is to derive a full classification of the structure of this bifurcated attractor when m = 2. More precisely, we rigorously prove that when m = 2, the bifurcated attractor is homeomorphic to a one‐dimensional circle consisting of exactly four or eight steady states and their connecting heteroclinic orbits. In addition, we show that the mixed modes can be stable steady states for small Prandtl numbers. Copyright © 2014 John Wiley & Sons, Ltd.     

Diffraction of creeping waves from a point of curvature jump on the boundary of a domain (The point of transition of the convex boundary into the concave boundary)

The problem of diffraction of a creeping wave propagating in a domain near the convex part of the boundary and overrunning a point where the convex boundary transforms to the concave one is studied. The tangent to the boundary is continuous at this point, but the derivative of the tangent has the jump. The Green's function to the right of the point of jump of curvature is a superposition of whispering gallery waves. The Dirichlet, Neumann, and impedance boundary conditions are considered. The formulas for the boundary current and for the diffraction coefficients related to the problem are obtained. Bibliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 288–299. Translated by N. Ya. Kirpichnikova     

An affirmative answer to a question of Ciliberto

We give an example of a nondegeneraten-dimensional smooth projective varietyX inP ²ⁿ⁺¹ with the canonical bundle ample a varietyX whose tangent variety TanX has dimension less than 2n over an algebraically closed field of any characteristic whenn≥9. This varietyX is not ruled by lines and the embedded tangent space at a general point ofX intersectsX at some other points, so that this yields an affirmative answer to a question of Ciliberto.     

The edge wave in the problem of diffraction on a boundary with jump of curvature

The problem of the diffraction of creeping waves on a point of transition of the convex boundary to the straight boundary of a domain is investigated. It is assumed that at the point of jump of curvature, the tangent to the boundary is continuous and its derivative has a jump. An expression for the edge wave is obtained and investigated. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 274–287. Translated by N. Ya. Kirpichnikova.     

A characterization of domains in <Emphasis Type="Bold">C</Emphasis><Superscript>2</Superscript> with noncompact automorphism group

Let D be a bounded domain in C ² with a non-compact group of holomorphic automorphisms. Model domains for D are obtained under the hypotheses that at least one orbit accumulates at a boundary point near which the boundary is smooth, real analytic and of finite type. The author was supported by DST (India) Grant No.: SR/S4/MS-283/05 and in part by a grant from UGC under DSA-SAP, Phase IV.     

Non‐classical boundary value problems of elastostatics and hydrostatics on Lipschitz domains

The main goal of this paper is to undertake a systematic study of the Stokes and Lamé systems with non‐classical boundary conditions in arbitrary Lipschitz subdomains of R ³. These include prescribing 〈 n , u 〉 in concert with n ×curl u on the boundary of the domain in question. Copyright © 2001 John Wiley & Sons, Ltd.     

On Rational Varieties Smooth Except at a Seminormal Singular Point

We construct a class of projective rational varieties X of any dimension m ≥ 1, which are smooth except at a point O, with the projective space ? ^m as normalization, having smooth branches, and reduced projectivized tangent cone in O. The Hilbert function of X is considered and is explicitly computed when the point O is seminormal. Indeed, we study seminormality, obtaining necessary and sufficient conditions for O to be seminormal and show that in such case the tangent cone is reduced and seminormal.     

Inverse Obstacle Problem for the Non-Stationary Wave Equation with an Unknown Background

We consider boundary measurements for the wave equation on a bounded domain M ? ?² or on a compact Riemannian surface, and introduce a method to locate a discontinuity in the wave speed. Assuming that the wave speed consist of an inclusion in a known smooth background, the method can determine the distance from any boundary point to the inclusion. In the case of a known constant background wave speed, the method reconstructs a set contained in the convex hull of the inclusion and containing the inclusion. Even if the background wave speed is unknown, the method can reconstruct the distance from each boundary point to the inclusion assuming that the Riemannian metric tensor determined by the wave speed gives simple geometry in M. The method is based on reconstruction of volumes of domains of influence by solving a sequence of linear equations. For τ ∈C(?M) the domain of influence M(τ) is the set of those points on the manifold from which the distance to some boundary point x is less than τ(x).     

The Singularities of the Distance Function Near Convex Boundary Points

In an open bounded set Ω, we consider the distance function from ∂Ω associated to a Riemannian metric with C ^1,1 coefficients. Assuming that Ω is convex near a boundary point x ₀, we show that the distance function is differentiable at x ₀ if and only if there exists the tangent space to ∂Ω at x ₀. Furthermore, if the distance function is not differentiable at x ₀ then there exists a Lipschitz continuous curve, with initial point at x ₀, such that the distance function is not differentiable along such a curve.      

Une étude mathématique d'un modèle de Rotenberg

In this paper, a cell proliferation model of Rotenberg is mathematically studied. Each cell of this population is distinguished by its degree of maturity μ and its maturation velocity v. We equipped this mathematical model by boundary conditions wich generalize all biological rules imposed by Rotenberg. We show that this model is governed by a strongly continuous semigroup. We also develop some properties of this semigroup and we study its asymptotic behaviour in the uniform topology.