1,348 results
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2. Self-similar Asymptotics for a Modified Maxwell–Boltzmann Equation in Systems Subject to Deformations
- Author
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Alexander Bobylev, Juan J. L. Velázquez, and Alessia Nota
- Subjects
010102 general mathematics ,Mathematical analysis ,Complex system ,FOS: Physical sciences ,Second moment of area ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,Collision ,01 natural sciences ,Maxwell–Boltzmann distribution ,Boltzmann equation ,symbols.namesake ,Mathematics - Analysis of PDEs ,Norm (mathematics) ,0103 physical sciences ,FOS: Mathematics ,symbols ,Higher order moments ,010307 mathematical physics ,0101 mathematics ,Mathematical Physics ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
In this paper we study a generalized class of Maxwell-Boltzmann equations which in addition to the usual collision term contains a linear deformation term described by a matrix A. This class of equations arises, for instance, from the analysis of homoenergetic solutions for the Boltzmann equation considered by many authors since 1950s. Our main goal is to study a large time asymptotics of solutions under assumption of smallness of the matrix A. The main result of the paper is formulated in Theorem 2.1. Informally stated, this Theorem says that, for sufficiently small norm of A, any non-negative solution with finite second moment tends to a self-similar solution of relatively simple form for large values of time. This is what we call "the self-similar asymptotics". We also prove that the higher order moments of the self-similar profile are finite under further smallness condition on the matrix A., Comment: 38 pages
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- 2020
3. Dynamics of closed singularities
- Author
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William P. Minicozzi and Tobias H. Colding
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Algebra and Number Theory ,Property (philosophy) ,010102 general mathematics ,Dynamics (mechanics) ,Mathematical analysis ,01 natural sciences ,Singularity ,Flow (mathematics) ,0103 physical sciences ,Gravitational singularity ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Special case ,Smoothing ,Mathematics - Abstract
Parabolic geometric flows have the property of smoothing for short time however, over long time, singularities are typically unavoidable, can be very nasty and may be impossible to classify. The idea of this paper is that, by bringing in the dynamical properties of the flow, we obtain also smoothing for long time for generic initial conditions. When combined with our earlier paper this allows us to show that, in an important special case, the singularities are the simplest possible. We take here the first steps towards understanding the dynamics of the flow. The question of the dynamics of a singularity has two parts. One is: What are the dynamics near a singularity? The second is: What is the long time behavior of the flow of things close to the singularity. That is, if the flow leaves a neighborhood of a singularity, is it possible for the flow to re-enter the same neighborhood at a much later time? The first part is addressed in this paper, while the second will be addressed elsewhere.
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- 2020
4. Rectifying and Osculating Curves on a Smooth Surface
- Author
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Absos Ali Shaikh and Pinaki Ranjan Ghosh
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Applied Mathematics ,General Mathematics ,Numerical analysis ,010102 general mathematics ,Mathematical analysis ,Osculating curve ,01 natural sciences ,Smooth surface ,0103 physical sciences ,Mathematics::Metric Geometry ,Mathematics::Differential Geometry ,010307 mathematical physics ,Tangent vector ,0101 mathematics ,Invariant (mathematics) ,Mathematics ,Geodesic curvature ,Osculating circle - Abstract
The main motive of the paper is to look on rectifying and osculating curves on a smooth surface. In this paper we find the normal and geodesic curvature for a rectifying curve on a smooth surface and we also prove that geodesic curvature is invariant under the isometry of surfaces such that rectifying curves remain. We find a sufficient condition for which an osculating curve on a smooth surface remains invariant under isometry of surfaces and also we prove that the component of the position vector of an osculating curve α(s) on a smooth surface along any tangent vector to the surface at α(s) is invariant under such isometry.
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- 2020
5. Courant-sharp Robin eigenvalues for the square: the case with small Robin parameter
- Author
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Katie Gittins, Bernard Helffer, Université de Neuchâtel (Université de Neuchâtel), Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), and Helffer, Bernard
- Subjects
Spectral theory ,General Mathematics ,Courant-sharp ,[MATH] Mathematics [math] ,01 natural sciences ,Domain (mathematical analysis) ,Square (algebra) ,Mathematics - Spectral Theory ,symbols.namesake ,Robin eigenvalues ,0103 physical sciences ,FOS: Mathematics ,[MATH.MATH-SP] Mathematics [math]/Spectral Theory [math.SP] ,Neumann boundary condition ,square ,[MATH]Mathematics [math] ,0101 mathematics ,Spectral Theory (math.SP) ,Eigenvalues and eigenvectors ,Mathematics ,35P99, 58J50, 58J37 ,010102 general mathematics ,Mathematical analysis ,Mathematics::Spectral Theory ,Robin boundary condition ,Number theory ,Dirichlet boundary condition ,symbols ,010307 mathematical physics ,[MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP] - Abstract
International audience; This article is the continuation of our first work on the determination of the cases where there is equality in Courant's Nodal Domain theorem in the case of a Robin boundary condition (with Robin parameter h). For the square, our first paper focused on the case where h is large and extended results that were obtained by Pleijel, Bérard-Helffer, for the problem with a Dirichlet boundary condition. There, we also obtained some general results about the behaviour of the nodal structure (for planar domains) under a small deformation of h, where h is positive and not close to 0. In this second paper, we extend results that were obtained by Helffer-Persson-Sundqvist for the Neumann problem to the case where h > 0 is small. MSC classification (2010): 35P99, 58J50, 58J37.
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- 2019
6. V.M. Miklyukov: from Dimension 8 to Nonassociative Algebras
- Author
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Vladimir G. Tkachev
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Mathematics - Differential Geometry ,Pure mathematics ,010102 general mathematics ,Dimension (graph theory) ,Geometry ,Quasielliptic equations ,Mathematics - Rings and Algebras ,Mathematical Analysis ,Nonassociative algebras ,Algebra and Logic ,01 natural sciences ,53A10, 17A01, 30A99 ,Differential Geometry (math.DG) ,Rings and Algebras (math.RA) ,Matematisk analys ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,0103 physical sciences ,FOS: Mathematics ,Entire solutions ,Geometri ,010307 mathematical physics ,0101 mathematics ,Algebraic number ,Algebra och logik ,Mathematics - Abstract
In this short survey we give a background and explain some recent developments in algebraic minimal cones and nonassociative algebras. A good deal of this paper is recollections of my collaboration with my teacher, PhD supervisor and a colleague, Vladimir Miklyukov on minimal surface theory that motivated the present research. This paper is dedicated to his memory., Comment: 19 pages
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- 2019
7. On the gap between the Gamma-limit and the pointwise limit for a nonlocal approximation of the total variation
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Massimo Gobbino, Clara Antonucci, and Nicola Picenni
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Gamma-convergence ,Computation ,nonlocal functional ,Variation (game tree) ,bounded-variation functions ,01 natural sciences ,nonconvex functional ,26B30, 46E35 ,0103 physical sciences ,FOS: Mathematics ,46E35 ,Limit (mathematics) ,0101 mathematics ,Mathematics - Optimization and Control ,Mathematics ,Pointwise ,Numerical Analysis ,Applied Mathematics ,Multivariable calculus ,Multiple integral ,010102 general mathematics ,Mathematical analysis ,Function (mathematics) ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,total variation ,Optimization and Control (math.OC) ,010307 mathematical physics ,26B30 ,Bounded-variation functions ,Nonconvex functional ,Nonlocal functional ,Total variation ,Analysis - Abstract
We consider the approximation of the total variation of a function by the family of non-local and non-convex functionals introduced by H. Brezis and H.-M. Nguyen in a recent paper. The approximating functionals are defined through double integrals in which every pair of points contributes according to some interaction law. In this paper we answer two open questions concerning the dependence of the Gamma-limit on the interaction law. In the first result, we show that the Gamma-limit depends on the full shape of the interaction law, and not only on the values in a neighborhood of the origin. In the second result, we show that there do exist interaction laws for which the Gamma-limit coincides with the pointwise limit on smooth functions. The key argument is that for some special classes of interaction laws the computation of the Gamma-limit can be reduced to studying the asymptotic behavior of suitable multi-variable minimum problems., 26 pages. In the second version we give a stronger negative answer to the open questions we address
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- 2020
8. Courant-sharp Robin eigenvalues for the square: the case of negative Robin parameter
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Bernard Helffer and Katie Gittins
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35P99, 58J50, 58J37 ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Mathematics::Spectral Theory ,01 natural sciences ,Domain (mathematical analysis) ,Square (algebra) ,Robin boundary condition ,Mathematics - Spectral Theory ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Laplace operator ,Computer Science::Operating Systems ,Spectral Theory (math.SP) ,Eigenvalues and eigenvectors ,Mathematics - Abstract
We consider the cases where there is equality in Courant’s nodal domain theorem for the Laplacian with a Robin boundary condition on the square. In our previous two papers, we treated the cases where the Robin parameter h > 0 is large, small respectively. In this paper we investigate the case where h < 0.
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- 2020
- Full Text
- View/download PDF
9. Curvature estimates for constant mean curvature surfaces
- Author
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William H. Meeks and Giuseppe Tinaglia
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Mathematics - Differential Geometry ,Minimal surface ,Mean curvature ,minimal surface ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,53A10 ,53C42 ,Curvature ,01 natural sciences ,49Q05 ,curvature estimates ,Differential Geometry (math.DG) ,0103 physical sciences ,FOS: Mathematics ,constant mean curvature ,010307 mathematical physics ,Mathematics::Differential Geometry ,0101 mathematics ,Constant (mathematics) ,Mathematics - Abstract
We derive extrinsic curvature estimates for compact disks embedded in $\mathbb{R}^3$ with nonzero constant mean curvature., Comment: We have separated the original paper into two parts. This new posting is the first part which is self-contained and deals with extrinsic curvature estimates for embedded nonzero constant mean curvature disks. The second part is now the paper arXiv:1609.08032
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- 2019
10. Inhomogeneous multi-parameter Lipschitz spaces associated with different homogeneities and their applications
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Jian Tan and Yanchang Han
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General Mathematics ,010102 general mathematics ,0103 physical sciences ,Mathematical analysis ,010307 mathematical physics ,0101 mathematics ,Lipschitz continuity ,01 natural sciences ,Multi parameter ,Mathematics - Abstract
This paper is motivated by Phong and Stein?s work in [11]. The purpose of this paper is to establish the inhomogeneous multi-parameter Lipschitz spaces Lip? com associated with mixed homogeneities and characterize these spaces via the Littlewood-Paley theory. As applications, the boundedness of the composition of Calder?n-Zygmund singular integral operators with mixed homogeneities has been considered.
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- 2018
11. Derived categories of moduli spaces of vector bundles on curves
- Author
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M.S. Narasimhan
- Subjects
Fourier–Mukai transform ,Degree (graph theory) ,010102 general mathematics ,Mathematical analysis ,General Physics and Astronomy ,Vector bundle ,Stable vector bundle ,01 natural sciences ,Moduli space ,Combinatorics ,Moduli of algebraic curves ,Mathematics::Algebraic Geometry ,Line bundle ,0103 physical sciences ,010307 mathematical physics ,Geometry and Topology ,Algebraic curve ,0101 mathematics ,Mathematical Physics ,Mathematics - Abstract
Let X be a smooth projective curve of genus g over \(\mathbb {C}\) and M be the moduli space of stable vector bundle of rank 2 and determinant isomorphic to a fixed line bundle of degree 1 on X. Let E be the Poincare bundle on \(X \times M\) and \(\Phi _E : D^b(X) \rightarrow D^b(M)\) Fourier–Mukai functor defined by E. It was proved in our earlier paper that \(\Phi _E\) is fully faithful for every smooth projective curve of genus \(g \ge 4.\) It is proved in this present paper that the result is also true for non-hyperelliptic curves of genus 3. Combining known results in the case of hyperelliptic curves, one obtains that \(\Phi _E\) is fully faithful for all X of genus \(g \ge 2\).
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- 2017
12. Equidistribution of the crucial measures in non-Archimedean dynamics
- Author
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Kenneth Jacobs
- Subjects
Algebra and Number Theory ,010102 general mathematics ,Diagonal ,Mathematical analysis ,Field (mathematics) ,Function (mathematics) ,01 natural sciences ,Measure (mathematics) ,Canonical measure ,Combinatorics ,Projective line ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Algebraically closed field ,Mathematics ,Probability measure - Abstract
Text Let K be a complete, algebraically closed, non-Archimedean valued field, and let ϕ ∈ K ( z ) with deg ( ϕ ) ≥ 2 . In this paper we consider the family of functions ord Res ϕ n ( x ) , which measure the resultant of ϕ n at points x in P K 1 , the Berkovich projective line, and show that they converge locally uniformly to the diagonal values of the Arakelov–Green's function g μ ϕ ( x , x ) attached to the canonical measure of ϕ. Following this, we are able to prove an equidistribution result for Rumely's crucial measures ν ϕ n , each of which is a probability measure supported at finitely many points whose weights are determined by dynamical properties of ϕ. Video For a video summary of this paper, please visit https://youtu.be/YCCZD1iwe00 .
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- 2017
13. Some notes on quasisymmetric flows of Zygmund vector fields
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Yulong He, Huaying Wei, and Yuliang Shen
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Mathematics::Functional Analysis ,Class (set theory) ,Pure mathematics ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Mathematics::Classical Analysis and ODEs ,01 natural sciences ,Sobolev space ,Flow (mathematics) ,0103 physical sciences ,Vector field ,010307 mathematical physics ,0101 mathematics ,Analysis ,Mathematics - Abstract
In an important paper [24] , Reimann showed that the flow mappings of a continuous vector field of Zygmund class Λ ⁎ are quasisymmetric homeomorphisms. In this paper, we will discuss the flow mappings when the vector field belongs to the smooth Zygmund class λ ⁎ or the Sobolev class H 3 2 .
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- 2017
14. From the Gross–Pitaevskii equation to the Euler Korteweg system, existence of global strong solutions with small irrotational initial data
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Boris Haspot and Corentin Audiard
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Small data ,Applied Mathematics ,010102 general mathematics ,Dimension (graph theory) ,Mathematical analysis ,Euler system ,Conservative vector field ,01 natural sciences ,Gross–Pitaevskii equation ,symbols.namesake ,0103 physical sciences ,Compressibility ,Euler's formula ,symbols ,Order (group theory) ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
In this paper we prove the global well-posedness for small data for the Euler Korteweg system in dimension N ≥ 3, also called compressible Euler system with quantum pressure. It is formally equivalent to the Gross-Pitaevskii equation through the Madelung transform. The main feature is that our solutions have no vacuum for all time. Our construction uses in a crucial way some deep results on the scattering of the Gross-Pitaevskii equation due to Gustafson, Nakanishi and Tsai in [28, 29, 30]. An important part of the paper is devoted to explain the main technical issues of the scattering in [29] and we give a detailed proof in order to make it more accessible. Bounds for long and short times are treated with special care so that the existence of solutions does not require smallness of the initial data in H s , s > N 2 . The optimality of our assumptions is also discussed.
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- 2017
15. On the Mathematical Description of Time-Dependent Surface Water Waves
- Author
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Wolf-Patrick Düll
- Subjects
History and Overview (math.HO) ,Mathematics - History and Overview ,010102 general mathematics ,Mathematical analysis ,Fluid Dynamics (physics.flu-dyn) ,FOS: Physical sciences ,Motion (geometry) ,Physics - Fluid Dynamics ,01 natural sciences ,Mathematical theory ,Physics - Atmospheric and Oceanic Physics ,Mathematics - Analysis of PDEs ,76B15, 35Q35, 35Q53, 35Q55 ,Atmospheric and Oceanic Physics (physics.ao-ph) ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Surface water ,Well posedness ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
This article provides a survey on some main results and recent developments in the mathematical theory of water waves. More precisely, we briefly discuss the mathematical modeling of water waves and then we give an overview of local and global well-posedness results for the model equations. Moreover, we present reduced models in various parameter regimes for the approximate description of the motion of typical wave profiles and discuss the mathematically rigorous justification of the validity of these models., This paper is an extension of the previous submission arXiv:1612.06242v1 with a slightly changed title. In its present form, the paper has been accepted for publication in: Jahresbericht der Deutschen Mathematiker-Vereinigung
- Published
- 2017
16. Analytic semigroups for the subelliptic oblique derivative problem
- Author
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Kazuaki Taira
- Subjects
oblique derivative problem ,Analytic semigroup ,Picard–Lindelöf theorem ,General Mathematics ,Open problem ,010102 general mathematics ,Mathematical analysis ,Oblique case ,analytic semigroup ,Differential operator ,01 natural sciences ,Agmon's method ,Topology of uniform convergence ,subelliptic operator ,Sobolev space ,Unit circle ,35J25 ,35P20 ,0103 physical sciences ,010307 mathematical physics ,35S05 ,0101 mathematics ,asymptotic eigenvalue distribution ,47D03 ,Mathematics - Abstract
This paper is devoted to a functional analytic approach to the subelliptic oblique derivative problem for second-order, elliptic differential operators with a complex parameter $\lambda$. We prove an existence and uniqueness theorem of the homogeneous oblique derivative problem in the framework of $L^{p}$ Sobolev spaces when $\vert\lambda\vert$ tends to $\infty$. As an application of the main theorem, we prove generation theorems of analytic semigroups for this subelliptic oblique derivative problem in the $L^{p}$ topology and in the topology of uniform convergence. Moreover, we solve the long-standing open problem of the asymptotic eigenvalue distribution for the subelliptic oblique derivative problem. In this paper we make use of Agmon's technique of treating a spectral parameter $\lambda$ as a second-order elliptic differential operator of an extra variable on the unit circle and relating the old problem to a new one with the additional variable.
- Published
- 2017
17. On the kinetic energy profile of Hölder continuous Euler flows
- Author
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Philip Isett and Sung-Jin Oh
- Subjects
Conservation of energy ,Conjecture ,Series (mathematics) ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Hölder condition ,Function (mathematics) ,Kinetic energy ,01 natural sciences ,symbols.namesake ,0103 physical sciences ,Euler's formula ,symbols ,Exponent ,010307 mathematical physics ,0101 mathematics ,Mathematical Physics ,Analysis ,Mathematics - Abstract
In [8], the first author proposed a strengthening of Onsager's conjecture on the failure of energy conservation for incompressible Euler flows with Holder regularity not exceeding 1/3. This stronger form of the conjecture implies that anomalous dissipation will fail for a generic Euler flow with regularity below the Onsager critical space L_t^∞(B_(3,∞)^(1/3) due to low regularity of the energy profile. The present paper is the second in a series of two papers whose results may be viewed as first steps towards establishing the conjectured failure of energy regularity for generic solutions with Holder exponent less than 1/5. The main result of this paper shows that any non-negative function with compact support and Holder regularity 1/2 can be prescribed as the energy profile of an Euler flow in the class C_(t,x)^(1/5 − ϵ). The exponent 1/2 is sharp in view of a regularity result of Isett [8]. The proof employs an improved greedy algorithm scheme that builds upon that in Buckmaster–De Lellis–Szekelyhidi [1].
- Published
- 2017
18. Global smooth solution of a two-dimensional nonlinear singular system of differential equations arising from geostrophics
- Author
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Dongfen Bian, Linghai Zhang, Yongqian Han, Boling Guo, and Daiwen Huang
- Subjects
Differential equation ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Cauchy distribution ,01 natural sciences ,Nonlinear system ,symbols.namesake ,Singularity ,0103 physical sciences ,Jacobian matrix and determinant ,symbols ,Order (group theory) ,010307 mathematical physics ,Uniqueness ,0101 mathematics ,Constant (mathematics) ,Analysis ,Mathematics - Abstract
Consider the Cauchy problems for the following two-dimensional nonlinear singular system of differential equations arising from geostrophics ∂∂t[γ(ψ1−ψ2)−△ψ1]+α(−△)ρψ1+β∂ψ1∂x+J(ψ1,γ(ψ1−ψ2)−△ψ1)=0,∂∂t[γδ(ψ2−ψ1)−△ψ2]+α(−△)ρψ2+β∂ψ2∂x+J(ψ2,γδ(ψ2−ψ1)−△ψ2)=0,ψ1(x,y,0)=ψ01(x,y),ψ2(x,y,0)=ψ02(x,y). In this system, α>0, γ>0, δ>0 and ρ>0 are positive constants, β≠0 is a real nonzero constant, the Jacobian determinant is defined by J(p,q)=∂p∂x∂q∂y−∂p∂y∂q∂x. The existence and uniqueness of the global smooth solution of the system of differential equations are very important in applied mathematics and geostrophics, but they have been open for a long time. The singularity generated by the linear parts, the strong couplings of the nonlinear functions and the fractional order of the derivatives make the existence and uniqueness very difficult to study. Very recently, we found that there exist a few special structures in the system. The main purpose of this paper is to couple together the special structures and an unusual method for establishing the uniform energy estimates to overcome the main difficulty to accomplish the existence and uniqueness of the global smooth solution: ψ1∈C∞(R2×R+) and ψ2∈C∞(R2×R+), for all ρ>3/2. The new energy method enables us to make complete use of the special structures of the nonlinear singular system. The results obtained in this paper provide positive solutions to very important open problems and greatly improve many previous results about nonlinear singular systems of differential equations arising from geostrophics.
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- 2017
19. The second moment of sums of coefficients of cusp forms
- Author
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Alexander Walker, Thomas A. Hulse, David Lowry-Duda, and Chan Ieong Kuan
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Cusp (singularity) ,Algebra and Number Theory ,Conjecture ,Mathematics - Number Theory ,010102 general mathematics ,Mathematical analysis ,Holomorphic function ,Function (mathematics) ,11F30 ,01 natural sciences ,Combinatorics ,symbols.namesake ,0103 physical sciences ,FOS: Mathematics ,symbols ,Number Theory (math.NT) ,010307 mathematical physics ,0101 mathematics ,Fourier series ,Dirichlet series ,Meromorphic function ,Mathematics ,Sign (mathematics) - Abstract
Let $f$ and $g$ be weight $k$ holomorphic cusp forms and let $S_f(n)$ and $S_g(n)$ denote the sums of their first $n$ Fourier coefficients. Hafner and Ivic [HI], building on Chandrasekharan and Narasimhan [CN], proved asymptotics for $\sum_{n \leq X} \lvert S_f(n) \rvert^2$ and proved that the Classical Conjecture, that $S_f(X) \ll X^{\frac{k-1}{2} + \frac{1}{4} + \epsilon}$, holds on average over long intervals. In this paper, we introduce and obtain meromorphic continuations for the Dirichlet series $D(s, S_f \times S_g) = \sum S_f(n)\overline{S_g(n)} n^{-(s+k-1)}$ and $D(s, S_f \times \overline{S_g}) = \sum_n S_f(n)S_g(n) n^{-(s + k - 1)}$. Using these meromorphic continuations, we prove asymptotics for the smoothed second moment sums $\sum S_f(n)\overline{S_g(n)} e^{-n/X}$, proving a smoothed generalization of [HI]. We also attain asymptotics for analogous smoothed second moment sums of normalized Fourier coefficients, proving smoothed generalizations of what would be attainable from [CN]. Our methodology extends to a wide variety of weights and levels, and comparison with [CN] indicates very general cancellation between the Rankin-Selberg $L$-function $L(s, f\times g)$ and shifted convolution sums of the coefficients of $f$ and $g$. In forthcoming works, the authors apply the results of this paper to prove the Classical Conjecture on $\lvert S_f(n) \rvert^2$ is true on short intervals, and to prove sign change results on $\{S_f(n)\}_{n \in \mathbb{N}}$., Comment: To appear in the Journal of Number Theory
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- 2017
20. Solitons and Scattering for the Cubic–Quintic Nonlinear Schrödinger Equation on $${\mathbb{R}^3}$$ R 3
- Author
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Tadahiro Oh, Oana Pocovnicu, Monica Visan, and Rowan Killip
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Partial differential equation ,Mechanical Engineering ,Operator (physics) ,010102 general mathematics ,Mathematical analysis ,Context (language use) ,Type (model theory) ,01 natural sciences ,Virial theorem ,Quintic function ,symbols.namesake ,Mathematics (miscellaneous) ,0103 physical sciences ,symbols ,010307 mathematical physics ,Soliton ,0101 mathematics ,Nonlinear Sciences::Pattern Formation and Solitons ,Nonlinear Schrödinger equation ,Analysis ,Mathematics - Abstract
We consider the cubic–quintic nonlinear Schrodinger equation: $$i\partial_t u = -\Delta u - |u|^2u + |u|^4u.$$ In the first part of the paper, we analyze the one-parameter family of ground state solitons associated to this equation with particular attention to the shape of the associated mass/energy curve. Additionally, we are able to characterize the kernel of the linearized operator about such solitons and to demonstrate that they occur as optimizers for a one-parameter family of inequalities of Gagliardo–Nirenberg type. Building on this work, in the latter part of the paper we prove that scattering holds for solutions belonging to the region \({{\mathcal{R}}}\) of the mass/energy plane where the virial is positive. We show that this region is partially bounded by solitons also by rescalings of solitons (which are not soliton solutions in their own right). The discovery of rescaled solitons in this context is new and highlights an unexpected limitation of any virial-based methodology.
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- 2017
21. Flattening of CR singular points and analyticity of the local hull of holomorphy II
- Author
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Wanke Yin and Xiaojun Huang
- Subjects
Pure mathematics ,Mathematics::Complex Variables ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Holomorphic function ,Codimension ,Singular point of a curve ,Submanifold ,01 natural sciences ,Plateau's problem ,Hypersurface ,Complex space ,Bounded function ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
This is the second article of the two papers, in which we investigate the holomorphic and formal flattening problem of a non-degenerate CR singular point of a codimension two real submanifold in C n with n ≥ 3 . The problem is motivated from the study of the complex Plateau problem that looks for the Levi-flat hypersurface bounded by a given real submanifold and by the classical complex analysis problem of finding the local hull of holomorphy of a real submanifold in a complex space. The present article is focused on non-degenerate flat CR singular points with at least one non-parabolic Bishop invariant. We will solve the formal flattening problem in this setting. The results in this paper and those in [23] are taken from our earlier arxiv post [22] . We split [22] into two independent articles to avoid it being too long.
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- 2017
22. Extremal function for capacity and estimates of QED constants in Rn
- Author
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Tao Cheng and Shanshuang Yang
- Subjects
Pure mathematics ,Extremal length ,Geometric function theory ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Boundary (topology) ,Conformal map ,01 natural sciences ,Upper and lower bounds ,Potential theory ,0103 physical sciences ,010307 mathematical physics ,Uniqueness ,0101 mathematics ,Constant (mathematics) ,Mathematics - Abstract
This paper is devoted to the study of some fundamental problems on modulus and extremal length of curve families, capacity, and n-harmonic functions in the Euclidean space R n . One of the main goals is to establish the existence, uniqueness, and boundary behavior of the extremal function for the conformal capacity cap ( A , B ; Ω ) of a capacitor in R n . This generalizes some well known results and has its own interests in geometric function theory and potential theory. It is also used as a major ingredient in this paper to establish a sharp upper bound for the quasiextremal distance (or QED) constant M ( Ω ) of a domain in terms of its local boundary quasiconformal reflection constant H ( Ω ) , a bound conjectured by Shen in the plane. Along the way, several interesting results are established for modulus and extremal length. One of them is a decomposition theorem for the extremal length λ ( A , B ; Ω ) of the curve family joining two disjoint continua A and B in a domain Ω.
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- 2017
23. Conical metrics on Riemann surfaces, II: spherical metrics
- Author
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Rafe Mazzeo and Xuwen Zhu
- Subjects
Mathematics - Differential Geometry ,General Mathematics ,Riemann surface ,010102 general mathematics ,Mathematical analysis ,Friedrichs extension ,Deformation theory ,01 natural sciences ,Moduli space ,Constant curvature ,symbols.namesake ,Mathematics - Analysis of PDEs ,Differential Geometry (math.DG) ,Conic section ,0103 physical sciences ,symbols ,FOS: Mathematics ,Gravitational singularity ,010307 mathematical physics ,0101 mathematics ,Laplace operator ,Mathematics ,Analysis of PDEs (math.AP) - Abstract
We continue our study, initiated in our earlier paper, of Riemann surfaces with constant curvature and isolated conic singularities. Using the machinery developed in that earlier paper of extended configuration families of simple divisors, we study the existence and deformation theory for spherical conic metrics with some or all of the cone angles greater than $2\pi$. Deformations are obstructed precisely when the number $2$ lies in the spectrum of the Friedrichs extension of the Laplacian. Our main result is that, in this case, it is possible to find a smooth local moduli space of solutions by allowing the cone points to split. This analytic fact reflects geometric constructions in papers by Mondello and Panov., Comment: Final version accepted by Int. Math. Res. Not
- Published
- 2019
- Full Text
- View/download PDF
24. Periodic orbits of oval billiards on surfaces of constant curvature
- Author
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Luciano Coutinho dos Santos and Sônia Pinto-de-Carvalho
- Subjects
Mathematics::Dynamical Systems ,General Mathematics ,Hyperbolic geometry ,010102 general mathematics ,Mathematical analysis ,Boundary (topology) ,Dynamical Systems (math.DS) ,Dynamical system ,01 natural sciences ,Computer Science Applications ,37E40 37J45 ,Nonlinear Sciences::Chaotic Dynamics ,Constant curvature ,Bounded function ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Diffeomorphism ,Mathematics - Dynamical Systems ,0101 mathematics ,Twist ,Dynamical billiards ,Mathematics - Abstract
In this paper we define and study the billiard problem on bounded regions on surfaces of constant curvature. We show that this problem defines a 2-dimensional conservative and reversible dynamical system, defined by a Twist diffeomorphism, if the boundary of the region is an oval. Using these properties and defining good perturbations for billiards, we show, in this new version, that having only a finite number of nondegenerate periodic orbits for each fixed period is an open property for billiards on surfaces of constant curvature and a dense one on the Euclidean and the hyperbolic planes. For the proof of the density, the techniques we use for the Euclidean and hyperbolic cases, do not work for the spherical case, due to a constraint (the perimeter of the polygonal trajectory being a multiple of {\pi}). We finish this paper studying the stability of these nondegenerate orbits., Comment: 11 pages, no figures
- Published
- 2016
25. Global existence for semilinear wave equations with the critical blow-up term in high dimensions
- Author
-
Hiroyuki Takamura and Kyouhei Wakasa
- Subjects
Work (thermodynamics) ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Wave equation ,Space (mathematics) ,01 natural sciences ,Square (algebra) ,Term (time) ,Nonlinear system ,Mathematics - Analysis of PDEs ,Quadratic equation ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,primary 35L71, 35E15, secondary 35A01, 35A09, 35B33, 35B44 ,Critical exponent ,Analysis ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
We are interested in almost global existence cases in the general theory for nonlinear wave equations, which are caused by critical exponents of nonlinear terms. Such situations can be found in only three cases in the theory, cubic terms in two space dimensions, quadratic terms in three space dimesions and quadratic terms including a square of unknown functions itself in four space dimensions. Except for the last case, criterions to classify nonlinear terms into the almost global, or global existence case, are well-studied and known to be so-called null condition and non-positive condition. Our motivation of this work is to find such a kind of the criterion in four space dimensions. In our previous paper, an example of the non-single term for the almost global existence case is introduced. In this paper, we show an example of the global existence case. These two examples have nonlinear integral terms which are closely related to derivative loss due to high dimensions. But it may help us to describe the final form of the criterion., 23pages. Some minor corrections are added in previous versions. arXiv admin note: substantial text overlap with arXiv:1404.4471
- Published
- 2016
26. Threshold results for semilinear parabolic systems
- Author
-
Qiuyi Dai, Haiyang He, and Junhui Xie
- Subjects
Statement (computer science) ,010102 general mathematics ,Mathematical analysis ,Geodetic datum ,01 natural sciences ,Computational Mathematics ,Parabolic system ,Computational Theory and Mathematics ,Modeling and Simulation ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Positive equilibrium ,Value (mathematics) ,Mathematics - Abstract
In this paper, we study initial-boundary value problem of semi-linear parabolic system (1.1) in Sectionź 1, and prove that any positive equilibrium of problemź (1.1) is an initial datum threshold for the existence and nonexistence of global solution to it. For the precise statement of this result, see Theoremź1.1 in Sectionź 1 of this paper.
- Published
- 2016
27. Proper Modifications of Generalized p-Kähler Manifolds
- Author
-
Lucia Alessandrini
- Subjects
Pure mathematics ,Class (set theory) ,Property (philosophy) ,010102 general mathematics ,Mathematical analysis ,Kähler manifold ,Submanifold ,01 natural sciences ,Cohomology ,Differential geometry ,0103 physical sciences ,Point (geometry) ,Mathematics::Differential Geometry ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Mathematics::Symplectic Geometry ,Geometry and topology ,Mathematics - Abstract
In this paper, we consider a proper modification $$f : \tilde{M} \rightarrow M$$ between complex manifolds, and study when a generalized p-Kahler property goes back from M to $$\tilde{M}$$ . When f is the blow-up at a point, every generalized p-Kahler property is conserved, while when f is the blow-up along a submanifold, the same is true for $$p=1$$ . For $$p=n-1$$ , we prove that the class of compact generalized balanced manifolds is closed with respect to modifications, and we show that the fundamental forms can be chosen in the expected cohomology class. We also get some partial results in the non-compact case; finally, we end the paper with some examples of generalized p-Kahler manifolds.
- Published
- 2016
28. On the inviscid limit of the 2D Navier–Stokes equations with vorticity belonging to BMO-type spaces
- Author
-
Tarek M. Elgindi, Sahbi Keraani, and Frédéric Bernicot
- Subjects
D'Alembert's paradox ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Banach space ,Non-dimensionalization and scaling of the Navier–Stokes equations ,Space (mathematics) ,01 natural sciences ,Euler equations ,Physics::Fluid Dynamics ,symbols.namesake ,Inviscid flow ,0103 physical sciences ,Hagen–Poiseuille flow from the Navier–Stokes equations ,symbols ,010307 mathematical physics ,0101 mathematics ,Navier–Stokes equations ,Mathematical Physics ,Analysis ,Mathematics - Abstract
In a recent paper [6] , the global well-posedness of the two-dimensional Euler equation with vorticity in L 1 ∩ LBMO was proved, where LBMO is a Banach space which is strictly imbricated between L ∞ and BMO. In the present paper we prove a global result on the inviscid limit of the Navier–Stokes system with data in this space and other spaces with the same BMO flavor. Some results of local uniform estimates on solutions of the Navier–Stokes equations, independent of the viscosity, are also obtained.
- Published
- 2016
29. The Construction of 3D Conformal Motions
- Author
-
Leo Dorst and Computer Vision (IVI, FNWI)
- Subjects
Primary field ,Extremal length ,Conformal field theory ,Applied Mathematics ,010102 general mathematics ,Dupin cyclide ,Mathematical analysis ,Conformal geometric algebra ,01 natural sciences ,Computational Mathematics ,Geometric algebra ,Computational Theory and Mathematics ,Conformal symmetry ,0103 physical sciences ,010307 mathematical physics ,Tangent vector ,0101 mathematics ,Mathematics - Abstract
This paper exposes a very geometrical yet directly computational way of working with conformal motions in 3D. With the increased relevance of conformal structures in architectural geometry, and their traditional use in CAD, its results should be useful to designers and programmers. In brief, we exploit the fact that any 3D conformal motion is governed by two well-chosen point pairs: the motion is composed of (or decomposed into) two specific orthogonal circular motions in planes determined by those point pairs. The resulting orbit of a point is an equiangular spiral on a Dupin cyclide. These results are compactly expressed and programmed using conformal geometric algebra (CGA), and this paper can serve as an introduction to its usefulness. Although the point pairs come in different kinds (imaginary, real, tangent vector, direction vector, axis vector and ‘flat point’), causing the great variety of conformal motions, all are unified both algebraically and computationally as 2-blades in CGA, automatically producing properly parametrized simple rotors by exponentiation. An additional advantage of using CGA is its covariance: conformal motions for other primitives such as circles are computed using exactly the same formulas, and hence the same software operations, as motions of points. This generates an interesting class of easily generated shapes, like spatial circles moving conformally along a knot on a Dupin cyclide.
- Published
- 2016
30. Resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds I: Resolvent construction at high energy
- Author
-
Andrew Hassell and Xi Chen
- Subjects
Series (mathematics) ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Semiclassical physics ,Hyperbolic manifold ,58J40 ,Resolvent formalism ,Mathematics::Spectral Theory ,01 natural sciences ,Fourier integral operator ,Multiplier (Fourier analysis) ,Mathematics - Analysis of PDEs ,0103 physical sciences ,Metric (mathematics) ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Analysis ,Analysis of PDEs (math.AP) ,Resolvent ,Mathematics - Abstract
This is the first in a series of papers in which we investigate the resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds with applications to the restriction theorem, spectral multiplier results and Strichartz estimates. In this first paper, we use semiclassical Lagrangian distributions and semiclassical intersecting Lagrangian distributions, along with Mazzeo-Melrose 0-calculus, to construct the high energy resolvent on general non- trapping asymptotically hyperbolic manifolds, generalizing the work due to Melrose, Sa Barreto and Vasy. We note that there is an independent work by Y. Wang which also constructs the high-energy resolvent., Comment: 49 pages
- Published
- 2016
31. Regularity of area minimizing currents II: center manifold
- Author
-
Camillo De Lellis, Emanuele Spadaro, and University of Zurich
- Subjects
Mathematics - Differential Geometry ,Current (mathematics) ,regularity ,minimal surfaces ,01 natural sciences ,510 Mathematics ,Mathematics (miscellaneous) ,Normal bundle ,0103 physical sciences ,FOS: Mathematics ,1804 Statistics, Probability and Uncertainty ,2613 Statistics and Probability ,0101 mathematics ,Mathematics ,Degree (graph theory) ,Series (mathematics) ,010102 general mathematics ,Mathematical analysis ,Codimension ,Lipschitz continuity ,10123 Institute of Mathematics ,Differential Geometry (math.DG) ,Gravitational singularity ,010307 mathematical physics ,Statistics, Probability and Uncertainty ,Center manifold - Abstract
This is the second paper of a series of three on the regularity of higher codimension area minimizing integral currents. Here we perform the second main step in the analysis of the singularities, namely the construction of a center manifold, i.e. an approximate average of the sheets of an almost flat area minimizing current. Such center manifold is complemented with a Lipschitz multi-valued map on its normal bundle, which approximates the current with a highe degree of accuracy. In the third and final paper these objects are used to conclude a new proof of Almgren's celebrated dimension bound on the singular set., In the new version the proofs and the structure are improved and some minor errors have been corrected
- Published
- 2016
- Full Text
- View/download PDF
32. Some results on the entropy of non-autonomous dynamical systems
- Author
-
Yuri Latushkin and Christoph Kawan
- Subjects
General Mathematics ,010102 general mathematics ,Mathematical analysis ,Topological entropy ,01 natural sciences ,Topological entropy in physics ,Joint entropy ,Computer Science Applications ,Rényi entropy ,Differential entropy ,0103 physical sciences ,Maximum entropy probability distribution ,010307 mathematical physics ,Statistical physics ,0101 mathematics ,Joint quantum entropy ,Entropy rate ,Mathematics - Abstract
In this paper, we advance the entropy theory of discrete non-autonomous dynamical systems that was initiated by Kolyada and Snoha in 1996. The first part of the paper is devoted to the measure-theoretic entropy theory of general topological systems. We derive several conditions guaranteeing that an initial probability measure, when pushed forward by the system, produces an invariant measure sequence whose entropy captures the dynamics on arbitrarily fine scales. In the second part of the paper, we apply the general theory to the non-stationary subshifts of finite type, introduced by Fisher and Arnoux. In particular, we give sufficient conditions for the variational principle, relating the topological and measure-theoretic entropy, to hold.
- Published
- 2015
33. Isolated singularities of polyharmonic operator in even dimension
- Author
-
R. Dhanya and Abhishek Sarkar
- Subjects
Delta ,Numerical Analysis ,Measurable function ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Sigma ,01 natural sciences ,Omega ,Polyharmonic spline ,Combinatorics ,Computational Mathematics ,Singularity ,0103 physical sciences ,Standard probability space ,Beta (velocity) ,010307 mathematical physics ,0101 mathematics ,Mathematics ,Analysis - Abstract
We consider the equation Delta(2)u = g(x, u) >= 0 in the sense of distribution in Omega' = Omega\textbackslash {0} where u and -Delta u >= 0. Then it is known that u solves Delta(2)u = g(x, u) + alpha delta(0) - beta Delta delta(0), for some nonnegative constants alpha and beta. In this paper, we study the existence of singular solutions to Delta(2)u = a(x) f (u) + alpha delta(0) - beta Delta delta(0) in a domain Omega subset of R-4, a is a nonnegative measurable function in some Lebesgue space. If Delta(2)u = a(x) f (u) in Omega', then we find the growth of the nonlinearity f that determines alpha and beta to be 0. In case when alpha = beta = 0, we will establish regularity results when f (t) 0. This paper extends the work of Soranzo (1997) where the author finds the barrier function in higher dimensions (N >= 5) with a specific weight function a(x) = |x|(sigma). Later, we discuss its analogous generalization for the polyharmonic operator.
- Published
- 2015
34. Local existence of a fourth-order dispersive curve flow on locally Hermitian symmetric spaces and its application
- Author
-
Eiji Onodera
- Subjects
Hermitian symmetric space ,Partial differential equation ,010102 general mathematics ,Mathematical analysis ,01 natural sciences ,Hermitian matrix ,Manifold ,Constant curvature ,Computational Theory and Mathematics ,Flow (mathematics) ,0103 physical sciences ,Initial value problem ,010307 mathematical physics ,Geometry and Topology ,Compact Riemann surface ,0101 mathematics ,Analysis ,Mathematics - Abstract
A fourth-order nonlinear dispersive partial differential equation arises in the field of mathematical physics, the solution of which is a curve flow on the two-dimensional unit sphere. In recent ten years, a geometric generalization of the sphere-valued physical model has been considered and the solvability of the initial value problem has been investigated. In particular, in the author's previous work, time-local existence and uniqueness result of the solution was established under the assumption that the solution is a closed curve flow on a compact Riemann surface with constant curvature. In the present paper, we propose a new geometric generalization of the sphere-valued physical model. As a main result, we show time-local existence of a solution to the initial value problem under the assumption that the solution is a closed curve flow on a compact locally Hermitian symmetric space. The proof is based on the geometric energy method combined with a gauge transformation to overcome the difficulty of the so-called loss of derivatives. Interestingly, the results can be applied to construct a generalized bi-Schrodinger flow proposed by Ding and Wang. The assumption on the manifold plays a crucial role both to enjoy a good solvable structure of the initial value problem and to reduce the generalized bi-Schrodinger flow equation to the one considered in the present paper.
- Published
- 2019
35. Rigidity of equality of Lyapunov exponents for geodesic flows
- Author
-
Clark Butler
- Subjects
Mathematics - Differential Geometry ,Mathematics::Dynamical Systems ,Geodesic ,Dynamical Systems (math.DS) ,Lyapunov exponent ,Curvature ,01 natural sciences ,symbols.namesake ,Rigidity (electromagnetism) ,0103 physical sciences ,FOS: Mathematics ,Mathematics - Dynamical Systems ,0101 mathematics ,Mathematics ,Algebra and Number Theory ,Invariance principle ,010102 general mathematics ,Mathematical analysis ,Lie group ,Nilpotent ,Differential Geometry (math.DG) ,Symmetric space ,symbols ,Mathematics::Differential Geometry ,010307 mathematical physics ,Geometry and Topology ,Analysis - Abstract
We study the relationship between the Lyapunov exponents of the geodesic flow of a closed negatively curved manifold and the geometry of the manifold. We show that if each periodic orbit of the geodesic flow has exactly one Lyapunov exponent on the unstable bundle then the manifold has constant negative curvature. We also show under a curvature pinching condition that equality of all Lyapunov exponents with respect to volume on the unstable bundle also implies that the manifold has constant negative curvature. We then study the degree to which one can emulate these rigidity theorems for the hyperbolic spaces of nonconstant negative curvature when the Lyapunov exponents with respect to volume match those of the appropriate symmetric space and obtain rigidity results under additional technical assumptions. The proofs use new results from hyperbolic dynamics including the nonlinear invariance principle of Avila and Viana and the approximation of Lyapunov exponents of invariant measures by Lyapunov exponents associated to periodic orbits which was developed by Kalinin in his proof of the Livsic theorem for matrix cocycles. We also employ rigidity results of Capogna and Pansu on quasiconformal mappings of certain nilpotent Lie groups., Comment: 35 pages.v4: Extensive revisions: New title (former title was "Lyapunov exponent rigidity for geodesic flows). Section 5 and Theorem 1.2 of the previous version have been removed from the paper; they will appear in a future paper. Theorems concerning hyperbolic spaces of nonconstant negative curvature have been strengthened slightly and the proofs have been reorganized
- Published
- 2018
36. A comparison principle for convolution measures with applications
- Author
-
René Quilodrán and Diogo Oliveira e Silva
- Subjects
Pointwise ,Paraboloid ,Computer Science::Information Retrieval ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Parabola ,Regular polygon ,01 natural sciences ,Measure (mathematics) ,Projection (linear algebra) ,Convolution ,symbols.namesake ,Fourier transform ,Mathematics - Classical Analysis and ODEs ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
We establish the general form of a geometric comparison principle for $n$-fold convolutions of certain singular measures in $\mathbb{R}^d$ which holds for arbitrary $n$ and $d$. This translates into a pointwise inequality between the convolutions of projection measure on the paraboloid and a perturbation thereof, and we use it to establish a new sharp Fourier extension inequality on a general convex perturbation of a parabola. Further applications of the comparison principle to sharp Fourier restriction theory are discussed in a companion paper., Comment: 17 pages, v2: updated reference to companion paper
- Published
- 2018
37. Stability of constant mean curvature surfaces in three dimensional warped product manifolds
- Author
-
Gregório Silva Neto
- Subjects
Mathematics - Differential Geometry ,Mean curvature ,Euclidean space ,010102 general mathematics ,Mathematical analysis ,Boundary (topology) ,53C42 (Primary), 53C21 (Secondary) ,01 natural sciences ,Ambient space ,Differential geometry ,Differential Geometry (math.DG) ,Norm (mathematics) ,0103 physical sciences ,Constant-mean-curvature surface ,FOS: Mathematics ,010307 mathematical physics ,Geometry and Topology ,Mathematics::Differential Geometry ,0101 mathematics ,Analysis ,Mathematics ,Scalar curvature - Abstract
In this paper we prove that stable, compact without boundary, oriented, nonzero constant mean curvature surfaces in the de Sitter-Schwarzschild and Reissner-Nordstrom manifolds are the slices, provided its mean curvature satisfies some positive lower bound. More generally, we prove that stable, compact without boundary, oriented nonzero constant mean curvature surfaces in a large class of three dimensional warped product manifolds are embedded topological spheres, provided the mean curvature satisfies a positive lower bound depending only on the ambient curvatures. We conclude the paper proving that a stable, compact without boundary, nonzero constant mean curvature surface in a general Riemannian is a topological sphere provided its mean curvature has a lower bound depending only on the scalar curvature of the ambient space and the squared norm of the mean curvature vector field of the immersion of the ambient space in some Euclidean space., 30 pages, 2 figures. Minor corrections. To appear in the Annals of Global Analysis and Geometry
- Published
- 2018
38. Damping estimates for oscillatory integral operators with real-analytic phases and its applications
- Author
-
Shaozhen Xu, Zuoshunhua Shi, and Dunyan Yan
- Subjects
Class (set theory) ,42B20 47G10 ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Oscillatory integral operator ,01 natural sciences ,Mathematics - Classical Analysis and ODEs ,0103 physical sciences ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Oscillatory integral ,Mathematics - Abstract
In this paper, we investigate sharp damping estimates for a class of one dimensional oscillatory integral operators with real-analytic phases. By establishing endpoint estimates for suitably damped oscillatory integral operators, we are able to give a new proof of the sharp $L^p$ estimates which have been proved by Xiao in Endpoint estimates for one-dimensional oscillatory integral operators, \emph{Advances in Mathematics}, \textbf{316}, 255-291 (2017). The damping estimates obtained in this paper are of independent interest., Comment: 30 pages
- Published
- 2018
- Full Text
- View/download PDF
39. Four-dimensional Painlevé-type equations associated with ramified linear equations II: Sasano systems
- Author
-
Hiroshi Kawakami
- Subjects
Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Mathematics - Classical Analysis and ODEs ,010102 general mathematics ,0103 physical sciences ,Mathematical analysis ,010307 mathematical physics ,0101 mathematics ,Type (model theory) ,01 natural sciences ,Linear equation ,Mathematics - Abstract
This is a continuation of the paper "Four-dimensional Painlev\'e-type equations associated with ramified linear equations I: Matrix Painlev\'e systems" (arXiv:1608.03927). In this series of three papers we aim to construct the complete degeneration scheme of four-dimensional Painlev\'e-type equations. In the present paper, we construct the degeneration scheme of what we call the Sasano system., Comment: 37 pages. arXiv admin note: text overlap with arXiv:1608.03927
- Published
- 2018
40. Soliton resolution along a sequence of times for the focusing energy critical wave equation
- Author
-
Thomas Duyckaerts, Hao Jia, Carlos E. Kenig, Frank Merle, Analyse, Géométrie et Modélisation (AGM - UMR 8088), and CY Cergy Paris Université (CY)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Sequence ,010102 general mathematics ,Mathematical analysis ,Wave equation ,Space (mathematics) ,01 natural sciences ,Upper and lower bounds ,Virial theorem ,Mathematics - Analysis of PDEs ,Bounded function ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Geometry and Topology ,Soliton ,0101 mathematics ,[MATH]Mathematics [math] ,Analysis ,Energy (signal processing) ,Mathematics ,Analysis of PDEs (math.AP) - Abstract
In this paper, we prove the soliton resolution conjecture for general type II solutions to the focusing energy critical wave equation, in space dimension 3,4 or 5, along a sequence of times. This is an important step towards the full soliton resolution in the nonradial case and without any size restrictions. This paper is an extension of the arXiv preprint 1510:00075 by the second author, where the finite time blow-up case is treated, with an error converging to zero in a weaker sense., Comment: Substantial changes in the introduction and ni Section 3. Accepted in GAFA
- Published
- 2017
41. Existence and regularity of solutions to a quasilinear elliptic problem involving variable sources
- Author
-
Ruimei Gao, Yan Sun, and Ying Chu
- Subjects
regularity ,Mathematics::Analysis of PDEs ,Fixed-point theorem ,lcsh:Analysis ,01 natural sciences ,Dirichlet distribution ,symbols.namesake ,0103 physical sciences ,Nabla symbol ,Boundary value problem ,0101 mathematics ,Mathematics ,variable exponent ,Algebra and Number Theory ,Partial differential equation ,quasilinear elliptic problem ,Operator (physics) ,010102 general mathematics ,Mathematical analysis ,existence ,lcsh:QA299.6-433 ,A priori estimate ,Ordinary differential equation ,nonlinear singular term ,symbols ,010307 mathematical physics ,Analysis - Abstract
The authors of this paper prove the existence and regularity results for the homogeneous Dirichlet boundary value problem to the equation $-\operatorname{div}(|\nabla u|^{p-2}\nabla u)=\frac{f(x)}{u ^{\alpha(x)}}$ with $f\in L^{m}(\Omega)$ ( $m\geqslant1$ ) and $\alpha(x)>0$ . Due to the nonlinearity of a p-Laplace operator and the anisotropic variable exponent $\alpha(x)$ , some classical methods may not directly be applied to our problem. In this paper, we construct a suitable test function and apply the Leray-Schauder fixed point theorem to prove the existence of positive solutions with necessary a priori estimate and compact argument. Furthermore, we also discuss the relationship among the regularity of solutions, the summability of f and the value of $\alpha(x)$ .
- Published
- 2017
42. Boundary behavior of solutions to the parabolic p-Laplace equation II
- Author
-
Benny Avelin
- Subjects
Laplace's equation ,Waiting time ,Series (mathematics) ,010102 general mathematics ,Mathematical analysis ,Boundary (topology) ,Type (model theory) ,01 natural sciences ,Measure (mathematics) ,Domain (mathematical analysis) ,Continuation ,Mathematics - Analysis of PDEs ,35K92, 35K65, 35K20, 35B33, 35B60 ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Analysis ,Mathematics ,Analysis of PDEs (math.AP) - Abstract
This paper is the second installment in a series of papers concerning the boundary behavior of solutions to the p-parabolic equations. In this paper we are interested in the short time behavior of the solutions, which is in contrast with much of the literature, where all results require a waiting time. We prove a dichotomy about the decay-rate of non-negative solutions vanishing on the lateral boundary in a cylindrical C 1 , 1 domain. Furthermore we connect this dichotomy to the support of the boundary type Riesz measure related to the p-parabolic equation in NTA-domains, which has consequences for the continuation of solutions.
- Published
- 2017
43. GAP PHENOMENA AND CURVATURE ESTIMATES FOR CONFORMALLY COMPACT EINSTEIN MANIFOLDS
- Author
-
Yuguang Shi, Gang Li, and Jie Qing
- Subjects
Mathematics - Differential Geometry ,gap phenomena ,Primary 53C25 ,General Mathematics ,Conformal map ,Einstein manifold ,Curvature ,01 natural sciences ,curvature estimates ,symbols.namesake ,Relative Volume ,Ricci-flat manifold ,0103 physical sciences ,FOS: Mathematics ,Gap theorem ,0101 mathematics ,Einstein ,Mathematical physics ,Mathematics ,Quantitative Biology::Biomolecules ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Conformally compact Einstein manifolds ,16. Peace & justice ,Pure Mathematics ,math.DG ,Differential Geometry (math.DG) ,rigidity ,Yamabe constants ,Secondary 58J05 ,symbols ,renormalized volumes ,010307 mathematical physics ,Mathematics::Differential Geometry ,Yamabe invariant ,Primary 53C25, Secondary 58J05 - Abstract
In this paper we first use the result in $[12]$ to remove the assumption of the $L^2$ boundedness of Weyl curvature in the gap theorem in $[9]$ and then obtain a gap theorem for a class of conformally compact Einstein manifolds with very large renormalized volume. We also uses the blow-up method to derive curvature estimates for conformally compact Einstein manifolds with large renormalized volume. The second part of this paper is on conformally compact Einstein manifolds with conformal infinities of large Yamabe constants. Based on the idea in $[15]$ we manage to give the complete proof of the relative volume inequality $(1.9)$ on conformally compact Einstein manifolds. Therefore we obtain the complete proof of the rigidity theorem for conformally compact Einstein manifolds in general dimensions with no spin structure assumption (cf. $[29, 15]$) as well as the new curvature pinch estimates for conformally compact Einstein manifolds with conformal infinities of very large Yamabe constant. We also derive the curvature estimates for conformally compact Einstein manifolds with conformal infinities of large Yamabe constant., Comment: 28 pages, 1 figure(with one sentence added)
- Published
- 2017
44. Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators using Caputo derivatives
- Author
-
Milton Ferreira and Nelson Vieira
- Subjects
Dirac operator ,01 natural sciences ,Caputo derivative ,symbols.namesake ,Operator (computer programming) ,0103 physical sciences ,Fundamental solution ,0101 mathematics ,Mathematics ,Eigenfunctions ,Numerical Analysis ,Laplace transform ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Eigenfunction ,Integral transform ,Fractional partial differential equations ,Fractional calculus ,Computational Mathematics ,symbols ,010307 mathematical physics ,Fractional Laplace and Dirac operators ,Laplace operator ,Analysis - Abstract
In this paper we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator ${}^C\!\Delta_+^{(\alpha,\beta,\gamma)}:= {}^C\!D_{x_0^+}^{1+\alpha} +{}^C\!D_{y_0^+}^{1+\beta} +{}^C\!D_{z_0^+}^{1+\gamma},$ where $(\alpha, \beta, \gamma) \in \,]0,1]^3$ and the fractional derivatives ${}^C\!D_{x_0^+}^{1+\alpha}$, ${}^C\!D_{y_0^+}^{1+\beta}$, ${}^C\!D_{z_0^+}^{1+\gamma}$ are in the Caputo sense. Applying integral transform methods we describe a complete family of eigenfunctions and fundamental solutions of the operator ${}^C\!\Delta_+^{(\alpha,\beta,\gamma)}$ in classes of functions admitting a summable fractional derivative. The solutions are expressed using the Mittag-Leffler function. From the family of fundamental solutions obtained we deduce a family of fundamental solutions of the corresponding fractional Dirac operator, which factorizes the fractional Laplace operator introduced in this paper. info:eu-repo/semantics/publishedVersion
- Published
- 2017
45. A geometric approach for sharp Local well-posedness of quasilinear wave equations
- Author
-
Qian Wang
- Subjects
Mathematics - Differential Geometry ,Wave packet ,Mathematics::Analysis of PDEs ,General Physics and Astronomy ,Type (model theory) ,01 natural sciences ,Mathematics - Analysis of PDEs ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Mathematical Physics ,Ricci curvature ,Mathematics ,Raychaudhuri equation ,Partial differential equation ,Eikonal equation ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Function (mathematics) ,Wave equation ,Differential Geometry (math.DG) ,010307 mathematical physics ,Geometry and Topology ,Analysis ,Analysis of PDEs (math.AP) - Abstract
This paper considers the problem of optimal well-posedness for general quasi-linear wave equations in $${{\mathbb {R}}}^{1+3} $$ of the type (1.1). In general, equations of this type are ill-posed with $$H^s$$ data for $$s\le 2$$ . The optimal result of the well-posedness with data in $$H^s, s>2$$ was proved by Smith–Tataru by constructing parametrices using wave packets. In this paper we give the proof by the vectorfield approach. This approach is initiated by Klainerman, developed by him and Rodnianski to achieve the result of $$s>2+\frac{2-\sqrt{3}}{2}$$ and then applied to the Einstein vacuum equations to achieve the result of $$s>2$$ . To achieve the optimal result for the general quasi-linear wave equations, one has to face the major hurdle caused by the Ricci tensor of the metric. This posed a question that if the geometric approach can provide the sharp result for the non-geometric equations. The new ingredients of this paper concern the regularity properties of the eikonal equation associated to (1.1). The optimal result for (1.1) is achieved based on geometric normalization and new observations on the Raychaudhuri equation and the mass aspect function.
- Published
- 2017
46. Flattening a non-degenerate CR singular point of real codimension two
- Author
-
Hanlong Fang and Xiaojun Huang
- Subjects
Mathematics - Differential Geometry ,Theory ,Mathematics - Complex Variables ,010102 general mathematics ,Mathematical analysis ,Degenerate energy levels ,Codimension ,Singular point of a curve ,Submanifold ,01 natural sciences ,Plateau's problem ,Flattening ,Differential Geometry (math.DG) ,Hull ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Geometry and Topology ,Complex Variables (math.CV) ,0101 mathematics ,Analysis ,Mathematics - Abstract
This paper continues the previous studies in two papers of Huang-Yin [HY3-4] on the flattening problem of a CR singular point of real codimension two sitting in a submanifold in ${\mathbb C}^{n+1}$ with $n+1\ge 3$, whose CR points are non-minimal. Partially based on the geometric approach initiated in [HY3] and a formal theory approach used in [HY4], we are able to provide a very general flattening theorem for a non-degenerate CR singular point. As an application, we provide a solution to the local complex Plateau problem and obtain the analyticity of the local hull of holomorphy near a real analytic definite CR singular point in a general setting., 44 pages
- Published
- 2017
47. Badly approximable points on manifolds and unipotent orbits in homogeneous spaces
- Author
-
Lei Yang
- Subjects
11J13, 11J83, 22E40 ,Distribution (number theory) ,Mathematics - Number Theory ,010102 general mathematics ,Mathematical analysis ,Order (ring theory) ,Unipotent ,16. Peace & justice ,Submanifold ,01 natural sciences ,Combinatorics ,Intersection ,Hausdorff dimension ,0103 physical sciences ,FOS: Mathematics ,Countable set ,Number Theory (math.NT) ,010307 mathematical physics ,Geometry and Topology ,Differentiable function ,0101 mathematics ,Analysis ,Mathematics - Abstract
In this paper, we study the weighted $n$-dimensional badly approximable points on manifolds. Given a $C^n$ differentiable non-degenerate submanifold $\mathcal{U} \subset \mathbb{R}^n$, we will show that any countable intersection of the sets of the weighted badly approximable points on $\mathcal{U}$ has full Hausdorff dimension. This strengthens a result of Beresnevich by removing the condition on weights and weakening the smoothness condition on manifolds. Compared to the work of Beresnevich, our approach relies on homogeneous dynamics. It turns out that in order to solve this problem, it is crucial to study the distribution of long pieces of unipotent orbits in homogeneous spaces. The proof relies on the linearization technique and representations of $\mathrm{SL}(n+1,\mathbb{R})$., 35 pages. The paper was revised according to referees' reports. The analyticity condition is weakened
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- 2017
48. On $T$-coercive interior transmission eigenvalue problems on compact manifolds with smooth boundary
- Author
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Naotaka Shoji
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Boundary (topology) ,eigenvalue free-region ,Type (model theory) ,interior transmission eigenvalue ,01 natural sciences ,non-scattering energy ,Mathematics - Spectral Theory ,81U40 ,47A40 ,0103 physical sciences ,FOS: Mathematics ,conductive boundary condition ,0101 mathematics ,Anisotropy ,Spectral Theory (math.SP) ,Eigenvalues and eigenvectors ,Mathematics ,scattering theory ,010102 general mathematics ,Mathematical analysis ,Order (ring theory) ,Transmission (telecommunications) ,$T$-coercivity ,non-scattering wave number ,010307 mathematical physics ,Scattering theory ,Mathematics::Differential Geometry ,Complex plane - Abstract
In this paper, we consider an interior transmission eigenvalue problem on two compact Riemannian manifolds with common smooth boundary. We suppose that a couple of these manifolds is equipped with locally anisotropic type Riemannian metric tensors, i.e., these two tensors are not equivalent in a neighborhood of common boundary. Here we note that we do not assume that these manifolds are diffeomorphic. In addition, we impose some conditions of the refractive indices in a neighborhood of common boundary. Then we prove that the set of ITEs form infinite discrete set and the existence of ITE-free region. In order to prove our results, we employ so-called the $T$-coercivity method., This paper has been withdrawn by the author due to the modification of the configuration
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- 2017
49. Toward a classification of killing vector fields of constant length on pseudo-riemannian normal homogeneous spaces
- Author
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Fabio Podestà, Joseph A. Wolf, and Ming Xu
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Mathematics - Differential Geometry ,General Mathematics ,math-ph ,FOS: Physical sciences ,Context (language use) ,01 natural sciences ,Combinatorics ,Killing vector field ,math.MP ,0103 physical sciences ,FOS: Mathematics ,Generalized flag variety ,0101 mathematics ,Moment map ,Mathematical Physics ,Mathematics ,Algebra and Number Theory ,math.SG ,Simple Lie group ,010102 general mathematics ,Mathematical analysis ,Homogeneous spaces, Killing fields, moment map ,Mathematical Physics (math-ph) ,Pure Mathematics ,Compact space ,math.DG ,Differential Geometry (math.DG) ,Mathematics - Symplectic Geometry ,Homogeneous space ,Symplectic Geometry (math.SG) ,010307 mathematical physics ,Geometry and Topology ,Constant (mathematics) ,Analysis - Abstract
In this paper we develop the basic tools for a classification of Killing vector fields of constant length on pseudo-Riemannian homogeneous spaces. This extends a recent paper of M. Xu and J. A. Wolf, which classified the pairs $(M,\xi)$ where $M = G/H$ is a Riemannian normal homogeneous space, G is a compact simple Lie group, and $\xi \in \mathfrak{g}$ defines a nonzero Killing vector field of constant length on $M$. The method there was direct computation. Here we make use of the moment map $M \to \mathfrak{g}^{*}$ and the flag manifold structure of $\mathrm{Ad} (G) \xi$ to give a shorter, more geometric proof which does not require compactness and which is valid in the pseudo-Riemannian setting. In that context we break the classification problem into three parts. The first is easily settled. The second concerns the cases where $\xi$ is elliptic and $G$ is simple (but not necessarily compact); that case is our main result here. The third, which remains open, is a more combinatorial problem involving elements of the first two.
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- 2017
50. Rectifiable measures, square functions involving densities, and the Cauchy transform
- Author
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Xavier Tolsa
- Subjects
Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Cauchy distribution ,01 natural sciences ,Square (algebra) ,Mathematics - Analysis of PDEs ,Mathematics - Classical Analysis and ODEs ,0103 physical sciences ,Radon measure ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,010307 mathematical physics ,28A75, 42B20 ,0101 mathematics ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
This paper is devoted to the proof of two related results. The first one asserts that if $\mu$ is a Radon measure in $\mathbb R^d$ satisfying $$\limsup_{r\to 0} \frac{\mu(B(x,r))}{r}>0\quad \text{ and }\quad \int_0^1\left|\frac{\mu(B(x,r))}{r} - \frac{\mu(B(x,2r))}{2r}\right|^2\,\frac{dr}r< \infty$$ for $\mu$-a.e. $x\in\mathbb R^d$, then $\mu$ is rectifiable. Since the converse implication is already known to hold, this yields the following characterization of rectifiable sets: a set $E\subset\mathbb R^d$ with finite $1$-dimensional Hausdorff measure $H^1$ is rectifiable if and only $$\int_0^1\left|\frac{H^1(E\cap B(x,r))}{r} - \frac{H^1(E\cap B(x,2r))}{2r}\right|^2\,\frac{dr}r< \infty \quad\mbox{ for $H^1$-a.e. $x\in E$.}$$ The second result of the paper deals with the relationship between a similar square function in the complex plane and the Cauchy transform $C_\mu f(z) = \int \frac1{z-\xi}\,f(\xi)\,d\mu(\xi)$. Suppose that $\mu$ has linear growth, that is, $\mu(B(z,r))\leq c\,r$ for all $z\in\mathbb C$ and all $r>0$. It is proved that $C_\mu$ is bounded in $L^2(\mu)$ if and only if $$ \int_{z\in Q}\int_0^\infty\left|\frac{\mu(Q\cap B(z,r))}{r} - \frac{\mu(Q\cap B(z,2r))}{2r}\right|^2\,\frac{dr}r\,d\mu(z)\leq c\,\mu(Q) \quad\mbox{ for every square $Q\subset\mathbb C$.} $$, Comment: Minor corrections and adjustments
- Published
- 2017
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