Mathematics Department Faculty Selected Works pagesCopyright (c) 2017 University of Massachusetts - Amherst All rights reserved.
http://scholarworks.umass.edu/math_sw
Recent documents in Mathematics Department Faculty Selected Works pagesen-usWed, 18 Jan 2017 01:51:34 PST3600Schrodinger Maps and Their Associated Frame Systems
http://works.bepress.com/andrea_nahmod/23
http://works.bepress.com/andrea_nahmod/23Thu, 31 Mar 2016 17:31:48 PDT
In this paper we establish the equivalence of solutions between Schr¨odinger maps into S 2 or H 2 and their associated gauge invariant Schr¨odinger equations. We also establish the existence of global weak solutions into H 2 in two space dimensions. We extend these ideas for maps into compact hermitian symmetric manifolds with trivial first cohomology.
]]>
Andrea Nahmod et al.The Cauchy Problem for the Hyperbolic-Elliptic Ishimori System and Schrodinger Maps
http://works.bepress.com/andrea_nahmod/25
http://works.bepress.com/andrea_nahmod/25Thu, 31 Mar 2016 17:31:48 PDT
We show an improved local in time existence and uniqueness result for Schrödinger maps and for the hyperbolic–elliptic nonlinear system proposed by Ishimori in analogy with the two-dimensional classical continuous isotropic Heisenberg spin (2d-CCIHS) chain. The proof uses fairly standard gauge geometric tools and energy estimates in combination with Kenig's version of the Koch–Tzvetkov method, to obtain a priori estimates for classical solutions to certain dispersive equations.
]]>
Carlos Kenig et al.On Schrodinger and wave maps
http://works.bepress.com/andrea_nahmod/21
http://works.bepress.com/andrea_nahmod/21Thu, 31 Mar 2016 17:31:48 PDTAndrea R. NahmodBoundedness of bilinear operators with nonsmooth symbols
http://works.bepress.com/andrea_nahmod/17
http://works.bepress.com/andrea_nahmod/17Thu, 31 Mar 2016 17:31:48 PDT
We announce the Lp-boundedness of general bilinear operators associated to a symbol or multiplier which need not be smooth. We establish a general result for multipliers that are allowed to have singularities along the edges of a cone as well as possibly at its vertex. It thus unifies ealier results of CoifmanMeyer for smooth multipliers and ones, such the Bilinear Hilbert transform of Lacey-Thiele, where the multiplier is not smooth.
]]>
John Gilbert et al.Bilinear Operators with Non-Smooth Symbol, I
http://works.bepress.com/andrea_nahmod/16
http://works.bepress.com/andrea_nahmod/16Thu, 31 Mar 2016 17:31:47 PDT
This paper proves the Lp-boundedness of general bilinear operators associated to a symbol or multiplier which need not be smooth. The Main Theorem establishes a general result for multipliers that are allowed to have singularities along the edges of a cone as well as possibly at its vertex. It thus unifies ealier results of Coifman-Meyer for smooth multipliers and ones, such the Bilinear Hilbert transform of Lacey-Thiele, where the multiplier is not smooth. Using a Whitney decomposition in the Fourier plane a general bilinear operator is represented as infinite discrete sums of time-frequency paraproducts obtained by associating wave-packets with tiles in phase-plane. Boundedness for the general bilinear operator then follows once the corresponding Lp-boundedness of time-frequency paraproducts has been established. The latter result is the main theorem proved in Part II, our subsequent paper [11], using phase-plane analysis.
]]>
John E. Gilbert et al.On the Well-Posedness of the Wave Map Problem in High Dimensions
http://works.bepress.com/andrea_nahmod/11
http://works.bepress.com/andrea_nahmod/11Thu, 31 Mar 2016 17:31:01 PDT
We construct a gauge theoretic change of variables for the wave map from R × Rn into a compact group or Riemannian symmetric space, prove a new multiplication theorem for mixed Lebesgue-Besov spaces, and show the global well-posedness of a modified wave map equation - n ≥ 4 - for small critical initial data. We obtain global existence and uniqueness for the Cauchy problem of wave maps into compact Lie groups and symmetric spaces with small critical initial data and n ≥ 4.
]]>
Andrea Nahmod et al.Sobolev Space Estimates and Symbolic Calculus for Bilinear Pseudodifferential Operators
http://works.bepress.com/andrea_nahmod/14
http://works.bepress.com/andrea_nahmod/14Thu, 31 Mar 2016 17:31:01 PDT
Bilinear operators are investigated in the context of Sobolev spaces and various techniques useful in the study of their boundedness properties are developed. In particular, several classes of symbols for bilinear operators beyond the so-called Coifman-Meyer class are considered. Some of the Sobolev space estimates obtained apply to both the bilinear Hilbert transform and its singular multipliers generalizations as well as to operators with variable dependent symbols. A symbolic calculus for the transposes of bilinear pseudodifferential operators and for the composition of linear and bilinear pseudodifferential operators is presented too.
]]>
Arpad Benyi et al.On Schrodinger Maps
http://works.bepress.com/andrea_nahmod/9
http://works.bepress.com/andrea_nahmod/9Thu, 31 Mar 2016 17:31:01 PDT
We study the question of well-posedness of the Cauchy problem for Schr¨odinger maps from R 1 ×R 2 to the sphere S 2 or to H2 , the hyperbolic space. The idea is to choose an appropriate gauge change so that the derivatives of the map will satisfy a certain nonlinear Schr¨odinger system of equations and then study this modified Schr¨odinger map system (MSM). We then prove local well posedness of the Cauchy problem for the MSM with minimal regularity assumptions on the data and outline a method to derive well posedness of the Schr¨odinger map itself from it. In proving well posedness of the MSM, the heart of the matter is resolved by considering truly quatrilinear forms of weighted L 2 functions.
]]>
Andrea R. Nahmod et al.Recent Advances in Harmonic Analysis and Partial Differential Equations in Contemporary Mathematics
http://works.bepress.com/andrea_nahmod/4
http://works.bepress.com/andrea_nahmod/4Wed, 30 Mar 2016 15:37:12 PDT
This volume is based on the AMS Special Session on Harmonic Analysis and Partial Differential Equations and the AMS Special Session on Nonlinear Analysis of Partial Differential Equations, both held March 12-13, 2011, at Georgia Southern University, Statesboro, Georgia, as well as the JAMI Conference on Analysis of PDEs, held March 21-25, 2011, at Johns Hopkins University, Baltimore, Maryland. These conferences all concentrated on problems of current interest in harmonic analysis and PDE, with emphasis on the interaction between them. This volume consists of invited expositions as well as research papers that address prospects of the recent significant development in the field of analysis and PDE. The central topics mainly focused on using Fourier, spectral and geometrical methods to treat wellposedness, scattering and stability problems in PDE, including dispersive type evolution equations, higher-order systems and Sobolev spaces theory that arise in aspects of mathematical physics. The study of all these problems involves state-of-the-art techniques and approaches that have been used and developed in the last decade. The interrelationship between the theory and the tools reflects the richness and deep connections between various subjects in both classical and modern analysis.
]]>
Andrea Nahmod et al.Bilinear Paraproducts Revisited
http://works.bepress.com/andrea_nahmod/3
http://works.bepress.com/andrea_nahmod/3Wed, 30 Mar 2016 15:37:11 PDTÁrpád Bényi et al.