Floquet transform, spectra of periodic elliptic operators, and quasi-periodic layer potentials

2009 ◽  
pp. 121-132
Author(s):  
Habib Ammari ◽  
Hyeonbae Kang ◽  
Hyundae Lee
Keyword(s):  
Elliptic Operators ◽  
Layer Potentials ◽  
Periodic Layer

scholarly journals Nontangential Estimates on Layer Potentials and the Neumann Problem for Higher-Order Elliptic Equations

2020 ◽  
Author(s):  
Ariel Barton ◽  
Steve Hofmann ◽  
Svitlana Mayboroda
Keyword(s):  
Neumann Problem ◽  
Elliptic Equations ◽  
Divergence Form ◽  
Higher Order ◽  
Elliptic Operators ◽  
Layer Potentials ◽  
The Neumann Problem

Abstract We solve the Neumann problem, with nontangential estimates, for higher-order divergence form elliptic operators with variable $t$-independent coefficients. Our results are accompanied by nontangential estimates on higher-order layer potentials.

Electron microscopy of crystalline structure in thermotropic random copolymers

1991 ◽  
Vol 49 ◽  
pp. 1054-1055
Author(s):  
Afzana Anwer ◽  
S. Eilidh Bedford ◽  
Richard J. Spontak ◽  
Alan H. Windle
Keyword(s):  
Electron Microscopy ◽  
Crystalline Structure ◽  
Liquid Crystalline ◽  
Elevated Temperatures ◽  
Random Copolymers ◽  
Molecular Characteristics ◽  
Scanning Calorimetry ◽  
X Ray ◽  
Identical Monomer ◽  
Periodic Layer

Random copolyesters composed of wholly aromatic monomers such as p-oxybenzoate (B) and 2,6-oxynaphthoate (N) are known to exhibit liquid crystalline characteristics at elevated temperatures and over a broad composition range. Previous studies employing techniques such as X-ray diffractometry (XRD) and differential scanning calorimetry (DSC) have conclusively proven that these thermotropic copolymers can possess a significant crystalline fraction, depending on molecular characteristics and processing history, despite the fact that the copolymer chains possess random intramolecular sequencing. Consequently, the nature of the crystalline structure that develops when these materials are processed in their mesophases and subsequently annealed has recently received considerable attention. A model that has been consistent with all experimental observations involves the Non-Periodic Layer (NPL) crystallite, which occurs when identical monomer sequences enter into register between adjacent chains. The objective of this work is to employ electron microscopy to identify and characterize these crystallites.

Degenerate Diffusion Operators Arising in Population Biology (AM-185)

2017 ◽  
Author(s):  
Charles L. Epstein ◽  
Rafe Mazzeo
Keyword(s):  
Adjoint Operator ◽  
Integral Kernel ◽  
Mathematical Finance ◽  
Elliptic Operators ◽  
Complete Analysis ◽  
Time Behavior ◽  
Degenerate Elliptic Operators ◽  
Regularity Properties ◽  
Degenerate Elliptic ◽  
Mathematical Foundations

This book provides the mathematical foundations for the analysis of a class of degenerate elliptic operators defined on manifolds with corners, which arise in a variety of applications such as population genetics, mathematical finance, and economics. The results discussed in this book prove the uniqueness of the solution to the martingale problem and therefore the existence of the associated Markov process. The book uses an “integral kernel method” to develop mathematical foundations for the study of such degenerate elliptic operators and the stochastic processes they define. The precise nature of the degeneracies of the principal symbol for these operators leads to solutions of the parabolic and elliptic problems that display novel regularity properties. Dually, the adjoint operator allows for rather dramatic singularities, such as measures supported on high codimensional strata of the boundary. The book establishes the uniqueness, existence, and sharp regularity properties for solutions to the homogeneous and inhomogeneous heat equations, as well as a complete analysis of the resolvent operator acting on Hölder spaces. It shows that the semigroups defined by these operators have holomorphic extensions to the right half plane. The book also demonstrates precise asymptotic results for the long-time behavior of solutions to both the forward and backward Kolmogorov equations.

scholarly journals Generalized principal eigenvalues of convex nonlinear elliptic operators in ℝN

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Anup Biswas ◽  
Prasun Roychowdhury
Keyword(s):  
Eigenvalue Problem ◽  
Principal Eigenvalue ◽  
Unbounded Domains ◽  
Elliptic Operators ◽  
Maximum Principles ◽  
Generalized Eigenvalues ◽  
Nonlinear Elliptic ◽  
Generalized Eigenvalue ◽  
General Convex ◽  
Appl Math

AbstractWe study the generalized eigenvalue problem in {\mathbb{R}^{N}} for a general convex nonlinear elliptic operator which is locally elliptic and positively 1-homogeneous. Generalizing [H. Berestycki and L. Rossi, Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains, Comm. Pure Appl. Math. 68 2015, 6, 1014–1065], we consider three different notions of generalized eigenvalues and compare them. We also discuss the maximum principles and uniqueness of principal eigenfunctions.

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