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Register a new userMetaharmonic functions: mean flux theorem, its converse and related properties
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Added by
Nikolay Kuznetsov G
Publication date
2020
fields
Physics
and research's language is
English
Authors
Nikolay Kuznetsov
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The mean flux theorems are proved for solutions of the Helmholtz equation and its modified version. Also, their converses are considered along with some other properties which generalise those that guarantee harmonicity.
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The Four Vertex Theorem, one of the earliest results in global differential geometry, says that a simple closed curve in the plane, other than a circle, must have at least four vertices, that is, at least four points where the curvature has a local maximum or local minimum. In 1909 Syamadas Mukhopadhyaya proved this for strictly convex curves in the plane, and in 1912 Adolf Kneser proved it for all simple closed curves in the plane, not just the strictly convex ones. The Converse to the Four Vertex Theorem says that any continuous real-valued function on the circle which has at least two local maxima and two local minima is the curvature function of a simple closed curve in the plane. In 1971 Herman Gluck proved this for strictly positive preassigned curvature, and in 1997 Bjorn Dahlberg proved the full converse, without the restriction that the curvature be strictly positive. Publication was delayed by Dahlbergs untimely death in January 1998, but his paper was edited afterwards by Vilhelm Adolfsson and Peter Kumlin, and finally appeared in 2005. The work of Dahlberg completes the almost hundred-year-long thread of ideas begun by Mukhopadhyaya, and we take this opportunity to provide a self-contained exposition.
We consider the propagation of wave packets for a one-dimensional nonlinear Schrodinger equation with a matrix-valued potential, in the semi-classical limit. For an initial coherent state polarized along some eigenvector, we prove that the nonlinear evolution preserves the separation of modes, in a scaling such that nonlinear effects are critical (the envelope equation is nonlinear). The proof relies on a fine geometric analysis of the role of spectral projectors, which is compatible with the treatment of nonlinearities. We also prove a nonlinear superposition principle for these adiabatic wave packets.
In cite{GUW} we introduced a class of semi-classical functions of isotropic type, starting with a model case and applying Fourier integral operators associated with canonical transformations. These functions are a substantial generalization of the oscillatory functions of Lagrangian type that have played major role in semi-classical and micro-local analysis. In this paper we exhibit more clearly the nature of these isotropic functions by obtaining oscillatory integral expressions for them. Then we use these to prove that the classes of isotropic functions are equivariant with respect to the action of general FIOs (under the usual clean-intersection hypothesis). The simplest examples of isotropic states are the coherent states, a class of oscillatory functions that has played a pivotal role in mathematics and theoretical physics beginning with their introduction by of Schrodinger in the 1920s. We prove that every oscillatory function of isotropic type can be expressed as a superposition of coherent states, and examine some implications of that fact. We also show that certain functions of elliptic operators have isotropic functions for Schwartz kernels. This lead us to a result on an eigenvalue counting function that appears to be new (Corollary ref{cor:altWeyl}).
The transform considered in the paper averages a function supported in a ball in $RR^n$ over all spheres centered at the boundary of the ball. This Radon type transform arises in several contemporary applications, e.g. in thermoacoustic tomography and sonar and radar imaging. Range descriptions for such transforms are important in all these areas, for instance when dealing with incomplete data, error correction, and other issues. Four different types of complete range descriptions are provided, some of which also suggest inversion procedures. Necessity of three of these (appropriately formulated) conditions holds also in general domains, while the complete discussion of the case of general domains would require another publication.
We consider the zeta function $zeta_Omega$ for the Dirichlet-to-Neumann operator of a simply connected planar domain $Omega$ bounded by a smooth closed curve of perimeter $2pi$. We prove that $zeta_Omega(0)ge zeta_{mathbb{D}}(0)$ with equality if and only if $Omega$ is a disk where $mathbb{D}$ denotes the closed unit disk. We also provide an elementary proof that for a fixed real $s$ satisfying $sle-1$ the estimate $zeta_Omega(s)ge zeta_{mathbb{D}}(s)$ holds with equality if and only if $Omega$ is a disk. We then bring examples of domains $Omega$ close to the unit disk where this estimate fails to be extended to the interval $(0,2)$. Other computations related to previous works are also detailed in the remaining part of the text.
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