Characterizing Two-Dimensional Maps Whose Jacobians Have Constant Eigenvalues

2003 ◽  
Vol 46 (3) ◽  
pp. 323-331 ◽  
Author(s):  
Marc Chamberland
Keyword(s):  
Nonlinear Pdes ◽  
Two Dimensional ◽  
Real Analytic ◽  
Monge Ampere

AbstractRecent papers have shown that C1 maps whose Jacobians have constant eigenvalues can be completely characterized if either the eigenvalues are equal or F is a polynomial. Specifically, F = (u, v) must take the formfor some constants a, b, c, d, e, f , α, β and a C1 function ϕ in one variable. If, in addition, the function ϕ is not affine, thenThis paper shows how these theorems cannot be extended by constructing a real-analytic map whose Jacobian eigenvalues are ±1/2 and does not fit the previous form. This example is also used to construct non-obvious solutions to nonlinear PDEs, including the Monge—Ampère equation.

A correction to ‘Analyticity and metric transitivity on the torus’

1999 ◽  
Vol 19 (1) ◽  
pp. 259-261
Author(s):  
SOL SCHWARTZMAN
Keyword(s):  
Differential Equations ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Topological Transitivity ◽  
Topological Disc ◽  
Real Analytic

In [2], flows on the standard two-dimensional torus given by the differential equations \begin{equation*} \frac{dx}{dt}=a-Fy(x,y),\quad \frac{dv}{dt}=b+Fx(x,y) \end{equation*} were considered. It was assumed that $F(x,y)$ was real analytic and of period one in both $x$ and $y$. A key step in proving the results in [2] was to show that one could conclude topological transitivity for the flow provided one assumed: \begin{enumerate} \item[(a)] $a/b$ is irrational; \item[(b)] there does not exist a topological disc on the torus that is invariant under the flow. \end{enumerate}

scholarly journals Fourier transforms of powers of well-behaved 2D real analytic functions

2018 ◽  
Vol 30 (3) ◽  
pp. 723-732
Author(s):  
Michael Greenblatt
Keyword(s):  
Analytic Functions ◽  
Fourier Transforms ◽  
Newton Polygon ◽  
Companion Paper ◽  
Two Dimensional ◽  
Compactly Supported ◽  
Sharp Estimates ◽  
Real Analytic Functions ◽  
Real Analytic ◽  
Compactly Supported Functions

AbstractThis paper is a companion paper to [6], where sharp estimates are proven for Fourier transforms of compactly supported functions built out of two-dimensional real-analytic functions. The theorems of [6] are stated in a rather general form. In this paper, we expand on the results of [6] and show that there is a class of “well-behaved” functions that contains a number of relevant examples for which such estimates can be explicitly described in terms of the Newton polygon of the function. We will further see that for a subclass of these functions, one can prove noticeably more precise estimates, again in an explicitly describable way.

scholarly journals Harmonic morphisms with one-dimensional fibres on conformally-flat Riemannian manifolds

2008 ◽  
Vol 145 (1) ◽  
pp. 141-151 ◽  
Author(s):  
RADU PANTILIE
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Harmonic Morphisms ◽  
Two Dimensional ◽  
Conformally Flat ◽  
Harmonic Morphism ◽  
One Dimensional ◽  
Dimension 2 ◽  
Real Analytic ◽  
Flat Riemannian Manifold

AbstractWe classify the harmonic morphisms with one-dimensional fibres (1) from real-analytic conformally-flat Riemannian manifolds of dimension at least four (Theorem 3.1), and (2) between conformally-flat Riemannian manifolds of dimensions at least three (Corollaries 3.4 and 3.6).Also, we prove (Proposition 2.5) an integrability result for any real-analytic submersion, from a constant curvature Riemannian manifold of dimensionn+2 to a Riemannian manifold of dimension 2, which can be factorised as ann-harmonic morphism with two-dimensional fibres, to a conformally-flat Riemannian manifold, followed by a horizontally conformal submersion, (n≥4).

scholarly journals Some remarks on contact manifolds, Monge–Ampère equations and solution singularities

2014 ◽  
Vol 11 (07) ◽  
pp. 1460026 ◽  
Author(s):  
A. M. Vinogradov
Keyword(s):  
Contact Geometry ◽  
Nonlinear Pdes ◽  
Contact Manifolds ◽  
Monge Ampere

We describe some natural relations connecting contact geometry, classical Monge–Ampère equations (MAEs) and theory of singularities of solutions to nonlinear PDEs. They reveal the hidden meaning of MAEs and sheds new light on some aspects of contact geometry.

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