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On functions, whose lines of steepest descent bend proportionally to level lines. (English) Zbl 0542.35007
The author investigates the geometry of solutions to special partial differential equations via the following Theorem. \(1)\quad 2| \partial \phi /\partial \bar z| \geq | \phi |^ 2\) at any point where \(\nabla u\) is not zero. \[ 2)\quad(\partial /\partial z)(| \phi |^ 2\pm i\sqrt{4| \partial \phi /\partial \bar z|^ 2-| \phi |^ 4}/\partial {\bar \phi}/\partial z)+\phi =0 \] at any point where \(\nabla u\), \(\partial {\bar \phi}/\partial z\) are not zero. In 1) and 2) u is a smooth real-valued function of two real variables x and y, \[ k=-| \nabla u|^{-3}[u^ 2yu_{xx}-2u_ yu_ xu_{xy}+u^ 2_ xu_{yy}],h=| \nabla u|^{- 3}[(u_{xx}-u_{yy})u_ xu_ y-u_{xy}(u^ 2_ x-u^ 2_ y)], \] \(\phi =k+ih\) (k is the curvature of the level lines of u and h the curvature of the lines of steepest descent of u) and \(\partial /\partial z={1\over2}(\partial /\partial x-i\partial /\partial y);\) \( \partial /\partial \bar z={1\over2}(\partial /\partial x+i\partial /\partial y) | \nabla u|^ 2=(u^ 2_ x+u^ 2_ y).\)
Reviewer: R.Salvi

MSC:
35A30 Geometric theory, characteristics, transformations in context of PDEs
35J60 Nonlinear elliptic equations
35A20 Analyticity in context of PDEs
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