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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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