# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] R. Courant - D. Hilbert , Methods of mathematical physics , Interscience , 1962 . · Zbl 0099.29504 [2] H. Federer , Geometric measure theory , Springer-Verlag , 1969 . MR 257325 | Zbl 0176.00801 · Zbl 0176.00801 [3] M. Longinetti , Sulla convessità delle linee di livello di funzioni armoniche , Boll. Un. Mat. It. , ser. 6 , vol. 2-A ( 1983 ), pp. 71 - 75 . Zbl 0525.35013 · Zbl 0525.35013 [4] F. John , Partial differential equations , Springer-Verlag , 1971 . MR 304828 | Zbl 0209.40001 · Zbl 0209.40001 [5] F. John , Formation of singularities in one-dimensional wave propagation , Comm. Pure Appl. Math. , 27 ( 1974 ), pp. 377 - 405 . MR 369934 | Zbl 0302.35064 · Zbl 0302.35064 · doi:10.1002/cpa.3160270307 [6] S. Klainerman - A. Majda , Formation of singularities for wave equation including the nonlinear vibrating string , Comm. Pure Appl. Math. , 33 ( 1980 ), pp. 241 - 263 . MR 562736 | Zbl 0443.35040 · Zbl 0443.35040 · doi:10.1002/cpa.3160330304 [7] P.D. Lax , Development of singularities of solutions of nonlinear hyperbolic partial differential equations , J. Math. Phys. , 5 ( 1964 ), pp. 611 - 613 . MR 165243 | Zbl 0135.15101 · Zbl 0135.15101 · doi:10.1063/1.1704154 [8] S.A. Levin , Nonlinear boundary problems for a quasilinear parabolic equation , J. Diff. Eq. , 5 ( 1969 ), pp. 32 - 37 . MR 233085 | Zbl 0169.13003 · Zbl 0169.13003 · doi:10.1016/0022-0396(69)90101-6 [9] G. Talenti , A note on the Gauss curvature of harmonic and minimal surfaces , Pacific J. Math. , 101 ( 1982 ), pp. 477 - 492 . Article | MR 675412 | Zbl 0496.53004 · Zbl 0496.53004 · doi:10.2140/pjm.1982.101.477 · minidml.mathdoc.fr
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.