A system of fifth-order partial differential equations describing a surface which contains many circles. (English) Zbl 1294.53008
In this work the authors consider a germ \(z=f(x,y)\) of a \(C^5\)-surface at the origin in \(\mathbb{R}^3\) containing several continuous families of circular arcs and introduce a system of fifth-order nonlinear partial differential equations for \(f\). They prove that this system describes such a surface germ completely. As applications, they obtain the analyticity of \(f,\) the finite dimensionality of the solution space of such a system of differential equations with an upper estimate of 21 for the dimension and a local characterization of Darboux cyclides by using this system of equations.
Reviewer: Esther Sanabria (Valencia)
MSC:
| 53A05 | Surfaces in Euclidean and related spaces |
| 35G50 | Systems of nonlinear higher-order PDEs |
| 35J62 | Quasilinear elliptic equations |
References:
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