A system of fifth-order partial differential equations describing a surface which contains many circles
A system of fifth-order partial differential equations describing a surface which contains many circles
复制标题
描述包含许多圆的表面的五阶偏微分方程组
DOI：
10.1016/j.bulsci.2012.09.002
复制
发表时间：
2013
期刊：
影响因子：
--
通讯作者：
Kiyoomi Kataoka
中科院分区：
文献类型：
--
作者：
Genki Matsuda;Shizuo Kaji and Hiroyiki Ochiai;T. Ishiwata;石渡 哲哉;J. Akahori;Hiroyuki Ochiai and Ken Anjyo;石渡 哲哉;S. Kondo and A. Tani;Jiro Akahori;Nobushige Kurokawa and Hiroyuki Ochiai;H. Honda and A. Tani;Hiroyuki Ochiai;Jiro Akahori;Nobushige Kurokawa and Hiroyuki Ochiai;H. Honda and A. Tani;Jiro Akahori;Hiroyuki Ochiai;Kiyoomi Kataoka
设z=f(x，y)是R3中包含多个连续圆弧族的原点处的C5曲面的芽。例如，一个有4个这样的族的普通环面和有6个这样的族的Blum Cycles，这是Darboux Cycldes的特例。我们引入了一个关于f的五阶非线性偏微分方程组，并证明了这个方程组完全描述了这样一个曲面胚芽。作为应用，我们得到了这类微分方程组的解空间的有限维f的解析性，其维度的上限估计为21。此外，在我们即将发表的文章中，我们利用这一方程组得到了Darboux圈的一些局部刻画：K.Kataoka，N.Takeuchi，描述包含6个圆族的曲面的一些五阶偏微分方程组的不可积性，Rims Kokyuroku Bessatsu京都大学，出版社[1]。
Let z=f(x,y) be a germ of a C5-surface at the origin in R3containing several continuous families of circular arcs. For examples, a usual torus with 4 such families and Blum cyclides with 6 such families, which are special cases of Darboux cyclides. We introduce a system of fifth-order nonlinear partial differential equations for f, and prove that this system describes such a surface germ completely. As applications, we obtain the analyticity of f, the finite dimensionality of the solution space of such a system of differential equations with an upper estimate 21 for the dimension. Further we obtained some local characterization of Darboux cyclides by using this system of equations in our forthcoming paper: K. Kataoka, N. Takeuchi, The non-integrability of some system of fifth-order partial differential equations describing surfaces containing 6 families of circles, RIMS Kokyuroku Bessatsu Kyoto University, in press [1].