Lyapunov surfaces and curves
A class of surfaces and curves that have quite good smoothness properties; it was introduced in potential theory by A.M. Lyapunov at the turn into the 20th century.
A surface 
in the three-dimensional Euclidean space 
is called a Lyapunov surface if it satisfies the following three conditions (Lyapunov's conditions): 1) at every point of 
there is a well-defined tangent plane, and consequently a well-defined normal; 2) there is a number 
, the same for all points of 
, such that if one takes the part 
of 
lying inside the Lyapunov sphere 
with centre at an arbitrary point 
and radius 
, then the lines parallel to the normal to 
at 
meet 
at most once; and 3) there are two numbers 
and 
, 
, the same for the whole of 
, such that for any two points 
,
 | (*) |
where 
is the angle between the normals to 
at 
and 
. Sometimes these three conditions are supplemented by the requirement that 
is closed and that the solid angle under which any part 
of 
is visible at an arbitrary point 
is uniformly bounded.
The Lyapunov conditions can be generalized to hypersurfaces in 
, 
.
Similarly, a simple continuous curve 
in the plane 
is called a Lyapunov curve if it satisfies the following conditions: 
) at every point of 
there is a well-defined tangent, and consequently a well-defined normal; and 
) there are two numbers 
and 
, 
, the same for the whole of 
, such that for any two points 
(*) holds, where 
is the angle between the tangents or normals to 
at 
and 
. Here Lyapunov's condition 2) follows from 
) and 
). The Lyapunov curves are a subclass of the simple smooth curves.
References
| [1] | A.M. Lyapunov, "On certain questions connected with the Dirichlet problem" , Collected works , 1 , Moscow (1954) pp. 45–47; 48–100 (In Russian) |
| [2] | S.L. Sobolev, "Partial differential equations of mathematical physics" , Pergamon (1964) (Translated from Russian) |
| [3] | V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) pp. Chapt. 5 (Translated from Russian) |
| [4] | N.I. Muskhelishvili, "Singular integral equations" , Wolters-Noordhoff (1972) pp. Chapt. 1 (Translated from Russian) |
Comments
A Lyapunov surface is necessarily 
, and on the other hand a compact surface of class 
is a Lyapunov surface. Lyapunov surfaces are used in the study of simple- and double-layer potentials.
Lyapunov surfaces and curves. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lyapunov_surfaces_and_curves&oldid=47731