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 $ S $ in the three-dimensional Euclidean space $ \mathbf R ^ {3} $ is called a Lyapunov surface if it satisfies the following three conditions (Lyapunov's conditions): 1) at every point of $ S $ there is a well-defined tangent plane, and consequently a well-defined normal; 2) there is a number $ r > 0 $, the same for all points of $ S $, such that if one takes the part $ \Sigma $ of $ S $ lying inside the Lyapunov sphere $ B ( y _ {0} , r ) $ with centre at an arbitrary point $ y _ {0} \in S $ and radius $ r $, then the lines parallel to the normal to $ S $ at $ y _ {0} $ meet $ \Sigma $ at most once; and 3) there are two numbers $ A > 0 $ and $ \lambda $, $ 0 < \lambda \leq 1 $, the same for the whole of $ S $, such that for any two points $ y _ {1} , y _ {2} \in S $,

$$ \tag{* } | \theta | < A | y _ {1} - y _ {2} | ^ \lambda , $$

where $ \theta $ is the angle between the normals to $ S $ at $ y _ {1} $ and $ y _ {2} $. Sometimes these three conditions are supplemented by the requirement that $ S $ is closed and that the solid angle under which any part $ \sigma $ of $ S $ is visible at an arbitrary point $ x \in \mathbf R ^ {3} $ is uniformly bounded.

The Lyapunov conditions can be generalized to hypersurfaces in $ \mathbf R ^ {n} $, $ n \geq 3 $.

Similarly, a simple continuous curve $ L $ in the plane $ \mathbf R ^ {2} $ is called a Lyapunov curve if it satisfies the following conditions: $ 1 ^ \prime $) at every point of $ L $ there is a well-defined tangent, and consequently a well-defined normal; and $ 3 ^ \prime $) there are two numbers $ A > 0 $ and $ \lambda $, $ 0 < \lambda \leq 1 $, the same for the whole of $ L $, such that for any two points $ y _ {1} , y _ {2} \in L $(*) holds, where $ \theta $ is the angle between the tangents or normals to $ L $ at $ y _ {1} $ and $ y _ {2} $. Here Lyapunov's condition 2) follows from $ 1 ^ \prime $) and $ 3 ^ \prime $). 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 $ C ^ {1} $, and on the other hand a compact surface of class $ C ^ {2} $ is a Lyapunov surface. Lyapunov surfaces are used in the study of simple- and double-layer potentials.