# 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)

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.