# Simple-layer potential

An expression

$$\tag{1 } u ( x) = \int\limits _ { S } h ( | x - y | ) f ( y) \ d \sigma ( y) ,$$

where $S$ is a closed Lyapunov surface (of class $C ^ {1 , \lambda }$, cf. Lyapunov surfaces and curves) in the Euclidean space $\mathbf R ^ {n}$, $n \geq 2$, separating $\mathbf R ^ {n}$ into an interior domain $D ^ {+}$ and an exterior domain $D ^ {-}$; $h ( | x - y | )$ is a fundamental solution of the Laplace operator:

$$h ( | x - y | ) = \ \left \{ \begin{array}{cc} \frac{1}{( n - 2 ) \omega _ {n} | x - y | ^ {n - 2 } } , & n \geq 3 ; \\ \frac{1}{2 \pi } \ \mathop{\rm ln} \frac{1}{| x - y | } , & n = 2 ; \\ \end{array} \right .$$

$\omega _ {n} = 2 \pi ^ {n/2} / \Gamma ( n / 2 )$ is the area of the unit sphere in $\mathbf R ^ {n}$; $| x - y |$ is the distance between two points $x$ and $y$; and $d \sigma ( y)$ is the area element on $S$.

If $f \in C ^ {(} 0) ( S)$, then $u$ is everywhere defined on $\mathbf R ^ {n}$. A simple-layer potential is a particular case of a Newton potential, generated by masses distributed on $S$ with surface density $f$, and with the following properties.

In $D ^ {+}$ and $D ^ {-}$ a simple-layer potential $u$ has derivatives of all orders, which can be calculated by differentiation under the integral sign, and satisfies the Laplace equation, $\Delta u = 0$, i.e. it is a harmonic function. For $n \geq 3$ it is a function regular at infinity, $u ( \infty ) = 0$. A simple-layer potential is continuous throughout $\mathbf R ^ {n}$, and $u \in C ^ {( 0 , \nu ) } ( \mathbf R ^ {n} )$ for any $\nu$, $0 < \nu < \lambda$. When passing through the surface $S$, the derivative along the outward normal $\mathbf n _ {0}$ to $S$ at a point $y _ {0} \in S$ undergoes a discontinuity. The limit values of the normal derivative from $D ^ {+}$ and $D ^ {-}$ exist, are everywhere continuous on $S$, and can be expressed, respectively, by the formula:

$$\tag{2 } \left . \lim\limits _ {x \rightarrow y _ {0} } \ \frac{du}{d \mathbf n _ {0} } \right | _ {i} = \ \frac{d u ( y _ {0} ) }{d \mathbf n _ {0} } - \frac{f ( y _ {0} ) }{2} ,\ x \in D ^ {+} ,$$

$$\left . \lim\limits _ {x \rightarrow y _ {0} } \frac{du}{d \mathbf n _ {0} } \right | _ {e} = \frac{d u ( y _ {0} ) }{d \mathbf n _ {0} } + \frac{f ( y _ {0} ) }{2} ,\ x \in D ^ {-} ,$$

where

$$\tag{3 } \frac{d u ( y _ {0} ) }{d \mathbf n _ {0} } = \ \int\limits _ { S } \frac \partial {\partial \mathbf n _ {0} } h ( | y - y _ {0} | ) f ( y) d \sigma ( y )$$

is the so-called direct value of the normal derivative of a simple-layer potential at a point $y _ {0} \in S$. Moreover, $( d u / d \mathbf n _ {0} ) ( y _ {0} ) \in C ^ {( 0 , \nu ) } ( S)$ for all $\nu$, $0 < \nu < \lambda$. If $f( y) \in C ^ {( 0 , \nu ) } ( S)$, then the partial derivatives of $u( x )$ can be continuously extended to $\overline{ {D ^ {+} }}\;$ and $\overline{ {D ^ {-} }}\;$ as functions of the classes $C ^ {( 0 , \nu ) } ( \overline{ {D ^ {+} }}\; )$ and $C ^ {( 0 , \nu ) } ( \overline{ {D ^ {-} }}\; )$, respectively. In this case one also has

$$\frac{d u }{d \mathbf n _ {0} } ( y _ {0} ) \in C ^ {( 0 , \lambda ) } ( S ) .$$

These properties can be generalized in various directions. E.g., if $f \in L _ {1} ( S)$, then $u \in L _ {1}$ inside and on $S$, formulas (2) hold almost everywhere on $S$, and the integral in (3) is summable on $S$. One has also studied properties of simple-layer potentials understood as integrals with respect to arbitrary Radon measures $\mu$ concentrated on $S$:

$$u ( x) = \int\limits h ( | x - y | ) \ d \mu ( y ) .$$

Here, also, $u$ is a harmonic function outside $S$, and formulas (2) hold almost everywhere on $S$ with respect to the Lebesgue measure, where $f ( y _ {0} )$ is replaced by the derivative $\mu ^ \prime ( y _ {0} )$ of the measure. In definition (1) one can replace the fundamental solution of the Laplace equation by an arbitrary Lewy function of a general second-order elliptic operator with variable coefficients of class $C ^ {( 0 , \lambda ) }$, replacing the normal derivative $d / d \mathbf n _ {0}$ by the derivative along the co-normal. The properties listed remain true in this case (cf. [2], [3], [4]).

Simple-layer potentials are used in solving boundary value problems for elliptic equations. The solution of a second boundary value problem with prescribed normal derivative is represented as a simple-layer potential with unknown density $f$; the use of (2) and (3) leads to a Fredholm integral equation of the second kind on $S$ for $f$( cf. [2][5]).

In solving boundary value problems for parabolic equations one uses simple-layer heat potentials, of the form

$$v ( x , t ) = \ \int\limits _ { 0 } ^ { t } \int\limits _ { S } G ( x , t ; y , \tau ) f ( y , \tau ) d \sigma ( y) d \tau ,$$

where

$$G ( x , t ; y , \tau ) = \ \frac{1}{( 2 \sqrt \pi ) ^ {n} ( t - \tau ) ^ {n/2} } \mathop{\rm exp} \ \left [ \frac{- | x - y | ^ {2} }{4 ( t - \tau ) } \right ]$$

is the fundamental solution of the heat equation in the $n$- dimensional space, and $f ( y , \tau )$ is the density. The function $v$ and its generalization to arbitrary second-order parabolic equations have properties analogous to those indicated for $u$( cf. [3], [4], [6]).

#### References

 [1] N.M. Günter, "Potential theory and its applications to basic problems of mathematical physics" , F. Ungar (1967) (Translated from Russian) [2] C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian) [3] A.N. Tikhonov, A.A. Samarskii, "Equations of mathematical physics" , Pergamon (1963) (Translated from Russian) [4] V.I. Smirnov, "A course of higher mathematics" , 4 , Addison-Wesley (1964) (Translated from Russian) [5] A. Friedman, "Partial differential equations of parabolic type" , Prentice-Hall (1964) [6] A.V. Bitsadze, "Boundary value problems for second-order elliptic equations" , North-Holland (1968) (Translated from Russian)

See [a1] for simple-layer potentials on more general open sets in $\mathbf R ^ {n}$.