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)

Comments

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

References

[a1]	J. Král, "Integral operators in potential theory" , Lect. notes in math. , 823 , Springer (1980)