Simple-layer potential

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An expression

 (1)

where is a closed Lyapunov surface (of class , cf. Lyapunov surfaces and curves) in the Euclidean space , , separating into an interior domain and an exterior domain ; is a fundamental solution of the Laplace operator:

is the area of the unit sphere in ; is the distance between two points and ; and is the area element on .

If , then is everywhere defined on . A simple-layer potential is a particular case of a Newton potential, generated by masses distributed on with surface density , and with the following properties.

In and a simple-layer potential has derivatives of all orders, which can be calculated by differentiation under the integral sign, and satisfies the Laplace equation, , i.e. it is a harmonic function. For it is a function regular at infinity, . A simple-layer potential is continuous throughout , and for any , . When passing through the surface , the derivative along the outward normal to at a point undergoes a discontinuity. The limit values of the normal derivative from and exist, are everywhere continuous on , and can be expressed, respectively, by the formula:

 (2)

where

 (3)

is the so-called direct value of the normal derivative of a simple-layer potential at a point . Moreover, for all , . If , then the partial derivatives of can be continuously extended to and as functions of the classes and , respectively. In this case one also has

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

Here, also, is a harmonic function outside , and formulas (2) hold almost everywhere on with respect to the Lebesgue measure, where is replaced by the derivative 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 , replacing the normal derivative 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 ; the use of (2) and (3) leads to a Fredholm integral equation of the second kind on for (cf. [2][5]).

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

where

is the fundamental solution of the heat equation in the -dimensional space, and is the density. The function and its generalization to arbitrary second-order parabolic equations have properties analogous to those indicated for (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)