MATHEMATICS

JOSEF KRÁL (IOSEF KRÁL)

ON THE POTENTIAL OF A DOUBLE LAYER IN MULTIDIMENSIONAL SPACE

(Presented by Academician V. I. Smirnov on 21 IX 1964)

In the present note we consider potentials of a double layer situated on a general hypersurface of a multidimensional Euclidean space. In particular, necessary and sufficient conditions for the continuous extendability of such potentials are indicated here.

Let \(E_{n+1}\) be an oriented \((n+1)\)-dimensional Euclidean space,
\[ E=\{x;\ x\in E_{n+1},\ |x|=1\}. \]
We regard the hypersphere \(E\) as positively oriented. By the symbol \(H_p M\) we shall denote the \(p\)-dimensional Hausdorff measure of a set \(M\subset E_{n+1}\). Throughout the article \(S\) will denote a closed hypersurface in \(E_{n+1}\), on which we choose a positive orientation. (Thus, the order of points interior with respect to \(S\) is equal to one.) For simplicity we shall assume that \(H_{n+1}S=0\). If \(x\in E_{n+1}\), then on \(S\setminus\{x\}=S(x)\) we define the mapping \(P_x\) by setting
\[ P_x(y)=\frac{y-x}{|y-x|},\qquad y\in S(x). \]

For \(U\subset S\) denote by \(bU\) the boundary of the set \(U\) relative to \(S\). Let \(B(x)\) be the system of all \(U\subset S\) for which \(\overline U\subset S(x)\) and \(P_x(bU)\) is contained in a union of a finite number of hyperplanes passing through the center of the hypersphere \(E\). If \(U\in B(x)\), then for each point \(z\in E\setminus P_x(bU)\) one can define the order \(d(z,P_x,U)\) of the point \(z\) with respect to the mapping \(P_x\), considered only on \(U\) (cf. \((^5)\)); the function \(d(z,P_x,U)\) of the variable \(z\) is defined almost everywhere \((H_n)\) on \(E\) and is integrable \((H_n)\). Put
\[ m_x(U)=\int_E d(z,P_x,U)\,dH_n(z),\qquad U\in B(x). \]

The quantity \(m_x(U)\) may be regarded as a measure of the solid angle under which \(U\) is seen from the point \(x\). It is an additive set function of \(U\in B(x)\). In order that it be possible to extend it to a completely finite generalized Borel measure on \(S(x)\) (which we shall again denote by \(m_x\)), it is necessary and sufficient that
\[ v(x)=\sup \sum_{j=1}^{q}|m_x(U_j)|<\infty, \]
where the supremum is taken over all finite systems of nonintersecting sets \(U_1,\ldots,U_q\in B(x)\). From the theory of continuous mappings (see \((^7)\)) one can obtain an explicit representation for \(v(x)\). The point of intersection of the hypersurface \(S\) with the half-line
\[ L_x(z)=\{x+zr;\ r<0\}\quad (z\in E) \]
will be called essential if it possesses arbitrarily small neighborhoods \(U\) on \(S\) satisfying the conditions
\[ z\notin P_x(bU),\qquad d(z,P_x,U)\ne 0. \]
The number of all essential points of intersection of \(S\) with \(L_x(z)\), whose distance from \(x\) is less than \(R>0\), will be denoted by \(v_R(z,x)\) \((0\le v_R(z,x)\le\infty)\).

Then \(v_R(z,x)\) is a measurable \((H_n)\) function of the variable \(z\) on \(E\), whence one may set
\[ v_R(x)=\int_E v_R(z,x)\,dH_n(z). \]

If \(R\) is so large that \(S\) is contained inside the sphere of radius \(R\) with center \(x\), then \(v_R(x)=v(x)\).

If \(v(x)<\infty\), then for every bounded Baire function \(f\) on \(S\) there exists the (finite) integral
\[ W(x,f)=\int_{S(x)} f\,dm_x, \]
which it is natural to call the value at the point \(x\) of the double-layer potential with density \(f\).

Let us first clarify the relation between \(v(x)\) and the area of the hypersurface \(S\). By the area \(p(S)\) of the hypersurface \(S\) we here mean its integral-geometric area in the sense of the definition proposed in \((^2)\) (cf. \((^3)\), p. 472).

Theorem 1. Suppose that the points \(x_1,\ldots,x_{n+2}\) of the space \(E_{n+1}\) do not lie on one hyperplane. If \(v(x_1)+\cdots+v(x_{n+2})<\infty\), then \(p(S)<\infty\). Conversely, if \(p(S)<\infty\), then for every point \(x\) at distance \(\rho(x)=\rho>0\) from the hypersurface \(S\) the estimate
\[ v(x)\leqslant \rho^{-n}p(S). \]
holds.

It is seen from this that, when considering the potentials \(W(x,f)\) for \(x\in E_{n+1}\setminus S\), it is natural to require that \(p(S)<\infty\). In what follows we always assume that this condition is satisfied. For every bounded Baire function \(f\), \(W(x,f)\) is a harmonic function of the variable \(x\) on \(E_{n+1}\setminus S\). Let us proceed to consider the behavior of the potential \(W(x,f)\) near \(S\). Denote by \(D_1\) (respectively \(D_2\)) the domain interior (respectively exterior) with respect to \(S\). If \(y\in S\) and \(v(y)<\infty\), then the set \(D_k\) has a definite density \(\delta_k(y)\) at the point \(y\) \((k=1,2)\), and we put
\[ \varepsilon_1(y)=\delta_2(y)H_nE,\qquad \varepsilon_2(y)=-\delta_1(y)H_nE. \]

Theorem 2. \(y\in S\). If for every continuous function \(f\) on \(S\) the relation
\[ \infty>\limsup_{x\to y}|W(x,f)|,\quad x\in D_k, \]
holds, then there exists in \(S\) a neighborhood \(U\) of the point \(y\) such that
\[ \sup_{x\in U} v(x)<\infty . \tag{1} \]

Conversely, if for some neighborhood \(U\) of the point \(y\) in \(S\) (1) holds, then \(v(x)\) is bounded in some spatial neighborhood of the point \(y\), and for every continuous function \(f\) on \(S\) there exist the limits
\[ W_k(y,f)=\lim_{x\to y} W(x,f),\qquad x\in D_k\quad (k=1,2) \]
and the equalities
\[ W_k(y,f)=W(y,f)+\varepsilon_k(y)f(y),\qquad k=1,2. \tag{2} \]
hold.

It follows from this that the condition
\[ \sup_{y\in S} v(y)<\infty \tag{3} \]
is necessary and sufficient in order that, for every continuous function \(f\) on \(S\), the potential \(W(x,f)\) admit a continuous extension from \(D_k\) to \(\overline D_k\).

It is now possible to apply the well-known Fredholm method for solving the first boundary-value problem of potential theory. In this connection it is useful to find an expression for the Fredholm radius of the operator

\[ T_k f(y)=W_k(y,f)+\frac12(-1)^k f(y)\,H_nE, \]

acting on the space \(C(S)\) of all continuous functions \(f\) on \(S\) with norm \(\|f\|=\max_{y\in S}|f(y)|\). The reciprocal \(\omega T_k\) of this radius is defined by the equality

\[ \omega T_k=\inf_V\|T_k-V\|, \]

where the lower bound is taken over all completely continuous operators \(V\) acting on \(C(S)\).

Theorem 3. The following equality holds:

\[ \omega T_k=\lim_{R\to 0+}\sup_{y\in S} \left(v_R(y)+\left|\frac12-\delta_k(y)\right|H_nE\right). \tag{4} \]

Let us note that

\[ \left|\frac12-\delta_1(y)\right|=\left|\frac12-\delta_2(y)\right|. \]

For estimating \(v_R(y)\) one may use the somewhat simpler function

\[ \overline{v}_R(y)=\int_E \overline{v}_R(z,y)\,dH_n(z), \]

where \(\overline{v}_R(z,y)\) is the number of all points of intersection of \(S\) with the segment \(\{y+rz,\ 0<r<R\}\) \((z\in E)\).

Obviously, \(v_R(y)\leq \overline{v}_R(y)\). In the plane case \((n=1)\) equality holds here. In the general case \(n>1\) it may happen that \(v_R(y)<\overline{v}_R(y)\). If \(S\) is a curve of bounded rotation in the plane \((n=1)\), then, by Radon’s theorem, in the right-hand side of equality (4) one may omit the term \(v_R(y)\) and the sign of passage to the limit (cf. \((^6)\)). It is interesting to note that, generally speaking, this can no longer be done if \(S\) satisfies only the weaker assumption (3).

The theorems indicated are a generalization of the results of the note \((^4)\), which concerns the logarithmic potential. A communication \((^1)\) is devoted to the consideration of double-layer potentials in three-dimensional space.

Karlov University
Mathematical Institute of the Academy of Sciences of the Czechoslovak SSR,
Prague

Received
12 IX 1964

CITED LITERATURE

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\(^2\) H. Federer, Proc. Nat. Acad. Sci., 37, No. 2, 90 (1951).
\(^3\) W. H. Fleming, Illinois J. Math., 4, No. 3, 452 (1960).
\(^4\) J. Král, Comm. Math. Univ. Carolinae, 3, No. 1, 3 (1962).
\(^5\) J. H. Michael, Proc. London Math. Soc., 7, No. 3, 616 (1957).
\(^6\) F. Riesz, B. Sz. Nagy, Leçons d’Analyse Fonctionnelle, Budapest, 1952.
\(^7\) T. Radó, P. V. Reichelderfer, Continuous Transformations in Analysis, Berlin, 1955.