Reports of the Academy of Sciences of the USSR

1. Volume 159, No. 6

MATHEMATICS

V. G. MAZ’YA, V. D. SAPOZHNIKOVA

SOLUTION OF THE DIRICHLET AND NEUMANN PROBLEMS FOR IRREGULAR DOMAINS BY METHODS OF POTENTIAL THEORY

(Presented by Academician V. I. Smirnov on 28 V 1964)

In the paper \((^1)\) the results of Radon \((^2)\) concerning the double-layer potential were generalized. In the present note a generalization of Radon’s theory is given in the part concerning the single-layer potential and the solvability of the Dirichlet and Neumann problems.

Below we shall use the notation and concepts introduced in \((^1)\). Let us only recall that an open domain \(\Omega \subset E^3\), bounded by a surface \(\Gamma\), is considered. By \(\omega(P,\mathcal E)\) we denote the solid angle under which the set \(\mathcal E \subset \Gamma\) is seen from the point \(P \in E^3\), where, by definition,
\[ \omega(P,P)=2\pi-\omega(P,\Gamma\setminus P). \]

As in \((^1)\), it is assumed that the absolute variation of the function of the set \(\omega(P,\mathcal E)\) is bounded by a constant independent of \(P\) (condition (A)).

We note that condition (A) is necessary in the following sense: if the surface \(\Gamma\) is quadrable* and there exist limiting values from within and from without of the double-layer potential with any continuous density, then this condition is satisfied, and moreover
\[ \left| \sup_{P\in E^3\setminus \Gamma}\operatorname{var}_{\mathcal E\subset\Gamma}\omega(P,\mathcal E) - \sup_{P\in\Gamma}\operatorname{var}_{\mathcal E\subset\Gamma}\omega(P,\mathcal E) \right|\leq 2\pi. \]

\(1^\circ\). The single-layer potential and its boundary flux.

The single-layer potential with charge \(\Phi(\mathcal E)\), where \(\Phi(\mathcal E)\) is a completely additive function of Borel sets on \(\Gamma\), is the integral
\[ V(P)=\frac{1}{2\pi}\int_{\Gamma}\frac{1}{r_{PX}}\,\Phi(dX),\qquad P\notin\Gamma. \]

It is obvious that \(V(P)\) is a harmonic function for \(P\notin\Gamma\).

Let us define the interior boundary flux of a function \(u(P)\) harmonic in \(\Omega\). Let \(\mathcal K(E^3)\) be the space of infinitely differentiable functions with compact supports in \(E^3\). Suppose that for any function \(\varphi\in\mathcal K(E^3)\) and for any sequence of domains tending to \(\Omega\), bounded by smooth surfaces \(\Gamma_m\subset\Omega\), there exists the limit
\[ l_u(\varphi)=\lim_{m\to\infty}\int_{\Gamma_m}\varphi(X)\frac{\partial u(X)}{\partial \nu}\,s(dX), \]
where \(\nu\) is the exterior normal to \(\Gamma_m\). Suppose, in addition, that the functional \(l_u(\varphi)\), defined on \(\mathcal K(E^3)\), is bounded in \(C(\Gamma)\). In this case the extension of \(l_u(\varphi)\) to \(C(\Gamma)\) can be represented by means of a completely additive set function \(\Sigma^{(i)}(\mathcal E)\) in the form
\[ \int_{\Gamma}\varphi(X)\,\Sigma^{(i)}(dX). \]

* The quadrability of \(\Gamma\) makes it possible to define the solid angle \(\omega(P,\mathcal E)\), and together with it also the double-layer potential \(W(P)\), for \(P\notin\Gamma\).

We shall call this function of sets \(\Sigma^{(i)}(\mathscr E)\) the interior boundary flux of the function \(u(P)\). The exterior flux \(\Sigma^{(e)}(\mathscr E)\) of the function \(u(P)\), harmonic in \(C\overline{\Omega}\), is defined analogously.

The assumptions made are in fact satisfied for the simple-layer potential; more precisely, the following holds:

Theorem 1. A simple-layer potential with charge \(\Phi(\mathscr E)\) has interior and exterior boundary fluxes, defined by the equalities
\[ \Sigma^{(i)}(\mathscr E) = -\Phi(\mathscr E) + \frac{1}{2\pi}\int_{\Gamma}\omega(X,\mathscr E)\,\Phi(dX), \]
\[ \Sigma^{(e)}(\mathscr E) = \Phi(\mathscr E) + \frac{1}{2\pi}\int_{\Gamma}\omega(X,\mathscr E)\,\Phi(dX). \]

In studying the solvability of boundary-value problems, the following properties of the simple-layer potential are used.

Lemma 1. Let \(V(P)\) be a simple-layer potential with absolutely continuous charge
\[ \Phi(\mathscr E)=\int_{\mathscr E} a(X)\,s(dX), \]
where \(a(X)\) is a bounded Borel-measurable function. Then at every point \(P\in\Gamma\) the limiting values of \(V(P)\) from outside and from inside \(\Omega\) exist and are equal.

Lemma 2. If the interior and exterior limiting values of the potential \(V(P)\) on \(\Gamma\) exist and are equal, then \(V(P)\in \mathscr L_{2}^{(1)}(E^{3})\), where \(\mathscr L_{2}^{(1)}(E^{3})\) is the closure of \(\mathscr K(E^{3})\) in the metric of the Dirichlet integral, and the equality
\[ \int_{E^{3}}|\operatorname{grad} V(X)|^{2}\,dX = \frac{1}{\pi}\int_{\Gamma}V(S)\,\Phi(dS) \]
holds.

2°. Solution of the Dirichlet and Neumann problems. As shown in (1), the interior and exterior Dirichlet problems reduce, respectively, to the equations
\[ W^{(i)}=f+Tf, \tag{D\(^{(i)}\)} \]
\[ W^{(e)}=-f+Tf, \tag{D\(^{(e)}\)} \]
where the operator
\[ (Tf)_S=\frac{1}{2\pi}\int_{\Gamma} f(X)\,\omega(S,dX) \]
acts in the space \(C(\Gamma)\)*.

The interior (exterior) Neumann problem is posed as follows: find a function \(u(P)\), harmonic in \(\Omega\) (in \(C\overline{\Omega}\)), whose interior (exterior) boundary flux exists and coincides with a prescribed completely additive function of sets on \(\Gamma\).

Thus the interior and exterior Neumann problems reduce to the functional equations
\[ \Sigma^{(i)}=-\Phi+T^{*}\Phi, \tag{N\(^{(i)}\)} \]
\[ \Sigma^{(e)}=\Phi+T^{*}\Phi, \tag{N\(^{(e)}\)} \]
where \(T^{*}\) is the operator adjoint to \(T\).

\[ \text{* It can be shown that the operator }T\text{ is also bounded in the space of functions defined on }\Gamma \]
satisfying a Lipschitz condition of order \(\alpha\in(0,1)\) in the sense of the metric \(E^{3}\).

Introduce the notation: \(\Gamma_\varepsilon(S)=\{X;\ r_{XS}\leqslant\varepsilon\}\cap\Gamma\). Let the surface \(\Gamma\) be such that

\[ \lim_{\varepsilon\to 0}\sup_{S\in\Gamma}\operatorname{Var}_{\mathscr{E}\subset\Gamma_\varepsilon(S)}\omega(S,\mathscr{E})<2\pi \]

(condition (B)).

If condition (B) is satisfied, then the Fredholm radius of the operator \(T\) is greater than one. Therefore, if the surface \(\Gamma\) satisfies conditions (A) and (B), then for equations \((D^{(i)})\), \((N^{(e)})\) (respectively, \((D^{(e)})\), \((N^{(i)})\)), the Fredholm alternative is valid*.

In the proof of uniqueness the following is used.

Lemma 3. If \(\Gamma\) satisfies conditions (A) and (B), then the solutions of the homogeneous equations \((N^{(i)})\) and \((N^{(e)})\) generate simple-layer potentials continuous in \(E^3\).

Following the classical scheme, we obtain the final result.

Theorem 2. If the surface \(\Gamma\) satisfies conditions (A) and (B), then: 1) the interior Dirichlet problem is solvable for any continuous function \(W^{(i)}(S)\), and the exterior Neumann problem is solvable for any absolutely additive function \(\Sigma^{(e)}(\mathscr{E})\) of Borel sets; 2) the exterior Dirichlet problem is solvable for any continuous boundary function \(W^{(e)}(S)\), and the interior Neumann problem is solvable for any absolutely additive function \(\Sigma^{(i)}(\mathscr{E})\) with zero total mass.

The authors express their sincere gratitude to N. D. Burago for discussion of the work.

Leningrad State University
named after A. A. Zhdanov

Received
4 V 1964

CITED LITERATURE

\(^1\) B. D. Burago, V. G. Maz’ya, V. D. Sapozhnikova, DAN, 147, No. 3, (1962). \(^2\) J. Radon, UMN, 1, issue 3–4 (1946). \(^3\) N. Dunford, J. T. Schwartz, Linear Operators, part 1, IL, 1962, p. 528.

* We take this occasion to note that the expression for the Fredholm radius given in \((^1)\) is, generally speaking, incorrect.