UDC 517.544

MATHEMATICS

E. I. MOISEEV

POTENTIAL THEORY FOR LYAPUNOV–DINI DOMAINS

(Presented by Academician A. N. Tikhonov on 8 XII 1969)

Definition. We shall say that a surface \(S \in A_{\varphi}^{(1)}\) if \(S \in C^{(1)}\) and, for any \(x\) and \(y \in S\), \(\theta \le a\varphi(r)\), where \(\theta\) is the angle between the normals to the surface at the points \(x\) and \(y\), \(r=|x-y|\), and \(a>0\) is a fixed constant (see, for example, \((^1)\)).

B. N. Khimchenko and the author proved the following theorem:

Theorem 1. Let \(S \in A_{\varphi}^{(1)}\) and let it bound a domain \(D\). Then the Neumann problem:

\[ \Delta U=0 \quad \text{in } D, \qquad \Delta=\sum_{i=1}^{m}\frac{\partial^2}{\partial x_i^2}, \]

\[ \left.\frac{\partial U}{\partial n}\right|_{S}=0,\qquad U\in C^{(0)}(\overline D)\cap C^{(2)}(D), \]

has only the trivial solution (see \((^2)\)).

In the proof the author constructed a harmonic function \(v(x_1,\rho)\) such that
\(\left.\partial v/\partial x_1\right|_{x_1=\rho=0}>0\), while on the lateral surface of the paraboloid of revolution \(x_1=\rho\varphi(\rho)\) the function \(v(x_1,\rho)\le 0\), \(v(0,0)=0\), where
\(\rho^2=\sum_{i=2}^{m}x_i^2\); for the construction the Poisson formula for the ball was used.

Relying on this theorem, the author transferred the principal results of potential theory to surfaces of class \(A_{\varphi}^{(1)}\). The proofs of all the following theorems are carried out essentially in the same way as for Lyapunov surfaces, provided the following facts are used:

1) Let \(S \in A_{\varphi}^{(1)}\); then there exists a constant \(d>0\) such that for an arbitrary point \(x\in S\) there exists a Lyapunov sphere.

2) If at a point \(O\in S\) a local coordinate system is introduced, with the direction of the outward normal coinciding with the axis \(OX_m\), then for points of the surface \(x\in S\cap K(0,d)\), where \(K(0,d)\) is a Lyapunov sphere with center at \(O\),

\[ |\cos(\nu,OX_k)|\le \sqrt{3}\,a\varphi(r),\quad k=1,2,\ldots,m-1; \]

\(\nu\) is the normal at the point \(x\);

\[ \cos(\nu,OX_m)\ge \tfrac12;\qquad |x_m|\le ar\varphi(r),\qquad |\cos(\nu,r)|\le c(a,m)\varphi(r) \]

(see, for example, \((^3)\)).

Let

\[ W(x)=\int_S \sigma(\xi)\frac{\partial}{\partial \nu}\left(\frac{1}{r^{m-2}}\right)\,dS, \qquad \sigma(\xi)\in C^{(0)}(S),\quad r=|x-\xi|,\quad S\in A_{\varphi}^{(1)}. \]

Theorem 2. \(W(x)\) exists for \(x\in S\), is a continuous function on \(S\), and the following relations hold:

\[ W_i(x_0)=\frac{(m-2)|S_1|}{2}\sigma(x_0)+\overline{W(x_0)}, \]

\[ W_e(x_0)=-\frac{(m-2)|S_1|}{2}\sigma(x_0)+\overline{W(x_0)}, \]

where \(W_i\) and \(W_e\) are the limiting values of \(W(x)\) as \(x\to x_0\in S\), respectively from the inside and from the outside; \(\overline{W(x_0)}=W(x_0)\), \(x_0\in S\), and the convergence is uniform with respect to \(x_0\in S\).

Theorem 3. Let

\[ |\sigma(x)-\sigma(x')|\le A|x-x'|,\qquad A=\mathrm{const};\quad x,x'\in S;\quad S\in A_{\varphi}^{(1)}. \]

If the potential \(W\) has one of the normal deriv-

\[ \frac{\partial W}{\partial n_e},\quad \frac{\partial W}{\partial n_i}; \]
at the point \(x_0\in S\), it also has the other normal derivative, and
\[ \left.\frac{\partial W}{\partial n_e}=\frac{\partial W}{\partial n_i}\right|_{x=x_0}. \]

Theorem 4. Suppose that the conditions of the preceding theorem are satisfied and
\[ \left|\int_0^{2\pi}(\sigma(x)-\sigma(x_0))\,d\varphi\right|\leq a\rho\varphi(\rho), \]
where \(|x-x_0|=\sqrt{\rho^2+z^2}\) in the local coordinate system; then \(W\) has a normal derivative at the point \(x_0\).

Let
\[ V(x)=\int_S \frac{\mu(\xi)}{r^{m-2}}\,dS,\qquad r=|x-\xi|. \]

Theorem 5. If \(S\in A_{\varphi}^{(1)}\), \(\mu\in C^{(0)}(S)\), then on the surface \(S\) the simple-layer potential has the normal derivative
\[ \frac{\partial V}{\partial n_i} = \frac{(m-2)|S_1|}{2}\mu(x_0)+\frac{\overline{\partial V}}{\partial n}, \]
\[ \frac{\partial V}{\partial n_e} = -\frac{(m-2)|S_1|}{2}\mu(x_0)+\frac{\overline{\partial V}}{\partial n}, \]
where \(\partial V/\partial n_i\), \(\partial V/\partial n_e\) are the limiting values of \(\partial V/\partial n\), respectively from inside and outside \(S\), and the convergence is uniform with respect to \(x_0\in S\);
\[ \frac{\overline{\partial V}}{\partial n} = \int_S \mu(\xi)\frac{\partial}{\partial n}\left(\frac{1}{r^{m-2}}\right)dS. \]

Theorem 6. If \(S\in A_{\varphi}^{(1)}\), \(|\mu(x)-\mu(x')|\leq \psi(|x-x'|)\),
\[ \int_0^1 \frac{\psi(x)}{x}\,dx<\infty, \]
then the derivatives \(\partial V/\partial x_1,\ldots,\partial V/\partial x_m\) are uniformly continuous functions both in the interior and in the exterior domain.

Theorem 7. Let \(\overline{\partial V}/\partial n=F(x)\), \(\mu\in C^{(0)}(S)\), \(S\in A_{\varphi_1}^{(1)}\),
\[ \int_0^1 \frac{dt}{t}\int_0^t \frac{\varphi_1(x)}{x}\,dx<\infty; \]
then
\[ |F(x)-F(x')|\leq B\psi(|x-x'|),\qquad x,x'\in S; \]
\[ B=\mathrm{const},\qquad \int_0^1 \frac{\psi(x)}{x}\,dx<\infty. \]

For the limiting values \(W(x)\), \(V(x)\), when \(\mu,\sigma\in L_1(S)\), exactly the same theorems are valid as in the case of Lyapunov surfaces, if one uses the strengthened theorem of F. Riesz:

Theorem 8. If \(m_0\) is a Lebesgue point of the summable function \(\mu(x)\) on \(S\), then
\[ \int_{(m_0,\delta)}\frac{|\mu-\mu_0|}{\rho^{m-1}}\varphi(\rho)\,dS \xrightarrow[\delta\to0]{}0, \qquad (m_0,\delta)=K(m_0,\delta)\cap S. \]

It is now clear that the first and second boundary-value problems for the Laplace equation with continuous boundary data on the surface \(S\in A_{\varphi}^{(1)}\) can be reduced to integral equations, and the following integral operators are obtained:
\[ (Ku)(x)=\int_{\Omega} K(x,\xi)u(\xi)\,d\xi,\qquad \Omega\subset E_{m-1}, \]
\[ K(x,\xi)=A(x,\xi)\varphi(r)/r^{m-1},\qquad |A(x,\xi)|\leq C. \]

Theorem 9. The integral operator \(K\) is defined on the entire space \(L_2(\Omega)\) and is bounded in it; moreover, it is completely continuous in \(L_2(\Omega)\).

Theorem 10. The integral operator \(K\) is completely continuous in the space \(C^{(0)}(\Omega)\) of functions continuous in \(\Omega\), if \(A(x,\xi)\) is continuous in \(\Omega\).

Hence we immediately obtain:

Theorem 11. If \(S \in A_{\varphi}^{(1)}\), then the interior and exterior Dirichlet and Neumann problems are solvable for arbitrary continuous boundary conditions, and the solutions can be represented, respectively, in the form of double- and single-layer potentials.

Moscow State University
named after M. V. Lomonosov

Received
19 XI 1969

References

1. K.-O. Widman, Math. Scand., 21, 1, 17 (1967).
2. B. N. Khimchenko, Differ. Equations, 5, 10, 1845 (1969).
3. N. M. Günter, Potential Theory and Its Applications to Basic Problems of Mathematical Physics, Moscow, 1953.