MATHEMATICS

N. I. MOZHEROVA

BOUNDARY PROPERTIES OF HARMONIC FUNCTIONS IN THREE-DIMENSIONAL SPACE

(Presented by Academician S. L. Sobolev on 31 V 1957)

In three-dimensional space \(x, y, z\) we consider a finite domain \(D\), bounded by a sufficiently smooth surface \(S\). For this domain the Dirichlet and Neumann problems are solved. We investigate the question: if the differential properties of the function \(f\) are known, then what differential properties are possessed by the functions \(u\) and \(v\), which are solutions of the Dirichlet problem

\[ \Delta u = 0,\qquad u\big|_S = f \]

or of the Neumann problem

\[ \Delta v = 0,\qquad \left.\frac{\partial v}{\partial n}\right|_S = f. \]

The first investigations in this direction belong to Lichtenstein \((^8)\) and A. M. Lyapunov. Lyapunov’s results were developed in the works of N. M. Günter and Kh. L. Smolitskii \((^1)\); the indicated authors proved the following theorems:

Theorem A. If the surface \(S\) is sufficiently smooth and \(f\) is bounded and has bounded continuous derivatives up to order \(l\) \((l \ge 0)\), and the derivatives of order \(l\) are properly continuous with exponent \(\lambda\) \((0 < \lambda \le 1)\), then the solution of the Dirichlet problem is bounded and has bounded continuous derivatives up to order \(l\), and the derivatives of order \(l\) are properly continuous with exponent \(\lambda'\), where \(0 < \lambda' < \lambda \le 1\).

Theorem B. If the surface \(S\) is sufficiently smooth and \(f\) is bounded and has bounded continuous derivatives up to order \(l\) \((l \ge 0)\), and the derivatives of order \(l\) are properly continuous with exponent \(\lambda\) \((0 < \lambda \le 1)\), then the solution of the Neumann problem is bounded, has bounded continuous derivatives up to order \(l+1\), and the \((l+1)\)-st derivatives are properly continuous with exponent \(\lambda'\) \((0 < \lambda' < \lambda \le 1)\).

We extend the results formulated to the case of the metric \(L_p\) \((1 \le p \le \infty)\) in terms of the classes \(W_p^{(r)}H^{(\alpha)}(M;D)\), in particular somewhat strengthening the case \(p=\infty\).

Definition 1. We shall say \((^3)\) that a measurable function \(f(x_1,x_2,\ldots,x_n)\), defined on the whole space \(R_n\), belongs to the class \(H_{px_1}^{(r)}(M)\), \(r>0\), \(1 \le p \le \infty\), if it satisfies the following condition. Represent \(r\) in the form \(r=\bar r+\alpha\), where \(\bar r\) is an integer, \(0<\alpha\le 1\). The function \(f\) is integrable with the \(p\)-th power on \(R_n\), together with its Sobolev-generalized \((^4)\) partial derivatives with respect to \(x_1\) up to order \(\bar r\) inclusive. In addition, the derivative \(\partial^{\bar r} f/\partial x_1^{\bar r}\) satisfies the inequality

\[ \left\| \frac{\partial^{\bar r} f(x_1+h,\ldots,x_n)}{\partial x_1^{\bar r}} -2\frac{\partial^{\bar r} f(x_1,\ldots,x_n)}{\partial x_1^{\bar r}} +\frac{\partial^{\bar r} f(x_1-h,\ldots,x_n)}{\partial x_1^{\bar r}} \right\|_{L_p} \le M |h|^{\alpha} \tag{1} \]

or

\[ \left\| \frac{\partial^{\bar r} f(x_2+h, x_2,\ldots,x_n)}{\partial x_1^{\bar r}} - \frac{\partial^{\bar r} f(x_1,x_2,\ldots,x_n)}{\partial x_1^{\bar r}} \right\|_{L_p} \leq M |h|^\alpha,\qquad 0<\alpha<1. \tag{2} \]

If the function belongs to \(H_{p x_i}^{(r)}\) for all \(i=1,\ldots,n\), then we shall say that it belongs to \(H_p^{(r)}(M)\).

Definition 2. Let \(D\) be a domain of the \(n\)-dimensional space \(R_n\) of points \((x_1,x_2,\ldots,x_n)\), and let \(D'\) be a domain whose closure belongs to \(D\). We shall call an \(n\)-dimensional vector \(h\) admissible, translating \(D'\) within \(D\), if all the vectors \(th\), where \(0\leq t\leq 1\), translate \(D'\) within \(D\). Let \(r\) be a nonnegative integer, \(0<\alpha\leq 1\), and \(M\) a positive constant. We shall say that a function \(f\), defined on \(D\), belongs to the class \(W_p^{(r)}H_p^{(r)}(M;D)\) if it is integrable in the \(p\)-th power together with its generalized derivatives (mixed and unmixed) up to order \(r\) inclusive, and if for every partial derivative of order \(r\), \(f^{(r)}\) (mixed or unmixed), the inequalities

\[ \|f^{(r)}(m+h)-f^{(r)}(m)\|_{L_p} = \]

\[ = \left(\int_{(D')} |f^{(r)}(m+h)-f^{(r)}(m)|^p\,dm\right)^{1/p} \leq M |h|^\alpha,\qquad 0<\alpha<1; \tag{3} \]

\[ \|f^{(r)}(m+h)-2f^{(r)}(m)+f^{(r)}(m-h)\|_{L_p} = \]

\[ = \left(\int_{(D')} |f^{(r)}(m+h)-2f^{(r)}(m)+f^{(r)}(m-h)|^p\,dm\right)^{1/p} \leq M |h|,\qquad \alpha=1, \tag{4} \]

hold for all \(D'\in D\) and all admissible \(h\). We shall agree to consider that, for \(0<\varepsilon<1\),
\[ W_p^{(r)}H_p^{(1+\varepsilon)}(D;M)=W_p^{(r+1)}H_p^{(\varepsilon)}(D;M). \]
The classes \(W_p^{(r)}H_p^{(\alpha)}(M;S)\) of functions given on a surface \(S\) are defined analogously. We shall not define them for the sake of brevity, but we note that in this case, when \(\alpha=1\), as well as when \(\alpha<1\), the definition of functions of the class \(W_p^{(r)}H^{(1)}(M;S)\) includes an inequality analogous to (3).

Theorem 1. Let
\[ S\in W_\infty^{(r+2)}H^{(\alpha)}(M_1),\qquad f\in W_p^{(r)}H_p^{(\alpha-1/p,\ \alpha-1/p)}(M;S), \]
\(M>0,\ M_1>0,\ 0<\alpha-1/p\leq 1,\ r\) an integer, \(1\leq p\leq\infty\). Then the harmonic function solving the Dirichlet problem
\[ \Delta u=0,\qquad u|_S=f, \]
belongs to the class
\[ W_p^{(\bar r,\bar r,\bar r)}H_p^{(\alpha,\alpha,\alpha)}(\bar M;D), \]
where
\[ \bar M\leq c\{M+\|f\|_{L_p(S)}\}, \]
and \(c\) depends on \(D\) and \(p\), but does not depend on \(M\) and \(\|f\|_{L_p(S)}\).

Theorem 2. Let
\[ S\in W_\infty^{(r+2)}H^{(\alpha)}(M_1),\qquad f\in W_p^{(r)}H_p^{(\alpha-1/p,\ \alpha-1/p)}(M;S), \]
\(M>0,\ 0<\alpha-1/p\leq 1,\ r\) an integer, \(1\leq p\leq\infty\). Then the harmonic function solving the Neumann problem
\[ \Delta v=0,\qquad \left.\frac{\partial v}{\partial n}\right|_S=f, \]
belongs to the class
\[ W_p^{(\bar r+1,\bar r+1,\bar r+1)}H_p^{(\alpha,\alpha,\alpha)}(\bar M;D), \]
where
\[ \bar M\leq c\{M+\|f\|_{L_p(S)}\}, \]
and \(c\) depends on \(D,p,\alpha\), but does not depend on \(M\) and \(\|f\|_{L_p(S)}\).

Let us note the works of S. M. Nikol’skii \((^5)\), T. I. Amanov \((^6)\), and O. V. Besov \((^7)\), in which analogous results were obtained in the case of the circle and the half-space. In the case \(p=\infty\) and when the boundary function \(f\) is properly continuous with exponent \(\alpha=1\), our result somewhat strengthens the corresponding result of Lyapunov, Günter, and Smolitskii.

Next, the solution of the Poisson equation is investigated, and the following theorem is established:

Theorem 3. If \(S \in W_{\infty}^{(r+2)}H^{(\alpha)}\), \(f \in W_{p}^{(\bar r,\bar r,\bar r)}H^{(\alpha,\alpha,\alpha)}(M; D)\), then \(u(x,y,z)\), the solution of the equation \(\Delta u=f\) under the conditions \(u|_{S}=0\) or \(\partial u/\partial n|_{S}=0\), belongs to the class \(W_{p}^{(\bar r+2,\bar r+2,\bar r+2)}H^{(\alpha,\alpha,\alpha)}(\overline M; D)\), where
\[ \overline M < c\{\|f\|_{L_p(D)}+M\}, \]
and \(c\) depends on \(D\), \(p\), \(\alpha\), but does not depend on \(M\) or \(\|f\|_{L_p(D)}\).

For \(p=\infty\) this theorem is proved in N. M. Günter’s book \((^{1})\).

As was said above, the results of the present article are proved for domains in three-dimensional space, but it is not difficult to extend them to the \(n\)-dimensional case. In obtaining them we mainly followed the method set forth in the book \((^{1})\), the method of potential theory.

Let us note that our results for \(\alpha\ne 1\), \(\alpha\ne 1/p\) (simultaneously) cannot be strengthened, as follows from the embedding theorem of S. M. Nikol’skii \((^{3})\) and from the circumstance (proved by S. M. Nikol’skii) that, for \(\alpha\ne 1\), the classes \(W_{p}^{(r)}H_{p}^{(\alpha)}(S)\) and \(H_{p}^{(r+\alpha)}(S)\) coincide, and any function \(f\in W_{p}^{(r)}H_{p}^{(\alpha)}(g)\) can be extended beyond \(g\) in such a way that the extended function \(\bar f\in H_{p}^{(r+\alpha)}\) \((^{2})\).

All-Union Institute
of Scientific and Technical Information

Received
28 IX 1956

CITED LITERATURE

¹ N. M. Günter, Potential Theory and Its Application to the Basic Problems of Mathematical Physics, Moscow, 1953.
² S. M. Nikol’skii, DAN, 88, No. 2, 213 (1953).
³ S. M. Nikol’skii, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 38, 244 (1951).
⁴ S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, Leningrad, 1950.
⁵ S. M. Nikol’skii, DAN, 83, No. 1, 23 (1952).
⁶ T. I. Amanov, DAN, 88, No. 1 (1953).
⁷ O. V. Besov, Izv. AN SSSR, ser. matem., 20, No. 4 (1956).
⁸ L. Lichtenstein, Encyklopädie der math. Wissenschaften, 2, 3, 1921.