MATHEMATICS

G. V. ZHIDKOV

BOUNDARY PROPERTIES OF DIFFERENTIABLE AND HARMONIC FUNCTIONS IN DOMAINS WITH ANGULAR POINTS

(Presented by Academician S. L. Sobolev, 27 VI 1957)

Consider in space a conical domain \(G\), formed by rotating the ray \(r\) about the axis \(x\). Rectangular coordinates are related to spherical ones by the relations

\[ z = r \sin \theta \sin \varphi, \qquad y = r \sin \theta \cos \varphi, \qquad x = r \cos \theta, \]

\[ 0 < r < \infty, \qquad 0 < \theta \leq \pi, \qquad 0 < \varphi \leq 2\pi . \]

On the boundary \(\Gamma\) of the domain \(G\) a function is given,

\[ \psi(\rho,\varphi) = \frac{1}{\sqrt{r}} f(\rho,\varphi), \qquad r = e^\rho . \]

Let us construct a harmonic function \(V(\rho,\varphi,\theta)\) inside the domain \(G\). The Dirichlet integral in the coordinates \((\rho,\varphi,\theta)\) for the function \(V(\rho,\varphi,\theta)\) has the form

\[ D[V] = \iiint_G \left[ \left(\frac{\partial V}{\partial \rho}\right)^2 + \frac{1}{\sin^2 \theta} \left(\frac{\partial V}{\partial \varphi}\right)^2 + \left(\frac{\partial V}{\partial \theta}\right)^2 \right] e^\rho \sin \theta \, d\rho \, d\varphi \, d\theta . \]

If the domain \(G\) is a cone bounded by two concentric spheres of radii \(r=1\) and \(r=e\), then the harmonic function has the form

\[ V(\rho,\varphi,\theta) = \frac{1}{\sqrt{e^\rho}} \sum_{k=0}^{\infty} \sum_{m=0}^{\infty} \sin k\pi \rho \left[ A_{km}\cos m\varphi + B_{km}\sin m\varphi \right] \frac{K_k^m(\cos \theta)}{K_k^m(\cos \theta_0)}, \tag{1} \]

where \(0 \leq \rho \leq 1\), \(0 < \theta \leq \theta_0\), \(0 < \varphi \leq 2\pi\); \(K_k^m(\cos \theta)\) are conical functions (*). \(A_{km}\) and \(B_{km}\) are the Fourier coefficients of the function

\[ f(\rho,\varphi) \sim \sum_{k=0}^{\infty} \sum_{m=0}^{\infty} \sin k\pi \rho \left[ A_{km}\cos m\varphi + B_{km}\sin m\varphi \right]. \]

In the case where the domain \(G\) is a full cone with vertex at the origin, symmetric with respect to the axis \(x\), the harmonic function \(V(\rho,\varphi,\theta)\) is representable by the expression

\[ V(\rho,\varphi,\theta) = \frac{1}{2\pi \sqrt{e^\rho}} \sum_{m=0}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \cos k(\rho-\rho') \left[ a_m(\rho')\cos m\varphi + \right. \]

\[ \left. {}+ b_m(\rho')\sin m\varphi \right] \frac{K_k^m(\cos \theta)}{K_k^m(\cos \theta_0)} \, dk \, d\rho' . \tag{2} \]

where \(-\infty<\rho<\infty\); \(0<\theta<\theta_0\); \(0<\varphi<2\pi\). Let \(A_m(k)\) and \(B_m(k)\) denote the Fourier transforms of the coefficients \(a_m(\rho)\) and \(b_m(\rho)\) of the function

\[ f(\rho,\varphi)\sim \sum_{m=0}^{\infty}\{a_m(\rho)\cos m\varphi+b_m(\rho)\sin m\varphi\}. \]

Theorem 1. If a function \(V(\rho,\varphi,\theta)\), harmonic in the domain \(G\), satisfies the conditions:

\[ 1)\quad \iint_{\Gamma}|V(\rho,\varphi,\theta)|^2\,d\Gamma \leqslant N \quad \text{for all } \theta<\theta_0; \]

\[ 2)\quad D[V]<\infty, \]

then there exists a boundary function \(\psi(\rho,\varphi)\), summable in \(L_2\), for which

\[ \lim_{\theta\to\theta_0}\iint_{\Gamma}|V(\rho,\varphi,\theta)-\psi(\rho,\varphi)|^2\,d\Gamma=0 \]

and the condition

\[ \iint_{\Gamma}|\psi(\rho+h_1,\varphi+h_2)-\psi(\rho-h_1,\varphi-h_2)|^2\,d\Gamma \leqslant M\sum_{k=1}^{2}h_k^{1+\varepsilon} \]

is fulfilled, where \(\Gamma\) is the boundary of the domain \(G\); \(\varepsilon=0\); \(M\) is a constant independent of \(h_k\).

Theorem 2. If the boundary function \(\psi(\rho,\varphi)\) is summable in \(L_2\) and satisfies the condition

\[ \iint_{\Gamma}|\psi(\rho+h_1,\varphi+h_2)-\psi(\rho-h_1,\varphi-h_2)|^2\,d\Gamma \leqslant M\sum_{k=1}^{2}h_k^{1+\varepsilon}, \tag{3} \]

where \(\varepsilon>0\), then the harmonic function in the domain \(G\) corresponding to it has the following properties:

\[ 1)\quad V(\rho,\varphi,\theta)\ \text{is harmonic in }G\text{ for }\theta<\theta_0; \]

\[ 2)\quad \iint_{\Gamma}|V(\rho,\varphi,\theta)|^2\,d\Gamma \leqslant N \quad \text{for } \theta<\theta_0; \]

\[ 3)\quad \lim_{\theta\to\theta_0}\iint_{\Gamma}|V(\rho,\varphi,\theta)-\psi(\rho,\varphi)|^2\,d\Gamma=0; \]

\[ 4)\quad D[V]<\infty. \]

Theorem 3. If the boundary function \(\psi(\rho,\varphi)\) is summable in \(L_2\) and satisfies the condition

\[ \iint_{\Gamma}|\psi(\rho+h_1,\varphi+h_2)-\psi(\rho-h_1,\varphi-h_2)|^2\,d\Gamma \leqslant M_1\sum_{k=1}^{2}h_k^{2\alpha}, \]

then, for the corresponding harmonic function \(V(\rho,\varphi,\theta)\) in the domain \(G\),

\[ \iiint_{G}|V(\rho+h_1,\varphi+h_2,\theta+h_3)-V(\rho-h_1,\varphi-h_2,\theta-h_3)|^2\,dG \leqslant M_2\sum_{k=1}^{3}h_k^{2\alpha+1}, \]

where \(M_1\) and \(M_2\) are constants independent of \(h_k\).

In the proof of Theorems 1 and 2 it is shown that, if \(V(\rho,\varphi,\theta)\) is representable by expression (1), then the finiteness of \(D[V]\) is equivalent to the convergence of the series

\[ \sum_{k=1}^{\infty}\sum_{m=1}^{\infty}(k+m)(A_{km}^{2}+B_{km}^{2}). \tag{4} \]

From condition (3) of Theorem 2, the convergence of the series (4) is proved. If \(V(\rho,\varphi,\theta)\) is represented by formula (2), then the finiteness of \(D[V]\) is equivalent to the convergence of the series

\[ \sum_{m=0}^{\infty}\int_{-\infty}^{\infty}(m+k)'\,[A_m^2(k)+B_m^2(k)]\,dk. \tag{5} \]

It is shown that, from condition (3) of Theorem 2, in this case the convergence of the series (5) follows.

In the proof of Theorem 3, the idea of the proof of a theorem of S. N. Bernstein \((^4)\) is used.

Consider the domain \(G:\ 0\leq x,y,z\leq \pi\). A function \(f(x,y,z)\in H_P^r(G)\) induces on the boundary \(\Gamma\) of the domain \(G\) the function \(f|_\Gamma=\varphi\). In this case the following theorems are proved:

Theorem 4. If \(0<r-1/P<1\) and the function \(f\) belongs to the class \(H_P^r(G)\), then the boundary function \(\varphi\in H_P^{r-1/P}(\overline M)\), where \(\overline M<C\|f\|_{L_P(G)}^r\); \(C\) is a constant independent of \(\|f\|_{L_P(G)}^r\).

Theorem 5. If \(0<r-1/P<1\) and on the boundary \(\Gamma\) a function \(\varphi\) is given which belongs to the class \(H_P^{r-1/P}(M)\), then there exists a function \(f\) of the class \(H_P^r(G)\), defined in the domain \(G\), such that \(f|_\Gamma=\varphi\),

\[ \|f\|_{L_P(G)}^r<C\|\varphi\|_{L_P(\Gamma)}^r. \]

Under the imposition of a continuity condition on the even derivatives of the function \(\varphi\) at the corner points, Theorems 4 and 5 are proved in the case \(r-1/P>1^*\).

Moscow State University
named after M. V. Lomonosov

Received
25 VI 1957

REFERENCES

\({}^1\) E. Hobson, Trans. Cambr. Phil. Soc., 14, 211 (1889);
\({}^2\) F. G. Mehler, Math. Ann., 18, 161 (1881).
\({}^3\) S. M. Nikolsky, Matem. sborn., 33 (75), 2 (1953); 35; (77), 2 (1954).
\({}^4\) A. Zygmund, Trigonometric Series, 1939, p. 138.
\({}^5\) E. Titchmarsh, Introduction to the Theory of Fourier Integrals, 1948.
\({}^6\) S. L. Sobolev, Some Applications of Functional Analysis, L., 1950.

* For the definition of \(H_P^r(G)\), \(\|f\|_{L_P(G)}^r\), \(\|\varphi\|_{L_P(\Gamma)}^r\), see \((^3)\).