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|^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|^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|^r), see ((^3)).