A. S. Fokht

ON AN ESTIMATE NEAR THE BOUNDARY OF A DOMAIN FOR A POLYHARMONIC FUNCTION AND ITS DERIVATIVES GIVEN ON A DISK

(Presented by Academician A. A. Dorodnitsyn, 19 V 1962)

In the literature \((^1)\) there are estimates for the growth of solutions of elliptic equations of the second order near the boundary of the domain on which they are defined, obtained in the \(C\) metric on the basis, chiefly, of the study of the corresponding Green’s function. S. M. Nikol’skii \((^2)\), by another method, obtained estimates, sharp in order, for the growth of a harmonic function and its derivatives near the boundary of a domain in the sense of \(L_p\). The purpose of the present paper is to obtain analogous estimates in the \(L_2\) metric for a polyharmonic function of arbitrary order on a disk (two-dimensional ball).

§ 1. Let \(u\) be an \(l\)-harmonic function in a disk of radius \(R\), i.e., let it satisfy the equation:

\[ \Delta^l u = 0; \tag{1,1} \]

\[ I=\int_0^R \int_0^{2\pi} u^2 \rho\, d\rho\, d\theta < +\infty; \tag{1,2} \]

\[ I_\rho^{(0)}=I_\rho=\int_0^{2\pi} u^2 \rho\, d\theta; \tag{1,3} \]

\[ I_\rho^{(q)}=\int_0^{2\pi} u(q)^2 \rho\, d\theta, \tag{1,4} \]

where \(q>0\) is an integer; \(u^{(q)}\) is any mixed partial derivative of the function \(u\) of order \(q\).

In the present paper the inequality

\[ I_\rho^{(q)} \leqslant \frac{C_{l,q}}{(R-\rho)^{2q+1}}\, I, \tag{1,5} \]

will be proved; it is sharp in order with respect to \(R-\rho\) \((0<\rho<R)\); \(C_{l,q}\) is a constant depending only on \(l\) and \(q\) and not depending on \(R\).

§ 2. We shall prove inequality (1,5) for \(R=1\). Then, by the change of variable \(\rho=Rr\), we shall obtain (1,5) for arbitrary \(R>0\). Let \(u(\rho,\theta)\) be an even \(l\)-harmonic function on the unit disk \((0\leq \rho\leq 1)\). The assumption of evenness does not affect the generality of the argument. Then, as is known, \(u\) will have the form

\[ u=\sum_{n=0}^{\infty}\left[a_{1n}+a_{2n}\rho^2+\cdots+a_{ln}\rho^{2(l-1)}\right]\rho^n \cos n\theta, \tag{2,1} \]

where \(a_{in}\) \((i=1,2,\ldots,l)\) are linear combinations of the Fourier coefficients of the boundary functions, and

\[ I=\int_0^1\int_0^{2\pi} u^2 \rho\, d\rho\, d\theta =\pi \sum_{n=0}^{\infty}\Phi_n, \tag{2,2} \]

where

\[ \Phi_n=2\int_0^1 [a_{1n}+a_{2n}\rho^2+\ldots+a_{ln}\rho^{2(l-1)}]^2\rho^{2n+1}\,d\rho . \tag{2,3} \]

In what follows we shall compare terms of the series for one and the same \(n\), and therefore, for brevity, instead of \(a_{in}\) we shall write \(a_i\) \((i=1,2,\ldots,l)\). We have:

\[ \Phi_n=\frac{A_1^2}{n+1} +\sum_{k=2}^{l} \frac{A_k^2[(k-1)!]^2} {(n+k)^2(n+k+1)^2\ldots(n+2k-2)^2(n+2k-1)}, \tag{2,3'} \]

where

\[ A_k=\sum_{i=k}^{l} \frac{(i-1)!}{(i-k)!(k-1)!}\, a_i\, \frac{(n+k)\ldots(n+2k-1)} {(n+i)\ldots(n+i+k-1)} \quad (1\leq k\leq l). \tag{2,4} \]

Consider also

\[ I_\rho=\int_0^{2\pi} u^2\rho\,d\theta =\pi\sum_{n=0}^{\infty} F_n\rho^{2n+1} \quad (0\leq \rho<1), \tag{2,5} \]

where

\[ F_n=[a_1+a_2\rho^2+\ldots+a_l\rho^{2(l-1)}]^2. \tag{2,6} \]

The inequality holds

\[ F_n\leq \sum_{k=1}^{l}2^k(1-\rho^2)^{2k-2}B_k^2, \tag{2,7} \]

where

\[ B_k\leq \sum_{i=k}^{l} a_i\,\frac{(i-1)!}{(k-1)!(i-k)!}. \tag{2,8} \]

Further:

\[ B_k^2\leq 2A_k^2+ \sum_{i=k}^{l-k} B_{k+i}^2\,2^{1+i} \frac{i^4(i+1)^4\ldots(i+k-1)^4} {(i!)^2(n+2k)^2(n+2k+1)^2\ldots(n+2k+i-1)^2}, \tag{2,9} \]

where the \(A_k\) are defined by formula (2,4).

From (2,9) it follows that

\[ B_k^2\leq 2A_k^2+ \sum_{i=k+1}^{l} \frac{A_i^2} {(n+2k)^2\ldots[n+2k+(i-k-1)]^2}\, \alpha_{k,i}, \tag{2,10} \]

where all \(\alpha_{k,i}\) are positive integers independent of the number \(n\).

Consider the auxiliary function

\[ \Psi_k(\rho)=(1-\rho)^k\rho^{2n+1} \quad (0\leq \rho\leq 1); \tag{2,11} \]

\(k,n\) are integers, \(n\geq 0\), \(k\geq 1\).

The inequality holds:

\[ \Psi_k(\rho)(1+\rho)^{k-1} < \frac{L_k}{(n+1)^k}, \tag{2,12} \]

where

\[ L_k=\frac{2^{k-1}k^k(1+k)^k}{e^k}. \tag{2,13} \]

Further, starting from (2.5) and taking into account (2.7), (2.10), and (2.12), we have

\[ F_n \rho^{2n+1}(1-\rho) \leqslant \sum_{k=1}^{l}\frac{A_k^2\beta_k}{(n+1)^{2k-1}}, \tag{2.14} \]

where \(\beta_k>0\) are bounded numbers independent of \(n\), of the form

\[ \beta_k=2^k L_{2k-1}\alpha_{k,k}+2^{k-1}L_{2k-3}\alpha_{k-1,k}+\cdots+2L_1\alpha_{1,k} \tag{2.15} \]

\[ (k=1,2,\ldots,l). \]

From (2.14) and (2.3) we find:

\[ C_l=C_{l,0}=\beta_l\,\frac{(2l-1)[(2l-2)!]^2}{[(l-1)!]^4}. \tag{2.16} \]

Thus, inequality (1.5) for \(q=0\) and \(R=1\) is proved.

§ 3. For \(R=1\)

\[ I_{\rho}^{(q)}=\int_{0}^{2\pi}\left(\frac{\partial^q u}{\partial \rho^q}\right)^2 \rho\,d\theta =\pi\sum_{n=q}^{\infty} W_{qn}\rho^{2(n-q)+1}n^2(n-1)^2\cdots(n-q+1)^2, \tag{3.1} \]

where (for brevity we shall write simply \(W_q\))

\[ W_q=\left[\sum_{k=1}^{l}a_k\rho^{2k-2} \frac{(n+2k-2)\cdots(n+2k-q-1)} {n(n-1)\cdots(n-q+1)}\right]^2. \tag{3.2} \]

The inequality holds

\[ D_k^2\leqslant \sum_{i=k}^{l} B_i^2\left(C_{i-1}^{k-1}\right)^2\cdot 2^{i-k+1}(1-\rho^2)^{2(i-k)}, \tag{3.3} \]

where \(B_i\) are defined by formula (2.8), and

\[ D_k=\sum_{i=k}^{l}a_i\rho^{2i-2k}C_{i-1}^{k-1}. \tag{3.4} \]

Further, the inequality holds

\[ W_q\leqslant \sum_{i=0}^{q}\frac{\gamma_{i+1}}{(n+1)^{2i}}\,D_{i+1}^2, \tag{3.5} \]

where \(\gamma_{i+1}\) are positive constants independent of \(n\).

Further, by virtue of (3.5), (3.3), and (2.12),

\[ W_q(1-\rho)^{2q+1}n^2(n-1)^2\cdots(n-q+1)^2 \leqslant \sum_{k=1}^{l}\frac{A_k^2\delta_{k,q}}{(n+1)^{2k-1}}, \tag{3.6} \]

where the constants \(\delta_{k,q}>0\) do not depend on \(n\).

Solving the inequality

\[ \sum_{k=1}^{l}A_k^2\frac{\delta_{k,q}}{(n+1)^{2k-1}} \leqslant C_{l,q}\Phi_n, \tag{3.7} \]

where \(\Phi_n\) are defined by formula (2.3), we find that

\[ C_{l,q}=\delta_{l,q}\,\frac{[(2l-2)!]^2}{[(l-1)!]^4}. \tag{3.8} \]

Thus, inequality (1.5) is proved for \(u^{(q)}=\partial^q u/\partial \rho^q\). For \(u^{(q)}=\partial^q u/\partial \theta^i\partial \rho^{q-i}\), \((i=1,2,\ldots,q)\), inequality (1.5) is obtained by the same method. By the change of variable \(R\rho=r\) we obtain inequality (1.5) in the case of arbitrary \(R\).

§ 4. From formula (3.8) it follows that

\[ \lim_{l\to\infty} C_{l,q}=+\infty . \tag{4.1} \]

But in the course of proving relation (1.5), roughened inequalities may have been introduced. However, one can effectively indicate a sequence \(l=l(N)\) of harmonic functions \((N=1,2,\ldots)\), where \(l(N)\to\infty\) as \(N\to\infty\), and such a positive constant \(A\), independent of \(N\), that for

\[ \rho=1-\frac{A}{N+1} \]

the inequality

\[ \frac{l_\rho^{(0)}(1-\rho)}{I}>N, \tag{4.2} \]

holds, whence, obviously, (4.1) follows.

Received
14 V 1962

CITED LITERATURE

¹ K. Miranda, Partial Differential Equations of Elliptic Type, IL, 1957. ² S. M. Nikol’skii, Siberian Mathematical Journal, 1, No. 1 (1960).