MATHEMATICS

T. I. AMANOV

ON THE SOLUTION OF THE BIHARMONIC PROBLEM

(Presented by Academician S. L. Sobolev on 26 X 1956)

1. Let \(r>0\), \(M>0\), and \(r=\bar r+\alpha\), where \(\alpha\) is an integer and \(0<\alpha\leqslant 1\). We shall say (as in [1]) that a measurable function \(f(\theta)\) of period \(2\pi\) belongs to the class \(H_2^{(r)}(M)\) if it: a) has a square summable over the period; b) has an absolutely continuous derivative of order \((\bar r-1)\); c) the derivative of order \(\bar r\), which exists by virtue of b) almost everywhere, has a square summable over the period and satisfies, for all \(h\), the condition

\[ \left(\int_0^\pi \left| f^{(\bar r)}(\theta+h)-f^{(\bar r)}(\theta)\right|^2\,d\theta\right)^{1/2} \leqslant M |h|^\alpha,\quad \text{if } 0<\alpha<1; \]

\[ \left(\int_0^{2\pi} \left| f^{(\bar r)}(\theta+h)-2f^{(\bar r)}(\theta)+f^{(\bar r)}(\theta-h)\right|^2\,d\theta\right)^{1/2} \leqslant M |h|,\quad \text{if } \alpha=1. \]

1. Let \(\sigma\) be the open unit disk with center at the origin. We shall say that a function \(f(x,y)\), measurable on \(\sigma\), belongs to the class \(W_2^{(m)}(\sigma)\) if: a) it has on \(\sigma\) all generalized, in the sense of S. L. Sobolev [2], partial derivatives up to order \(m\) inclusive; b) \(f\) itself and all these derivatives have squares summable on \(\sigma\), i.e.

\[ D_0[f]\equiv \iint_\sigma f^2(x,y)\,dx\,dy<\infty, \]

\[ D_j[f]=\iint_\sigma \sum_{\alpha+\beta=j}\frac{j!}{\alpha!\beta!} \left(\frac{\partial^j f}{\partial x^\alpha \partial y^\beta}\right)^2 \,dx\,dy<\infty \quad (j=1,2,\ldots,m). \]

Let us note that, in passing to polar coordinates, the classical formula for change of variables in the integrals \(D_j[f]\) remains valid also in the case of generalized derivatives. This is easily proved by introducing the function \(f_h(x,y)\), the mean of \(f(x,y)\) in the sense of S. L. Sobolev [2].

In what follows, instead of \(\lim_{\rho\to 1-0}F(\rho,\theta)=\varphi(\theta)\) almost everywhere, we shall write \(F_{\rho=1}=\varphi\).

1. In this note the following theorems will be proved, established for \(m=2\) in [3].

Theorem 1. Let \(u(\rho,\theta)\) be biharmonic in \(\sigma\), belonging to the class \(W_2^{(m)}(\sigma)\) \((m\geqslant 2)\); \(u|_{\rho=1}=\varphi(\theta)\); \(\left.\dfrac{\partial u}{\partial \rho}\right|_{\rho=1}=\psi(\theta)\). Then

\[ \varphi\in H_2^{(m-\frac12)}(M_1),\quad \psi(\theta)\in H_2^{(m-\frac32)}(M_2), \]

where \(M_1, M_2\) are completely determined constants.

Theorem 2. Let

\[ \varphi(\theta)\in H_{2}^{(m-1)/2+\varepsilon_{1}}(M_{1}),\qquad \psi(\theta)\in H_{2}^{(m-3)/2+\varepsilon_{2}}(M_{2}), \tag{1} \]

where \(0\leqslant \varepsilon_{1}<1/2,\ 0\leqslant \varepsilon_{2}<1/2\). If \(u(\rho,\theta)\) is a biharmonic function in \(\sigma\) satisfying the boundary conditions

\[ u\big|_{\rho=1}=\varphi,\qquad \frac{\partial u}{\partial \rho}\bigg|_{\rho=1}=\psi, \]

then

a) for \(\varepsilon_{1}>0,\ \varepsilon_{2}>0\)

\[ u\in W_{2}^{(m)}(\sigma); \]

\[ D_m[u]\leqslant \begin{cases} C_{1}\left[\dfrac{M_{1}}{\sqrt{\varepsilon_{1}}}+ \dfrac{M_{2}+\|\psi\|_{L_2}}{\sqrt{\varepsilon_{2}}}\right]^2, & m=2,\\[1.2em] C_{2}\left[\dfrac{M_{1}}{\sqrt{\varepsilon_{1}}}+ \dfrac{M_{2}}{\sqrt{\varepsilon_{2}}}\right]^2, & m>2, \end{cases} \tag{2} \]

where \(C_{1}, C_{2}\) are constants independent of \(\varepsilon_{1},\varepsilon_{2},M_{1},M_{2}\);

b) for \(\varepsilon_{1}=\varepsilon_{2}=0\), among functions belonging to the classes \(H_{2}^{*(m-1)/2}\) and \(H_{2}^{*(m-3)/2}\), but not belonging to \(H_{2}^{m'}\) and \(H_{2}^{m''}\), where \(m'>m-1/2,\ m''>m-3/2\), there exist both functions for which the corresponding integral \(D_m[u]<\infty\), and functions for which \(D_m[u]=\infty\).

1. Consider, for given \(\varepsilon_{1}>0,\ \varepsilon_{2}>0\), the classes of functions \(\varphi\) and \(\psi\) satisfying conditions (1). Denote by \(M_{\varepsilon_{1}}(\varphi)\) and \(M_{\varepsilon_{2}}(\psi)\) the corresponding least constants for which conditions (1) are satisfied. These classes form spaces of type \((B)\), respectively with metrics \(\|\varphi\|_{\varepsilon_{1}}^{*}=M_{\varepsilon_{1}}(\varphi)\), \(\|\psi\|_{\varepsilon_{2}}^{*}=M_{\varepsilon_{2}}(\psi)\) (for \(m=2\), \(\|\psi\|_{\varepsilon_{2}}^{*}=M_{\varepsilon_{2}}(\psi)+\|\psi\|_{L_2}\)). Denote these spaces by \(H_{\varepsilon_{1}}\) and \(H_{\varepsilon_{2}}\). Then Theorem 3 follows from Theorem 2.

Theorem 3. Let \(u(\rho,\theta)\) be a biharmonic function in \(\sigma\) satisfying the boundary conditions

\[ u\big|_{\rho=1}=\varphi\in H_{\varepsilon_{1}},\qquad \frac{\partial u}{\partial \rho}\bigg|_{\rho=1}=\psi\in H_{\varepsilon_{2}}. \]

Then

\[ D_m[u]\leqslant C\left[ \dfrac{\|\varphi\|_{\varepsilon_{1}}^{*}}{\sqrt{\varepsilon_{1}}} + \dfrac{\|\psi\|_{\varepsilon_{2}}^{*}}{\sqrt{\varepsilon_{2}}} \right]^2, \tag{3} \]

where \(C\) is a positive constant independent of \(\varepsilon_{1},\varepsilon_{2}\).

Thus, the solution of the biharmonic problem is stable in the following sense: if \(\varphi\) and \(\psi\) are arbitrarily small in norm, respectively in the metrics of \(H_{\varepsilon_{1}}\) and \(H_{\varepsilon_{2}}\), then for the biharmonic function \(u(\rho,\theta)\) in \(\sigma\) satisfying the boundary conditions

\[ u\big|_{\rho=1}=\varphi,\qquad \frac{\partial u}{\partial \rho}\bigg|_{\rho=1}=\psi, \]

the integral \(D_m[u]\) will be arbitrarily small.

1. Theorems 1 and 2 can be obtained with the aid of certain auxiliary propositions, starting from Theorems 1 and 2 of S. M. Nikol’skii \((^{4})\). However, in this case the integral \(D_m[u]\) is estimated from above with another constant, in which the quantities \(\varepsilon_{1}\) and \(\varepsilon_{2}\) appearing in the denominator of (2) are not singled out. Theorem 1 can also be obtained by combining the theorems of S. L. Sobolev and V. I. Kondrashov \((^{2})\), if one takes into account that a function of the class \(W_{2}^{(m)}(\sigma)\) can be extended beyond \(\sigma\) with preservation of the class.

2. In what follows we shall need Lemmas 1 and 2.

Lemma 1 \((^{5})\). Let the function \(u(x,y)\) have sufficiently many continuous partial derivatives in \(\sigma\). If \(x+iy=\rho e^{i\theta}\), then

\[ \frac{\partial^{n}u}{\partial x^{\alpha}\partial y^{\beta}} = \sum_{j=1}^{n}\sum_{s+t=j} \frac{P_{\alpha,\beta}^{(s,t)}(\theta)}{\rho^{\,n-j+s}} \frac{\partial^{j}u}{\partial \rho^{t}\partial \theta^{s}}, \tag{4} \]

\[ S_n[u]\equiv \sum_{\alpha+\beta=n}\frac{n!}{\alpha!\beta!} \left(\frac{\partial^{n}u}{\partial x^{\alpha}\partial y^{\beta}}\right)^2 \geqslant \left(\frac{\partial^{n}u}{\partial \rho^{n}}\right)^2. \tag{5} \]

where \(P_{\alpha,\beta}^{(s,t)}(\theta)\) are trigonometric polynomials of order \(\alpha+\beta=n\), independent of \(u\).

Lemma 2. Let the \(2\pi\)-periodic function

\[ f(\theta)=\left(\frac{\gamma_0}{2}+\sum_{k=1}^{\infty}\gamma_k\cos k\theta+\eta_k\sin k\theta\right)\in H_2^{(n-\frac12+\varepsilon)}(M), \]

where \(n\geqslant 1,\ 0\leqslant \varepsilon<1/2\). Then: a) if \(\varepsilon>0\), then

\[ \sigma_n(f)\equiv\sum_{k=1}^{\infty} k^{2n-1}\left(\gamma_k^2+\eta_k^2\right)\leqslant \frac{\pi M^2}{2}; \tag{6} \]

b) if \(\varepsilon=0\), then among the boundary functions of the class \(H_2^{(n-\frac12)}\) there are both functions for which \(\sigma_n(f)<\infty\), and functions for which \(\sigma_n(f)=\infty\) (for the definition of boundary functions see (6)); c) if \(\sigma_n(f)<\infty\), then

\[ f\in H_2^{(n-\frac12)}\left(\sqrt{2\pi\sigma_n(f)}\right). \tag{7} \]

Proof. a) By hypothesis

\[ \left(\int_0^{2\pi}\left|f^{(n-1)}(\theta+h)-f^{(n-1)}(\theta)\right|^2\,d\theta\right)^{1/2} \leqslant M|h|^{1/2+\varepsilon}, \]

which is equivalent to the inequality

\[ \sum_{k=1}^{\infty} k^{2(n-1)}\left(\gamma_k^2+\eta_k^2\right)\sin^2 \frac12 kh \leqslant \frac{M^2}{4\pi}|h|^{1+2\varepsilon}, \]

whence, by virtue of the inequality

\[ \int_0^1\left(\frac{\sin^2 \frac12 kh}{kh}\right)^2\,dh\geqslant \frac{1}{k\pi^2} \]

(6) follows.

b) The functions

\[ f_1(\theta)=\sum_{k=2}^{\infty}\frac{\cos(k\theta+(n-1)\pi/2)}{k^n\ln^{(1+\alpha)/2}k}, \qquad f_2(\theta)=\sum_{\nu=1}^{\infty}\frac{\cos N^{2\nu}(\theta)}{N^{(2n-1)\nu}}, \]

where \(0<\alpha<1/2,\ N>1\) is an integer, are boundary functions in the class \(H_2^{(n-\frac12)}\), and for them \(\sigma_n(f_1)<\infty,\ \sigma_n(f_2)=\infty\).

c) follows from the identity

\[ \left(\int_0^{2\pi}\left|f^{(n-1)}(\theta+h)-f^{(n-1)}(\theta)\right|^2\,d\theta\right)^{1/2} = \left(4\pi\sum_{k=1}^{\infty} k^{2(n-1)}\left(\gamma_k^2+\eta_k^2\right)\sin^2\frac12 kh\right)^{1/2}. \]

7. Proof of Theorem 1. Let \(u\) be a biharmonic in \(\sigma\) function belonging to \(W_2^{(m)}(\sigma)\). As shown in note \((^3)\),

\[ u(\rho,\theta)=\frac{(\rho^2-1)a_0+2a_0}{4}+ \]

\[ +\sum_{k=1}^{\infty}\rho^k\left\{\left[(\rho^2-1)\frac{\alpha_k-ka_k}{2}+a_k\right]\cos k\theta+ \left[(\rho^2-1)\frac{\beta_k-kb_k}{2}+b_k\right]\sin k\theta\right\}, \tag{8} \]

where \(a_k,b_k\) and \(\alpha_k,\beta_k\) are the Fourier coefficients of the functions \(\varphi=u|_{\rho=1}\) and \(\psi=u_\rho|_{\rho=1}\).

From (8) we find that

\[ \iint \left(\frac{\partial^m u}{\partial \rho^m}\right)^2 \rho\, d\rho\, d\theta \geq C_3 \sum_{k=m}^{\infty} k^{2m-1}(a_k^2+b_k^2) + k^{2m-3}(\alpha_k^2+\beta_k^2), \]

which, with the aid of (5) and (7), proves Theorem 1.

Proof of Theorem 2. a) Let \(0<\varepsilon_1<1/2,\ 0<\varepsilon_2<1/2\). Then

\[ \sum k^{2m-1}(a_k^2+b_k^2)+k^{2m-3}(\alpha_k^2+\beta_k^2)<\infty. \]

Put

\[ u_\rho=\frac{(\rho^2-1)a_0+2a_0}{4}+ \]

\[ +\sum_{k=1}^{\mu}\rho^k \left\{ \left[(\rho^2-1)\frac{a_k-ka_k}{2}+a_k\right]\cos k\theta + \left[(\rho^2-1)\frac{\beta_k+kb_k}{2}+b_k\right]\sin k\theta \right\}. \]

Then

\[ D_j[u_{\mu+\nu}-u_\mu]\leq d_j\sum_{\mu+1}^{\mu+\nu} k^{2m-1}(a_k^2+b_k^2) + k^{2m-3}(\alpha_k^2+\beta_k^2) \underset{\mu\to\infty}{\longrightarrow}0 \]

uniformly with respect to \(\nu\) \((j=0,1,\ldots,m)\).

By virtue of Lemma 4 ((1), p. 260), it follows from this that \(u\in W_2^{(m)}(\sigma)\). Since

\[ D_2[u]\leq C_4\left(a_0^2+\sum_{k=1}^{\infty} k^3(a_k^2+b_k^2)+k(\alpha_k^2+\beta_k^2)\right), \]

\[ D_m[u]\leq C_5\sum_{k=m-2} k^{2m-1}(a_k^2+b_k^2)+k^{2m-3}(\alpha_k^2+\beta_k^2), \]

where \(C_4\) and \(C_5\) are positive constants independent of \(\varepsilon_1,\varepsilon_2,M_1,M_2\), it follows, by virtue of (6), that (2) holds.

a) follows from part b) of Lemma 2.

Semipalatinsk State
Pedagogical Institute

Received
29 V 1956

REFERENCES

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3. T. I. Amanov, DAN, 88, No. 3, 389 (1953).
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