Abstract
Full Text
Mathematics
S. M. NIKOLSKII
ON A VARIATIONAL PROBLEM OF HILBERT
(Presented by Academician M. A. Lavrentiev, 8 VI 1957)
D. Hilbert, in one of his papers \((^1)\), considered the following problem.
In the complex plane \(R\) (or on a Riemann surface with a finite number of branch points and a finite number of sheets) a closed curve \(C\) is given, and a class \(\mathfrak M\) of functions \(f\) possessing the following properties:
-
The function \(f \in \mathfrak M\) is defined, continuous, and piecewise analytic on \(R - C\).
-
The quantity
\[ D[f]=\iint_R \left[\left(\frac{\partial f}{\partial x}\right)^2+ \left(\frac{\partial f}{\partial y}\right)^2\right]\,dx\,dy<\infty \]
is finite.
-
On passing from one side of the curve \(C\) to the other, the function \(f\) undergoes a jump equal to one.
-
The mean value of \(f\) over some segment \(\gamma\), not intersecting \(C\), is equal to zero.
Hilbert proves, restricting himself to considering as the curve \(C\) a closed polygon with sides parallel to the axes \(x\) and \(y\) (not passing, in the case where \(R\) is a Riemann surface, through the branch points of \(R\)), that in the class \(\mathfrak M\) there exists a function \(u\) for which the minimum is attained
\[ \min_{f\in\mathfrak M} D[f] = D[u]. \tag{1} \]
Here \(u\) is a harmonic function on \(R-C\), regular at infinity.
The results formulated below show that similar facts also hold in \(n\)-dimensional space (we do not have in mind, however, any generalizations of a Riemann surface). In addition, we investigate the extremal function \(u\) and give for it another equivalent definition in terms not connected with the variational problem.
Let \(g\) be a domain of the real space \(R_n\), bounded by a twice continuously differentiable surface \(\Lambda\). Define on \(R_n-\Lambda\) a function \(f\) having generalized (in the sense of Sobolev) first-order derivatives satisfying the condition
\[ D[f]=\int_g \sum_1^n \left(\frac{\partial f}{\partial x_i}\right)^2\,dg<\infty. \tag{2} \]
For such a function, for almost all points \(Q\in\Lambda\) there exist the limits
\[ \lim_{P\to +Q} f(P)=f_+(Q), \qquad \lim_{P\to -Q} f(P)=f_-(Q) \]
respectively from the side of the inner (positive) and outer (negative) normals to \(\Lambda\).
Denote by \(H\) the set of functions \(\varphi=f_+\), where \(f\) is an arbitrarily prescribed function on \(R_n-\Lambda\) satisfying condition (2). Also denote by \(\Lambda_\delta\) and \(\Lambda_{-\delta}\) the surfaces at distance \(\delta>0\) from \(\Lambda\) along the normals to \(\Lambda\), respectively on the positive and negative sides. For sufficiently small \(\delta\), these surfaces have a meaning under the conditions imposed on \(\Lambda\).
The layer \(V_\delta\in R_n\), bounded by the surfaces \(\Lambda_\delta\) and \(\Lambda_{-\delta}\), consists of points \(P\) whose position is completely determined by a pair \((Q,h)\), where \(Q\in\Lambda\) is the point on the normal to \(\Lambda\) on which \(P\) lies, and \(h\) is a number equal to the distance from \(P\) to \(Q\), taken with the corresponding sign.
It can be proved that a function \(f\) satisfying condition (2) can be modified on a set of \(n\)-dimensional measure zero, and after this
\[ \int_{\Lambda} [f(Q,\delta)-f_+(Q)]^2\,d\Lambda=o(\delta)\qquad(\delta\to0,\ \delta>0). \]
Here \(f_+\in L_2(\Lambda)\), i.e. it has an integrable square on \(\Lambda\).
Let \(a(Q)\ne0\) and \(b(Q)\) be continuously differentiable functions prescribed on \(\Lambda\), and let \(\varkappa(Q)\in H\), and let \(\gamma\) be an \((n-1)\)-dimensional parallelepiped lying in \(R_n\) but outside \(\Lambda\).
By definition, a function \(f\) belongs to the class \(\mathfrak M\) if the following properties hold for it:
1) The function \(f\) is defined on \(R_n-\Lambda\), has generalized derivatives there, and inequality (2) is satisfied.
2) On \(\Lambda\) one has
\[
af_+-bf_-=\varkappa.
\]
3)
\[
\int_{\gamma} f\,d\gamma=0.
\]
If \(\varkappa\equiv0\) on \(\Lambda\), then the class \(\mathfrak M\) will be denoted by \(\mathfrak M_0\).
Theorem 1. Among the functions \(f\) of the class \(\mathfrak M\) there exists, moreover a unique, function \(u\) for which the minimum of the variational problem (1) is attained. The function \(u\) is harmonic on \(R_n-\Lambda\).
The following assertions are equivalent:
a) the function \(u\) satisfies the conditions of Theorem 1;
b) the function \(u\in\mathfrak M\) and for it one has
\[ D[u,f]=\int_R \sum_1^n \frac{\partial u}{\partial x_i}\frac{\partial f}{\partial x_i}\,dR=0 \]
for all \(f\in\mathfrak M_0\);
c) the function \(u\in\mathfrak M_0\) is harmonic on \(R_n-\Lambda\) and for all \(\varphi\in H\)
\[ \lim_{\delta\to0}\int_{\Lambda}\left\{b\,\frac{\partial u}{\partial n}(Q,\delta)-a\,\frac{\partial u}{\partial n}(Q,-\delta)\right\}\varphi(Q)\,d\Lambda=0, \]
\[ \int_{\sigma_\rho}\left(\frac{\partial u}{\partial \rho}\right)^2\,d\sigma_\rho=O(\rho^{1-n})\qquad(\rho\to\infty), \]
where \(\sigma_\rho\subset R\) is the surface of radius \(\rho\) with center at the origin of coordinates;
d) for \(n=2\), the function \(u\), harmonic on \(R_n-\Lambda\), belongs to \(\mathfrak M\), and for any conjugate function \(v\) to it, for almost all \(s\) (\(s\) is the length of the arc \(\Lambda\)) there exist limiting values \(v_+\) and \(v_-\) on \(\Lambda\), satisf—
satisfying the equalities
\[ \int_0^l \mu(s)\,ds=0,\qquad \nu(s)+\int_0^s \mu(t)\,dt+\lambda_0=0, \]
where
\[ \mu(s)=v_-a'-v_+b',\qquad \nu(s)=v_-a-v_+b, \]
and \(\lambda_0\) is a constant depending on \(v\).
In the two-dimensional case, the variational problem considered above is equivalent to a special (particular) Hilbert problem in the theory of functions of a complex variable, well studied by means of the apparatus of singular integral equations \({}^{2}\).
Let us note several lemmas on which the proof of the stated assertions was based.
Lemma 1. For every ball \(\omega \subset R\) there exists a constant \(C_\omega\) such that, for all \(f\in \mathfrak M_0\),
\[ \|f\|_{L_2(\omega)}=\left(\int_\omega |f|^2\,d\omega\right)^{1/2}<C_\omega\sqrt{D[f]}. \]
Lemma 2. For a function \(f\) with finite \(D[f]\) the estimates
\[ \int_{\sigma_\rho} f^2\,d\sigma_\rho < \begin{cases} C\sqrt{\rho\ln\rho}, & \text{for } n=2,\\ C\sqrt{\rho^{\,n-1}}, & \text{for } n>2, \end{cases} \qquad (\rho\ge 1) \]
hold, where \(\sigma_\rho\) is the surface of the ball of radius \(\rho\) with center at the origin, and \(C\) does not depend on \(\rho\).
Lemma 3. Let \(u(P)\) be a continuous function defined in a domain \(g\) with boundary \(\Lambda\), satisfying the conditions stated above. Moreover,
\[ \int_g u^2\,dg=M^2 \]
and there exists a constant \(K\) such that, if \(S\) is a spherical surface with center at an arbitrary point \(P\in g\), entirely contained in \(g\), then the inequality
\[ |u(P)|\le \frac{K}{|S|}\int_{|S|}|u|\,dS \]
holds, where \(|S|\) is the area of \(S\). Then there exists a constant \(C_g\) such that, for sufficiently small \(\delta>0\),
\[ \int_\Lambda u^2(Q;h)\,d\Lambda<C_gKM^2h^{-1}\qquad (0<h<\delta). \]
In conclusion I wish to express my gratitude to A. V. Bitsadze, whose general considerations on Hilbert’s problem, expressed by him in our conversation, were very valuable to me.
Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR
Received
22 V 1957
CITED LITERATURE
\({}^{1}\) D. Hilbert, Math. Ann., 59, 161 (1904); D. Hilbert, Gesammelte Abhandlungen, 3, 1935, pp. 15–27.
\({}^{2}\) N. I. Muskhelishvili, Singular Integral Equations, 1946.