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Reports of the Academy of Sciences of the USSR
- Volume 148, No. 5
MATHEMATICS
S. M. NIKOLSKII
UNIQUENESS OF THE SOLUTION OF A BOUNDARY-VALUE PROBLEM FOR A CONVEX DOMAIN
(Presented by Academician I. N. Vekua on 15 VIII 1962)
The present article continues my preceding notes \((^{1,2})\). In \((^1)\) a regular domain \(G \subset R_n\) was defined. For it (in particular for \(p=2\)) stable (more precisely, locally stable) boundary values were studied for a function \(\Phi\) with finite integral
\[ D_G(\Phi)=\int_G \sum_1^N |\Phi^{(k)}|^2\,dG, \tag{1} \]
where
\[ \mathbf{k}^1,\ldots,\mathbf{k}^N \tag{2} \]
are given nonnegative vectors. Let the smallest convex body of the (integer) vectors spanned by the system (2), to which the vector \(0\) is adjoined, be \(\mathscr E\). Together with \(\mathbf{k}\), let \(\mathscr E\) contain the projections of \(\mathbf{k}\) onto coordinate subspaces of arbitrary dimensions. To a function \(\Phi\) there is associated a definite set of boundary functions. This made it possible to speak of classes \(\mathfrak M=\mathfrak M(G;\Phi)\) and \(\mathfrak M_0=\mathfrak M_0(G)\) of functions \(f\) with \(D_G(f)<\infty\), possessing respectively the same sets as \(\Phi\) and \(0\).
In \((^2)\) the following differential equation was considered in \(G\):
\[ Lu=\sum_{\mathbf{k},\mathbf{l}\in\mathscr E}(-1)^{(k)}D^{(k)}\bigl(\alpha_{\mathbf{kl}}(\bar x)u^{(l)}\bigr)=0, \tag{3} \]
\[ \bigl(\alpha_{\mathbf{kl}}(\bar x)=\alpha_{\mathbf{lk}}(\bar x),\ |\alpha_{\mathbf{kl}}(\bar x)|\leq M,\ x\in G\bigr), \]
for which, for example, it was assumed that
\[ \sum_{\mathbf{k},\mathbf{l}\in\mathscr E}\alpha_{\mathbf{kl}}(\bar x)i^{|k|-|l|}\xi^{(k+l)} \geq \chi\sum_1^N\bigl(\xi^{(k^s)}\bigr)^2, \tag{4} \]
\[ \xi^{(k)}=\xi_1^{k_1}\cdots \xi_n^{k_n},\quad \mathbf{k}=(k_1,\ldots,k_n),\quad |k|=\sum_1^n k_j, \]
where \(\chi>0\) does not depend on \(\bar x\) and \(\xi\). It was shown that for equation (3) there exists a unique generalized solution, and conditions were studied under which the existence of a classical solution of (3) is guaranteed. Equation (3) covers a broad class of equations of hypoelliptic (in particular elliptic) type, but in general goes beyond the limits of the hypoelliptic type.
The question arises of the uniqueness of a classical solution of (3). It is solved here in the case of a bounded convex domain \(G\).
Let \(\alpha_{\mathbf{kl}}\) be continuously differentiable in the domain \(G\) (open) \(\mathbf{k}\) times, \(\mathbf{k}\in\mathscr E\).
We shall say, for example, that \(u\) is a classical solution of (3) if \(u\) has in \(G\) continuous partial derivatives of orders \(\mathbf{k}+\mathbf{l}\) \((\mathbf{k},\mathbf{l}\in\mathscr E)\) and satisfies (3).
Theorem 1. A bounded convex domain \(G\) is regular. Consequently (see \((^2)\)), in \(\mathfrak M\) there exists, and moreover is unique, a generalized solution \(u\) of equation (3).
Theorem 2. Let \(\Phi\) have a finite integral and, for every \(\mathbf k\in \mathcal E\) for which \(\alpha_{\mathbf k1}(x)\not\equiv 0\), let \(\Phi^{(\mathbf k)}\in L_2(G)\), where \(G\) is a bounded convex domain. Then in the class \(\mathfrak M=\mathfrak M(G;\Phi)\) there can exist one classical solution of equation (3).
In the proof of the theorems one introduces \(\widetilde G_s^j\)—the projection of \(G\) onto some (with number \(s\)) coordinate plane \(R_{n-j}^{(s)}\) of dimension \(n-j\) \((j=0,1,\ldots,n-1)\); \(\widetilde G_{s,\delta}^j\) is the set of \(\bar x\in \widetilde G_s^j\) lying at a distance greater than \(\delta>0\) from the boundary of \(\widetilde G_s^j\); \(G_{s,\delta}^j\) is the cylindrical body constructed on \(\widetilde G_{s,\delta}^j\) with generators belonging to \(R_j^{(s)}\), where \(R_n=R_j^{(s)}\times R_{n-j}^{(s)}\). Here \(\widetilde G_1^0=G_1^0=G\), \(\widetilde G_{1,\delta}^0=G_{1,\delta}^0=G_\delta\)—the set of \(\bar x\in G\) lying at a distance greater than \(\delta\) from the boundary \(\Gamma\) of the domain \(G\). The corresponding cylinder constructed on \(G_\delta\) coincides with \(G_\delta\) (here only \(s=1\) is possible).
Theorem 1 follows for \(n=3\) from the following facts. Let \(\Gamma_s^1\) and \(\Gamma_s^2\) \((s=1,2,3)\) be the intersections of \(\Gamma\), respectively, with the boundary of \(G_s^1\), \(G_s^2\). The points \(\Gamma-\bigcup_{s=1}^3\Gamma_s^1\) and the interior (with respect to \(\Gamma_s^1,\Gamma_s^2\)) points of \(\Gamma_s^1,\Gamma_s^2\) are regular. For any \(s=1,2,3\) the edge (boundary of a two-dimensional surface) \(\Gamma_s^1,\Gamma_s^2\) has projection onto any plane \(x_j=0\) of two-dimensional measure zero.
Lines passing through the points \(\bar x\in G_s^1\) intersect \(\Gamma\) in two points—the upper and the lower (since \(G\) is open, so are \(\widetilde G_s^j\), \(G_s^j\)). The upper points form the set \(\Gamma_s^1\), the lower the set \(\Gamma_s^0\). Assign positive numbers \(\delta,\sigma_1,\ldots,\sigma_n\) and put \(\omega_s^0=\Gamma_s^0G_{s,\sigma_s}^{0*}\), \(\omega_s^1=\Gamma_s^1G_{s,\sigma_s}^{1*}\) \((2\sigma_s^*=\sigma_s)\). The sets \(\omega_s^0,\omega_s^1\) are at a distance greater than some positive number. Let \(\Omega_s=(G-G_{s,[\delta]})G_{s,\sigma_s}^1\). For sufficiently small \(\delta_0\), depending on \(\sigma_s\), \(\Omega_s\) for \(0<\delta<\delta_0\) decomposes into two nonintersecting sets \(\Omega_s=\Omega_s^0+\Omega_s^1\), adjacent respectively to \(\omega_s^0,\omega_s^1\). Introduce the sets \(\omega_{i_1,\ldots,i_n}=\omega_1^{i_1}\cdots\omega_n^{i_n}\), \(\Omega_{i_1,\ldots,i_n}=\Omega_1^{i_1}\cdots\Omega_n^{i_n}\), where \(i_s=0,1\). There are decompositions into sums of pairwise nonintersecting sets
\[ \omega=\Gamma G_{1,\sigma_1}'\cdots G_{n,\sigma_n}'=\sum_{i_s=0,1}\omega_{i_1,\ldots,i_n} \]
and, for sufficiently small \(\delta_0\), depending on \(\sigma_1,\ldots,\sigma_n\),
\[ \Omega^0=(G^0-G_{3\delta}^0)G_{1,\sigma_1}'\cdots G_{n,\sigma_n}' =\sum \Omega_{i_1,\ldots,i_n}\qquad (0<\delta<\delta_0). \tag{5} \]
\[ G-G_{3\delta}=G_1^0-G_{1\delta}^0. \]
For a function \(v\in\mathfrak M_0(G)\) and \(\mathbf k\ll 1\in\mathcal E\), the basic inequality is proved:
\[ \left(\int_{\Omega_0}|v^{(1-\mathbf k)}|^2\,dG\right)^{1/2} \le \sum_{\Omega_{i_1,\ldots,i_n}} \left(\int |v^{(1-\mathbf k)}|^2\,dG\right)^{1/2} \le \]
\[ \le c\delta^{|\mathbf k|} \left(\int_G |v^{(1)}|^2\,dG\right)^{1/2} \qquad (0<\delta<\delta_0), \tag{6} \]
where \(c\) does not depend on \(\delta\) and \(v\).
Let us now take as the initial set, instead of \(G\), a certain \(\widetilde G_s^j\). It belongs to some \(R_{n-j}^{(s)}\) (depending on \(s\)). We shall denote its projections onto the coordinate \((n-j-1)\)-dimensional subspaces \(R_{n-j}^{(s)}\) by \(\widetilde G_{s,j+1}^{j+1}, \ldots, \widetilde G_{s,n}^{j+1}\) (with indices \(j+1,\ldots,n\)). Let
\[ \Omega_s^j=(G_s^j-G_{s,3\delta}^j)G_{(s,j+1),\sigma_{j+1}}^{j+1}\cdots G_{(s,n),\sigma_n}^{j+1}, \]
\[ R_n=R_j^{(s)}\times R_{n-j}^{(s)} \]
and let \(\mathbf k\leq \mathbf 1\in \mathcal E\), and all projections \(k_i\) of the vector \(\mathbf k\) onto \(R_j^{(s)}\) be equal to zero. Then, if \(v\in \mathfrak M_0(G)\), the following estimate, more general than (6), holds:
\[ \int_{\Omega_s^j}|v^{(\mathbf 1-\mathbf k)}|^2\,dG \leq c\delta^{|\mathbf k|}\int_G |v^{(\mathbf 1)}|^2\,dG \qquad (0<\delta<\delta_0;\ j=0,1,\ldots,n-1), \tag{7} \]
where \(c\) does not depend on \(\delta\) or \(v\).
Let now \(u\in \mathfrak M\) be a classical solution of equation (3) in \(G\). If it is proved that \(u\) is a generalized solution of (3), i.e. that \(E_G(u,v)=0\) (see (2)) for all \(v\in\mathfrak M_0\), then this will prove uniqueness, since the generalized solution is unique.
Restricting ourselves to the three-dimensional case, introduce the positive numbers \(\sigma_0,\sigma_1,\sigma_2,\sigma_3,\sigma_1^2,\sigma_2^2,\sigma_3^2\) and the functions \(\alpha_{\sigma_0}^0,\alpha_{1,\sigma_1}',\alpha_{2,\sigma_2}',\alpha_{3,\sigma_3}',\alpha_{1,\sigma_1^2}^2,\alpha_{2,\sigma_2^2}^2,\alpha_{3,\sigma_3^2}^2\), depending on \(\bar x\), equal respectively to \(1\) on \(G_{2\sigma_0},G_{2\sigma_1}',\ldots,G_{2\sigma_3^2}^2\) and to \(0\) outside these sets. Let the functions of \(\bar x\)
\(\eta=\eta_{\sigma_0}^0,\eta_1=\eta_{\sigma_1},\ldots,\eta_{3,\sigma_3^2}^2\) be respectively their \(\sigma_0,\ldots,\sigma_3^2\)-Sobolev averages (see (3)). Let \(v\in\mathfrak M_0\) and
\[ v_0=\eta\eta_1\eta_2\eta_3\eta_1^2\eta_2^2\eta_3^2 =\eta v_1=\eta\eta_1 v_2=\cdots \]
not only belong to \(\mathfrak M_0\), but also be equal to zero along the strip in \(G\) adjacent to \(\Gamma\). Therefore \(E_G(u,v_0)=0\), and, taking into account that \(\eta=\eta_1=\cdots=\eta_3^2=1\) on
\[ Q_0=G_{3\sigma_0}G_{1,3\sigma_1}\cdots G_{3,3\sigma_3^2}^2 =G_{3\sigma_0}Q_1=G_{3\sigma_0}G_{1,3\sigma_1}Q_2=\cdots, \]
one may easily conclude that uniqueness will be proved if it is proved that
\[ \overline{\lim_{\sigma_3^2\to 0}}\ \overline{\lim_{\sigma_2^2\to 0}}\ \cdots\ \overline{\lim_{\sigma_0\to 0}} E_{G-Q_0}(u,v_0)=0. \]
This method, in the simplest case, when \(v_0=\eta_\delta v\), was applied by S. L. Sobolev (3) to prove uniqueness of the solution of the polyharmonic problem. In the present complicated case one has to pass to the limit successively \(n+1\) times. Directly, by a single passage to the limit, the desired property cannot be detected.
From the condition imposed in Theorem 2 on \(\Phi\) it follows that \(u\) also possesses this property. This leads to the fact that, for uniqueness, it is sufficient to prove that
\[ \overline{\lim_{\sigma_3^2\to 0}}\ \cdots\ \overline{\lim_{\sigma_0\to 0}} \int_{G-Q_0}|v_0^{(\mathbf 1)}|^2\,dG=A<\infty. \]
We have
\[ \left(\int_{G-Q_0}|v_0^{(\mathbf 1)}|^2\,dG\right)^{1/2} \leq c\sum_{\mathbf k\leq \mathbf 1} \left( \int_{(G-G_{3\sigma_0})G_{1,\sigma_1}G_{2,\sigma_2}G_{3,\sigma_3}} |\eta^{(\mathbf k)}v_1^{(\mathbf 1-\mathbf k)}|^2\,dG \right)^{1/2} + \]
\[ +\left( \int_{(G-Q_0)G_{3\sigma_0}} |v_1^{(\mathbf 1)}|^2\,dG \right)^{1/2}, \]
and since \(|\eta^{(k)}| \ll c\sigma_0^{-|k|}\), it follows from (7), for \(j=0\), that the sum \(\sum_{k\leqslant l}\) is bounded, whence
\[ \overline{\lim_{\sigma_0\to 0}}\int_{G-Q_0} |v_0^{(1)}|^2\,dG = \int_{G-Q_1} |v_1^{(1)}|^2\,dG+B_1, \qquad B_1<\infty . \]
Further,
\[ \int_{G-Q_1} |v_1^{(1)}|^2\,dG \ll c\sum_{k\leqslant l} \left( \int_{(G-G_{1,3\sigma_1})G^2_{1,\sigma_1}G^2_{2,\sigma_2}G^2_{3,\sigma_3}} |\eta_1^{(k)}v_2^{(1-k)}|^2\,dG \right)^{1/2} + \left( \int_{(G-Q_1)G_{1,3\sigma_1}} |v_2^{(1)}|^2\,dG \right)^{1/2}. \tag{8} \]
The function \(\eta_1\) does not depend on \(x_1\); therefore, if \(k=(k_1,\ldots,k_n)\) and \(k_1>0\), then \(\eta_1^{(k)}\equiv 0\); but if \(k_1=0\), then, taking into account that
\[ (G-G_{1,3\sigma_1})G^2_{1,\sigma_1}G^2_{2,\sigma_2}G^2_{3,\sigma_3} \subset (G'_1-G'_{1,3\sigma_1})\widetilde G^2_{2,\sigma_2}\widetilde G^2_{3,\sigma_3} \]
(\(G^2_2,\ G^2_3\) are cylinders constructed on \(\widetilde G^2_2,\ \widetilde G^2_3\), the projections of \(\widetilde G_1\) onto the axes \(x_2,x_3\)), then, by virtue of the inequality \(|\eta_1^{(k)}|\ll c\sigma_1^{|k|}\) and (7) for \(j=1\), we conclude that the sum \(\sum_{k\leqslant l}\) is bounded, while the second term on the right-hand side of (8), as \(\sigma_1\to 0\), tends to
\[ \left(\int_{G-Q_2}|v_2^{(1)}|^2\,dG\right)^{1/2}. \]
We reason in the same spirit in the cases where \(\sigma_2,\sigma_3,\ldots\) tend to zero. Here one must take into account that each of the functions \(\eta_1^2,\eta_2^2,\eta_3^2\) does not depend on two variables, and in the corresponding place (7) is applied with \(j=2\).
Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR
Received
7 VIII 1962
CITED LITERATURE
\(^{1}\) S. M. Nikol’skii, DAN, 146, No. 3 (1962).
\(^{2}\) S. M. Nikol’skii, DAN, 146, No. 4 (1962).
\(^{3}\) S. M. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, Leningrad, 1950.