Reports of the Academy of Sciences of the USSR
1962, Volume 142, No. 6

MATHEMATICS

V. I. KONDRASHOV

ON THE THEORY OF BOUNDARY-VALUE PROBLEMS WITH BOUNDARY CONDITIONS CONTAINING PARAMETERS

(Presented by Academician S. L. Sobolev on 20 VII 1961)

We shall regard the functions \(W_p^\nu\) as defined in such domains \(D\) (the space \(x_1,\ldots,x_n\)) with boundary \(S\), for which the theory of these functional spaces applies \(\left({}^{1-8,16,18}\right)\).

Assuming \(S=\sum_{s=1}^{n} S_{n-s}\), we divide it into two parts \(S^3\) and \(S^c\):

\[ S=S^c+S^3=\sum_{s=1}^{n} S_{n-s}^c+\sum_{s=1}^{n} S_{n-s}^3. \]

Then \(W_p^\nu(S^3,0)\) is the set of functions from \(W_p^\nu\), each of which, together with its derivatives up to order \(\nu-[s/p]-1\) inclusive, vanishes in the mean, with certain exponents \(\left({}^{1,5}\right)\), on the manifolds \(S_{n-s}^3\). \(W_p^\nu(S^3,0)\) is closed in \(W_p^\nu\) \(\left({}^{5}\right)\). \(W_p^\nu(0)=W(S^3,0)\) for \(S_{n-s}^c=0\); \(W_p^\nu(0)\in W_p^\nu(S^3,0)\). I shall give the formulation of the principal boundary-value problem for a certain special type of the equations and functionals under consideration. At the same time the results of the work also extend to functionals of a more general form, for example as in the papers \(\left({}^{7,8}\right)\).

Basic problem I. In \(W_p^\nu\), find a function \(u\) satisfying the variational equation

\[ \int_D \cdots \int \left( \sum_{l=0}^{\nu} \sum_{\sum \alpha=l} \frac{\partial F_\nu^p(u)}{\partial u_{\alpha_1,\ldots,\alpha_n}^{l}} \,\xi_{\alpha_1,\ldots,\alpha_n}^{l} \right)\,dv = \int_D \cdots \int F_\nu^{p-1,1}(u,\xi)\,dv=0, \tag{1} \]

where \(\xi\) is an arbitrary function from \(W_p^\nu(0)\), under the following boundary conditions:

\[ \int_D \cdots \int F_\nu^{p-1,1}(u,\eta)\,dv - \mu \sum_{s=1}^{n}\lambda_s \int_{S_{n-s}^c}\cdots\int \rho_s(x_1,\ldots,x_n)\, F_{\nu_s}^{q_s-1,1}(u,\eta)\,dS_{n-s} =0; \tag{2} \]

here \(\eta\) ranges over the entire set \(W_p^\nu(S^3,0)\);

\[ \sum_{s=1}^{n} \int_{S_{n-s}^c}\cdots\int \rho_s(x_1,\ldots,x_n)\, F_{\nu_s}^{q_s}(u)\,dS_{n-s} =1 \tag{3} \]

\(\rho_s \geq 0\) on the manifolds \(S_{n-s}^c\).

1) \(F_\nu^p(u)\) is an admissible \(p\)-form in \(W_p^\nu\) \(\left({}^{7}\right)\); 2) \(F^{q_s}\) is an admissible form on \(S_{n-s}\) in \(W_\nu^p\) of the connection \(S_{n-s}\) \(\left({}^{7,8}\right)\); 3) \(\mu\) and \(\lambda_s\) are constants; 4a) \(0\leq \nu_s\leq \nu-[s/p]-1\); 4b) \(1<q_s\leq p(n-s)/(n-p(\nu-\nu_s))\) (\(q_s\) is any number \(>1\)) if \(n=p(\nu-\nu_s)\).

Remark. This problem is also posed and solved in the “weighted” classes \(W_{p,b_1,\ldots,b_s,\ldots,b_n}^{\nu}\), which denotes the space whose elements have in the domain \(D\) generalized (in the sense of S. L. Sobolev) derivatives up to order ...

including \(v\). Derivatives of order \(v-1\) and lower are summable in \(D\), and

\[ \underbrace{\int\cdots\int}_{D}^{n}\prod_{s=1}^{n} r_{n-s}^{b_s} \sum_{\sum\alpha=v} \left| \frac{\partial^{v}u}{\partial x_1^{\alpha_1}\cdots \partial x_n^{\alpha_n}} \right|^{p}\,d\omega \tag{3\(_1\)} \]

has meaning. Here \(r_{n-s}^{b_s}\) is the distance from points of the domain \(D\) to points of the boundary manifold \(S_{n-s}\), raised to the power \(b_s\); \(b_s\) is any number—the exponent of degeneration*. (For details on these spaces, see below.)

Some problems of type I have also been considered earlier \((^{8-10,\,17})\). In the present work, problems with linear and nonlinear equations of different orders, with boundary conditions mixed with nonlinearity, in domains with a degenerate contour, and also in “weighted” classes, are studied and solved for the first time. I shall explain the connection of the basic problem with boundary-value problems for differential equations by the simplest example. Let \(n=p=2\), \(q<4/b\), \(q_1<2/b\), \(b<1\).

Proposition. There exists a number \(\mu\) such that the differential equation**

\[ \sum_{i=1}^{2}\frac{\partial}{\partial x_i} \left(r^{b}\frac{\partial u}{\partial x_i}\right) +f(x_1,x_2,u)=0 \tag{1′} \]

has a solution \(u\in W_{2,b}^{1}\), satisfying the following boundary conditions:

\[ \int_{S_1^c} \left(r^{b}\frac{\partial u}{\partial n} -\mu\beta(S)u|u|^{q-2}\right)\eta\,dS=0;\qquad \int_{S_2^c} r^{b}\frac{\partial u}{\partial n}\eta\,dS=0;\qquad u|_{S^3}=0. \tag{2′} \]

For any \(\eta\) from \(W_{2,b}^{1}(S^3,0)\) (generalized boundary condition)

\[ \int_{S_0^c} r^{b}\frac{\partial u}{\partial n}\eta\,dS \to \int_{S_1^c} r^{b}\frac{\partial u}{\partial n}\eta\,dS \]

as \(S_0\to S_1^c\); \(S_0^c\) are boundaries of closed domains \(D_0\) such that \(D_0\to D\). If \(b=0\) and the function \(u\) has a normal derivative on the contour \(S^c\), then the boundary conditions (2′) take the form

\[ \left.\frac{\partial u}{\partial n}\right|_{S_1^c} =\mu\beta(S)u|u|^{q-2};\qquad \left.\frac{\partial u}{\partial n}\right|_{S_2^c}=0;\qquad u|_{S^3}=0; \]

here \(S_1^c+S_2^c+S^3=S\), the boundary of \(D\). The problem can be expressed in terms of integral equations.

The following variational problem corresponds to Problem I. In \(W_p^v(S^3,0)\), find a function \(\varphi=u\) realizing the minimum of the integral

\[ \underbrace{\int\cdots\int}_{D}^{n} F_v^p(\varphi)\,dv \tag{4} \]

under condition (3).

Basic proposition \(A_1\). For arbitrary admissible forms of the kernel \(F_v^p\) (in \(D\)) and forms of the bond \(F_{\nu_1}^{q}\) (on the manifolds \(S_{n-s}\)) there exists an infinite sequence of solutions \(u_k\) of the following recurrent variational problems. In \(W_p^v\), find a function \(\varphi=u_k\) realizing the minimum

* Instead of \(r_{n-s}^{b_s}\), one may take more complicated distance functions.

** The equation may degenerate also on part of the boundary of the domain \(D\); in this case the boundary conditions change correspondingly. \(f(x_1,x_2,u)\) is a certain polynomial in \(u\) of degree \(\le q\) or a function of growth \(|u|^{q}\). Analogous nonlinearities may also occur in the boundary conditions.

of the integral (1) under the additional conditions:

\[ 1)\quad \sum_{s=1}^{n}\left(\int_{S_{n-s}^{c}}\!\!\cdots\!\!\int \rho_s F_{\nu_s}^{q_s}(\varphi)\,dS_{n-s}\right)^{p/q_s}=1; \]

\[ 2)\quad \sum_{s=1}^{n}\lambda_s\int_{S_{n-s}^{c}}\!\!\cdots\!\!\int \rho_s F_{\nu_s}^{q_s-1,1}(u_j,\varphi)\,dS_{n-s}=0;\qquad j=1,2,\ldots,k-1. \]

The functions \(u_k\) then satisfy equation (1) in the form

\[ \int_D\!\!\cdots\!\!\int F_\nu^{p-1,1}(u_k,\eta_k)\,dv -\mu_k\sum_{s=1}^{n}\lambda_s \int_{S_{n-s}}\!\!\cdots\!\!\int F_{\nu_s}^{q_s-1,1}(u_k,\eta_k)\,dS_{n-s}=0, \]

where

\[ \int_D\!\!\cdots\!\!\int F_\nu^p(u_k)\,dv=\mu_k, \]

\(\eta_1\) ranges over the whole set \(W_p^\nu(S^3,0)\).

In the case of quadratic kernel forms and connection functions, \(\eta_k\) range over the whole set \(W_p^\nu(S^3,0)\).

Proposition \(A_2\). The functions \(u_k\) (eigenfunctions in \(A_1\)) are “orthogonal” and “normalized” as follows:

\[ 1)\quad \sum_{s=1}^{n}\lambda_s \int_{S_{n-s}^{c}}\!\!\cdots\!\!\int \rho_s F_{\nu_s}^{q_s-1,1}(u_j,u_i)\,dS_{n-s} = \begin{cases} 1, & j=i,\\ 0, & j<i; \end{cases} \]

\[ 2)\quad \int_D\!\!\cdots\!\!\int F_\nu^{p-1,1}(u_j,u_i)\,dv = \begin{cases} \mu_i, & j=i,\\ 0, & j<i. \end{cases} \]

Proposition \(A_3\). The eigenvalues of equation (1) form an increasing sequence \(\mu_k\to\infty\) as \(k\to\infty\). The spectrum is countable and discrete.

Theorems \(A_1\), \(A_2\), and \(A_3\) also extend to the corresponding problems considered in “weighted” classes. Here, instead of the subspaces \(W_p^\nu(S^3,0)\) and \(W_p^\nu(S,0)\), one uses the subspaces \(W_{p,b_1\ldots b_n}^\nu(S^3,0)\) and \(W_{p,b_1\ldots b_n}^\nu(0)\), whose definition is obvious.

Let the kernel form in the principal problem be quadratic and of the same type as in (7).

Proposition \(A_4\). There exists an infinite sequence of numbers \(\mu_1,\mu_2,\ldots,\mu_k,\ldots\)* such that the differential equation

\[ L_{b_1,b_2,\ldots,b_n}^{\nu}(u) + f\left( x_1,\ldots,x_n,u,\frac{\partial u}{\partial x_1},\ldots, \frac{\partial^{2m-2}u}{\partial x_1^{\alpha_1}\cdots\partial x_n^{\alpha_n}} \right)=0, \]

\[ L_{b_1,b_2,\ldots,b_n}^{\nu}(u) = \sum_{\sum\alpha=\nu} \frac{\partial^\nu}{\partial x_1^{\alpha_1}\cdots\partial x_n^{\alpha_n}} \left( \prod_{s=1}^{n} r_{n-s}^{b_s} A_{\alpha_1\ldots\alpha_n}(x_1,\ldots,x_n) \frac{\partial^\nu u}{\partial x_1^{\alpha_1}\cdots\partial x_n^{\alpha_n}} \right), \]

where \(f\) is the nonlinear part of (7), has a solution \(u_k\in W_{2,b_1,b_2,\ldots,b_n}^{\nu}\) satisfying generalized boundary conditions of the form (2), (3).

In the course of proving the propositions stated above, the properties of the functions \(W_p^\nu\), as well as additional integral

* The numbers \(\mu_k\) enter into the boundary conditions (see (2) and \((2')\)).

inequality *. In the case of equations that “degenerate” on the boundary of the domain \(D\) or of a part of it, the corresponding “embedding” and “compactness” theorems were also applied. The “weighted” classes for the case \(0>b_s>-1\) were studied by the author already in \((^2)\). The first systematic study of these classes for positive exponents in a certain class of domains (mainly direct and inverse “embedding” theorems) was carried out by L. D. Kudryavtsev in terms of the \(H\)-classes of S. M. Nikol’skii \((^{12})\), and subsequently by A. A. Vasharin \((^{14})\), P. I. Lizorkin \((^{15})\), and others (direct and inverse embedding theorems with integral-Hölder conditions on the boundary functions). The metric in \(W^\nu_{p,b_1\ldots b_n}\) can be introduced analogously to the way this was done for the classes \(W\) in \((^{4,6})\), with the factor \(\prod_{s=1}^n r_{ns}^{b_s}\) for the highest derivatives, or in accordance with the problems under consideration. Then this space will be complete. In this connection the following holds:

Proposition \(A_5\). a) A set \(\rho(0,u)\le A\), bounded in \(W^\nu_{p,b_1\ldots b_n}\), on the manifolds \(S_{n-s}\in \overline D\), where \(s<pl-b_s\), is compact in the spaces into which it is embedded:

1) in \(W^{\nu-k}_q\), where

\[ q<q^*=\frac{p(n-s)}{\,n-kp+b_s\,}, \quad \text{if } n>kp-b_s \ **; \]

2) in \(W^{\nu-k}_{q^{**}}\), where \(q^{**}\) is any number \(>1\), when \(n=kp-b_s\);

3) in

\[ C^{\nu-k}=C^{\nu-\left[\frac{s+b_s}{p}\right]-1} \]

when \(n<kp-b\) \ **.

b) The limiting functions \(u\) for a sequence \(u_k\), selected according to the indicated compactness, belong to \(W^\nu_{p,b_1,\ldots,b_n}\).

The convergence of the \(\nu\)-th derivatives of the functions \(u_k\) to the corresponding derivatives of \(u\) is weak (in the sense of \(L_p\)) with a weight factor (see 3\(_1\)) \ ****.

The proofs of the stated theorems were carried out on the basis of a development of the methods of investigation used in \((^{1-8,16,17})\).

Moscow Engineering-Physics
Institute

Received
8 VII 1960

CITED LITERATURE

\(^1\) S. L. Sobolev, Matem. sborn., 2 (44), 3, 465 (1937).
\(^2\) V. I. Kondrashov, DAN, 18, 236 (1938).
\(^3\) S. L. Sobolev, Matem. sborn., 4 (46), No. 3 (1938).
\(^4\) V. I. Kondrashov, DAN, 48, No. 8 (1945).
\(^5\) V. I. Kondrashov, DAN, 52, No. 6 (1950).
\(^6\) S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, 1950.
\(^7\) V. I. Kondrashov, DAN, 90, No. 2 (1953).
\(^8\) V. I. Kondrashov, Doctoral dissertation, Mathematical Institute named after V. A. Steklov, Academy of Sciences of the USSR, 1948.
\(^9\) R. Courant, D. Hilbert, Methods of Mathematical Physics, 1, 2, Moscow, 1951.
\(^ {10}\) L. A. Lyusternik, Matem. sborn., 4 (46), 227 (1938).
\(^ {11}\) L. D. Kudryavtsev, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 55 (1959).
\(^ {12}\) S. M. Nikol’skii, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 38, 244 (1951).
\(^ {13}\) M. I. Vishik, Matem. sborn., 35, No. 3 (1953).
\(^ {14}\) A. A. Vasharin, Izv. AN SSSR, ser. matem., 23, 421 (1959).
\(^ {15}\) P. I. Lizorkin, DAN, 126, No. 4 (1959).
\(^ {16}\) E. Gagliardo, Ric. di mat., 7, No. 1, 102 (1958); E. Gagliardo, Sborn. per. Matematika, 5, 4 (1961).
\(^ {17}\) L. Sandgren, Meddelanden fran Lunds Univ. Mat. seminarium, 13 (1955).
\(^ {18}\) V. P. Il’in, DAN, 96, No. 5, 905 (1954).
\(^ {19}\) S. M. Nikol’skii, Matem. sborn., 33, 2 (1953).

* For example,

\[ \int_D \cdots \int |u|^q\,d\nu \le c\left( \int_D \cdots \int \sum_{i=1}^n \left|\frac{\partial u}{\partial x_i}\right|^p\,d\nu \right)^{q/p}; \qquad \int_D \cdots \int u\,d\omega=0. \]

** The embedding and compactness theorems in the case \(W_{2,b_s}\), \(0<b_s<1\), the domain bounded, degeneration on all of \(S_{n-s}\) (a hyperplane), \(q\) not equal to \(q^*\) in the embedding theorems, were indicated by I. A. Solomesh (without the second part of theorem b).

*** In the embedding theorems: 1) \(q=q^*\), for \(W^\nu_p\) this was discovered for a number of cases by the author \((^5)\), and then in the general case obtained by V. P. Il’in \((^{18})\); 2) the domain \(D\) may be unbounded under the same conditions as for \(W^\nu_p\).

**** This proposition is a development of the author’s result \((^4)\).