A. NARCHAEV

THE FIRST BOUNDARY-VALUE PROBLEM FOR ELLIPTIC EQUATIONS DEGENERATING ON THE BOUNDARY OF THE DOMAIN

(Presented by Academician S. L. Sobolev on 6 I 1964)

The first boundary-value problem for a second-order elliptic equation degenerating on the boundary of a domain was first investigated by M. V. Keldysh \((^{1})\). Later a number of papers appeared \((^{2,3})\) and others, devoted to degenerating second-order elliptic equations. Among elliptic equations of higher orders degenerating on the boundary of a domain, equations to which the variational method is applicable have been considered. Fourth-order elliptic equations degenerating on the boundary of a domain, to which the variational method is not applicable, were considered by V. K. Zakharov \((^{4})\).

In the present paper an elliptic equation on part of the boundary of a domain degenerates into a so-called quasi-parabolic equation. It is proved that the formulation of the problem depends on the sign of the coefficient of the highest derivative of odd order with respect to the variable with respect to which the highest even derivative degenerates.

Let a bounded domain \(Q \subset R^n\) be situated in the half-space \(x_n > 0\) and let part \(\Gamma_0\) of its boundary \(\Gamma\) adjoin the plane \(x_n = 0\). We denote the remaining part of the boundary by \(\Gamma_1\): \(\Gamma_1 \cup \Gamma_0 = \Gamma\), \(\overline Q = Q + \Gamma\). We assume that the boundary \(\Gamma_1\) is such that Sobolev’s embedding theorems hold for it \((^{5})\). In the domain \(Q\) consider the equation

\[ Lu = L_0u + Au = h(x), \tag{1} \]

where

\[ L_0u \equiv \sum_{i,j=1}^{n} \frac{\partial^2}{\partial x_i \partial x_j} \left( A_{ij}(x)\frac{\partial^2 u}{\partial x_i \partial x_j} \right) + b\,\frac{\partial^3 u}{\partial x_n^3} \]

is the principal part of the operator \(L\);

\[ Au = \sum_{i/4+s/3<1} a_{(i,s)}(x)\, \frac{\partial^s}{\partial x_n^s}D^i u \]

is an operator subordinate to the principal part. Here \(s \ge 0\), \(i \ge 0\) are some integers; \(x=(x_1,\ldots,x_n)\); \(i=i_1+\cdots+i_{n-1}\); \(D^i=\partial^i/\partial x_1^{i_1}\cdots \partial x_{n-1}^{i_{n-1}}\); \(b=\pm 1\); the coefficients \(A_{ij}(x)=A_{ji}(x)\), \(a_{(i,s)}(x)\) are sufficiently smooth functions in \(\overline Q\), and

\[ \sum_{i,j=1}^{n-1} A_{ij}(x)\xi_i^2\xi_j^2 \ge \theta^2 > 0 \quad \text{for all } \xi_i:\ \sum_{i=1}^{n-1}\xi_i^2 \ne 0; \tag{2} \]

\[ c_{i1}^2 x_n^{\alpha_i} \le A_{in}(x) \le c_{i2}^2 x_n^{\alpha_i} \quad (i=1,\ldots,n); \]

\(\alpha_i\) are some nonnegative numbers.

For equation (1) the following boundary-value problems are posed:

Problem D. Find a solution \(u(x)\) of equation (1) vanishing on \(\Gamma\) together with its first derivatives, if: a) \(b=-1\), \(\alpha_n\) is arbitrary, or b) \(b=+1\), \(\alpha_n<1\).

Problem E. Find a solution \(u(x)\) of equation (1) vanishing on \(\Gamma_1\) together with its first derivatives, if \(b=+1\), \(\alpha_n \ge 1\); on \(\Gamma_0\) in this case only the function \(u(x)\) vanishes.

Let us complete \(C_0^4\)—the set of all functions, four times continuously differentiable in \(Q\), vanishing near \(\Gamma\)—in the metric

\[ \|u\|_4^2 = \iint_Q \left[ \sum_{i,j=1}^{n} A_{ij}(x) \left( \frac{\partial^2 u}{\partial x_i \partial x_j} \right)^2 + \left( \frac{\partial u}{\partial x_n} \right)^2 \right]\,dQ . \tag{3} \]

We denote the resulting space by \(\dot W_2^2(\alpha)\). The corresponding embedding theorems, analogous to the theorems in \((^2,^4)\), are proved; in particular, for any function \(u(x)\in \dot W_2^2(\alpha)\) the estimate

\[ \iint_Q \sigma(x)u^2(x)\,dQ \leq c_3\|u\|_+^2, \]

holds, where \(c_3\) is a constant independent of \(u(x)\), and

\[ \sigma(x)= \begin{cases} x_n^{\alpha_n-4}|\ln x_n|^{-1-\varepsilon_0}, & \text{for } \alpha_n \leq 2,\\ x_n^{-2}|\ln x_n|^{-1-\varepsilon_0}, & \text{for } \alpha_n \geq 2, \end{cases} \qquad (\varepsilon_0>0). \]

To prove the existence of the required solutions, known theorems from functional analysis are used (see, for example, \((^6)\)). The method of \((^6)\) for proving the existence of a generalized solution of degenerating elliptic equations of the second order was used in \((^7)\). Denote by \(\mathcal L_2(\sigma^{-1})\) the set of functions \(f(x)\) for which

\[ \iint_Q \sigma^{-1}f^2(x)\,dQ < +\infty, \]

and by \(W_2^{-2}(\alpha)\) the closure of the set \(\mathcal L_2(\sigma^{-1})\) with respect to the norm

\[ \|f\|_-=\sup_{u\in \dot W_2^2(\alpha)} \frac{|(f,u)|}{\|u\|_+}, \]

where \((\cdot,\cdot)\) denotes the scalar product in \(\mathcal L_2(Q)\).

Lemma 1. The space \(W_2^{-2}(\alpha)\) is isometrically isomorphic to the space of all linear bounded functionals on the Hilbert space \(\dot W_2^2(\alpha)\).

Theorem 1. Every linear bounded functional \(m_u(f)\) on the space \(W_2^{-2}(\alpha)\) can be represented as

\[ m_u(f)=(u,f), \quad u\in \dot W_2^2(\alpha). \]

The proofs of Lemma 1 and Theorem 1 are carried out analogously to \((^6)\).

Denote by \(\dot W_2^4(Q)\) the set of functions \(v(x)\in W_2^4(Q)\) that vanish on the boundary \(\Gamma\) together with their first derivatives, and by \(\dot{\dot W}_2^4(Q)\) the set of functions \(v(x)\in \dot W_2^4(Q)\) satisfying on the boundary the following conditions:

\[ v|_{\Gamma}=0,\qquad \left.\frac{\partial v}{\partial x_i}\right|_{\Gamma_1}=0 \quad (i=1,\ldots,n). \]

Definition 1. A weak solution of problem D for equation (1), for \(\alpha_n\geq 1\), will mean a function \(u(x)\in \dot W_2^2(\alpha)\) that satisfies the equality

\[ (h,v)=(u,L^*v) \tag{4} \]

for all functions \(v(x)\in \dot{\dot W}_2^4(Q)\), where \(L^*\) is the “formally adjoint” operator to \(L\).

Weak solutions of problem D for \(\alpha_n<1\), and of problem E with \(v(x)\in \dot W_2^4(Q)\), are defined in an analogous way.

Theorem 2. If the coefficients of equation (1) for \(b=-1\) are such that

\[ (L^*v,v)\geq \mathrm{const}\,\|v\|_+^2 \tag{5} \]

for any \(v(x)\in \dot{\dot W}_2^4(Q)\), then for every function \(h(x)\in W_2^{-2}(\alpha)\) there exists a weak solution of problem D for \(\alpha_n\geq 1\).

To prove this theorem, the generalized Schwarz inequality is applied to (5), and then the Hahn—Banach theorem and the theore-

ma 1. The existence of a weak solution of problem D for \(\alpha_n<1\) and of problem E is proved in the same way. We note that \(\partial u/\partial x_n\) in the case of problem D \((\alpha_n\geqslant 1)\) vanishes on \(\Gamma_0\) in the following sense:

\[ \lim_{\delta\to 0}\int_{\Gamma_\delta} \frac{\partial u}{\partial x_n}\frac{\partial v}{\partial x_n}\,d\Gamma_\delta=0 \qquad (\Gamma_\delta=Q\cap (x_n=\delta)) \]

for any function \(v(x)\in \dot W^{4}_{2}(Q)\) that vanishes near \(\Gamma_1\). The remaining boundary values are assumed in the mean.

Next suppose that \(\alpha_n\geqslant 3\) and, for simplicity, \(A_{in}(x)=0\) for \(i=1,\ldots,n-1\). We shall prove the uniqueness of the weak solution of problem D in the cylinder \(Q\), whose lateral surface, upper and lower bases we denote respectively by \(S,\Omega_T,\Omega_0\), and \(\Gamma=S\cup\Omega_T\cup\Omega_0\). Choose \(v(x)\) as the solution of the following boundary-value problem:

\[ \sum_{i,j=1}^{n-1}\frac{\partial^2}{\partial x_i\partial x_j} \left(A_{ij}(x)\frac{\partial^2 v}{\partial x_i\partial x_j}\right) +\frac{\partial^3 v}{\partial x_n^3} +\frac12 a_{(0,0)}v=u(x); \tag{6} \]

\[ v\big|_{\Gamma}=0,\qquad \left.\frac{\partial v}{\partial x_i}\right|_{S}=0 \quad (i=1,\ldots,n-1),\qquad \left.\frac{\partial v}{\partial x_n}\right|_{\Omega_T}=0. \tag{7} \]

A solution \(v(x)\in \dot W^{4}_{2}(Q)\) of problem (6)—(7) exists and is unique for \(a_{(0,0)}>0\) \(({}^{8,9})\), and from (4) for \(h=0\) we obtain:

\[ \begin{aligned} \iint_Q u^2\,dQ &+\iiint_Q\left[ \sum_{i,j=1}^{n-1}\frac{\partial^2}{\partial x_i\partial x_j} \left(A_{ij}\frac{\partial^2 v}{\partial x_i\partial x_j}\right) +\frac{\partial^3 v}{\partial x_n^3} +\frac12 a_{(0,0)}v \right]\times \\ &\quad \times \frac{\partial^2}{\partial x_n^2} \left(A_{nn}(x)\frac{\partial^2 v}{\partial x_n^2}\right)\,dQ \\ &+\iiint_Q\left[ \sum_{i,j=1}^{n-1}\frac{\partial^2}{\partial x_i\partial x_j} \left(A_{ij}\frac{\partial^2 v}{\partial x_i\partial x_j}\right) +\frac{\partial^3 v}{\partial x_n^3} +\frac12 a_{(0,0)}v \right] \left[A^*v-\frac12 a_{(0,0)}v\right]\,dQ=0, \end{aligned} \]

where \(A^*\) is the “formally adjoint” operator to \(A\).

Transforming the integrals by integration by parts, estimating them from below by known inequalities, and applying Kudryavtsev’s lemma \(({}^{2})\), we find:

\[ \left(1-\sum_{i=1}^{N}\varepsilon_i\right)\iint_Q u^2\,dQ +\iint_Q\left[\frac14 a_{(0,0)}^2-\sum_{i=1}^{N}c(\varepsilon_i)\right]v^2\,dQ \leqslant 0, \tag{8} \]

where the \(\varepsilon_i\) are chosen so that \(1-\sum_{i=1}^{N}\varepsilon_i>0\), and \(N\) is a certain integer.

Thus, from (8) we obtain the following uniqueness theorem:

Theorem 3. If

\[ a_{(0,0)}^2\geqslant 4\sum_{i=1}^{N}c(\varepsilon_i) \tag{9} \]

then the weak solution of problem D is unique.

Consider equation (1) for \(b=+1\) in the cylinder \(Q\). The method of proving the uniqueness theorem used for \(b=-1\) does not essentially work for \(b=+1\), since the corresponding boundary-value problem for \(v(x)\) is not solvable. The initial domain of definition of the operator of the left-hand side consists of functions in \(C^4\) satisfying the conditions:

\[ u\big|_{\Gamma}=0,\qquad \left.\frac{\partial u}{\partial x_i}\right|_{S}=0 \quad (i=1,\ldots,n-1),\qquad \left.\frac{\partial u}{\partial x_n}\right|_{\Omega_T}=0. \tag{10} \]

The closure of the set of smooth functions satisfying conditions (10) in the norm (3) gives \(\dot W_2^2(\alpha)\). We extend the operator \(L\), defining it on all \(u(x)\in \dot W_2^2(\alpha)\) satisfying the additional condition

\[ \frac{\partial^2 u}{\partial x_n^2}\in {\mathcal L}_2(Q) \tag{11} \]

as a functional on \(\dot W_2^2(\alpha)\), by setting

\[ \langle Lu,v\rangle = -\iint_Q \frac{\partial^2 u}{\partial x_n^2} \frac{\partial v}{\partial x_n}\,dQ + \iint_Q \sum_{i,j=1}^n A_{ij}\, \frac{\partial^2 u}{\partial x_i\partial x_j}\, \frac{\partial^2 v}{\partial x_i\partial x_j}\,dQ + A(u,v) \tag{12} \]

for any \(v(x)\in \dot W_2^2(\alpha)\), where \(A(u,v)\) is the expression obtained formally from \(\iint_Q Au\cdot v\,dQ\) by integration by parts. The extension constructed is closed in \(W_2^{-2}(\alpha)\).

Definition 2. We shall say that \(u(x)\in B_L^C\) (the domain of definition of the strong extension of the operator \(L\)) if \(u\in \dot W_2^2(\alpha)\) and there exist an element \(h(x)\in W_2^{-2}(\alpha)\) and a sequence \(u_i\) of functions belonging to \(\dot W_2^2(\alpha)\) and satisfying the additional condition (11), such that \(\|u-u_i\|_+\to 0\), \(\|Lu_i-h\|_-\to 0\) as \(i\to\infty\), where \(L\) is understood in the sense of (12). The corresponding function will be called a strong solution of problem E.

Theorem 4. The strong solution of problem E, for \(a_{0,0}\) satisfying condition (9), exists and is unique.

Uniqueness follows from the following lemma:

Lemma 2. For strong solutions of problem E the inequality

\[ \|u\|_+ \leq \mathrm{const}\,\|Lu\|_- \]

holds.

The existence of a strong solution of problem E is proved in the same way as in \((^{10})\), using Theorem 3.

Remark 1. The uniqueness of the generalized solution of problem D for \(\alpha_n<1\) was proved by another method in \((^4)\).

Remark 2. The results obtained also extend to the corresponding elliptic equations of higher orders that degenerate on the boundary of the domain; we have considered fourth-order equations only for simplicity of exposition.

In conclusion I express my sincere gratitude to my scientific adviser V. N. Maslennikova.

Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
24 XII 1963

REFERENCES

\(^1\) M. V. Keldysh, DAN, 77, No. 2 (1951).
\(^2\) M. I. Vishik, Matem. sborn., 35 (77), issue 3 (1954).
\(^3\) L. D. Kudryavtsev, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 55 (1959).
\(^4\) V. K. Zakharov, DAN, 114, No. 3 (1957); 114, No. 4 (1957).
\(^5\) S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, L., 1950.
\(^6\) P. Lax, Collection of Translations: Mathematics, 1, 1 (1957).
\(^7\) V. P. Didenko, Candidate’s dissertation, Novosibirsk, 1962.
\(^8\) V. P. Mikhailov, DAN, 147, No. 3 (1962).
\(^9\) L. N. Slobodetskii, Uch. zap. Leningrad State Pedagogical Institute named after A. I. Herzen, 197, 54 (1958).
\(^ {10}\) A. A. Dezin, DAN, 123, No. 4 (1958).