MATHEMATICS

A. Kh. Gudiev

ON THE SOLVABILITY OF THE FIRST BOUNDARY-VALUE PROBLEM FOR A SECOND-ORDER ELLIPTIC EQUATION WITH DIVERGENT PRINCIPAL PART IN \(W_2^1\)

(Presented by Academician S. L. Sobolev on 23 III 1967)

In the paper \((^1)\), for the uniformly elliptic equation

\[ Lu \equiv \frac{\partial}{\partial x_i}\left(a_{ij}u_{x_j}+a_i u\right)+b_i u_{x_i}+au=f \tag{1} \]

in the case when \(a_i^2,\ b_i^2,\ a \in L_p\), and \(f \in L_{2n/(n+2)}\) for \(p>n/2\) \((n\ge 3)\), conditions are given which ensure the existence and uniqueness of a generalized solution from \(W_2^1\) of the first boundary-value problem for equation (1). In the present paper, conditions are given which ensure the existence and uniqueness of a generalized solution from \(W_2^1\) for \(p\le n/2\).

First the problem is considered for the case when \(a_i^2,\ b_i^2,\ a\) belong to spaces \(L_{(r_1,r_2)}\) with mixed norm. We shall denote each point \(x\) of \(n\)-dimensional Euclidean space \(R^n\) in the form of the pair \((\bar x_s,\bar x_{n-s})\), where \(\bar x_s(x_1,x_2,\ldots,x_s)\), \(\bar x_{n-s}(x_{s+1},\ldots,x_n)\). By \(R^s\) \((R^{n-s})\) we denote the \(s\)-dimensional \(((n-s)\)-dimensional) space of points \(\bar x_s\) \((\bar x_{n-s})\), \(1\le s\le n\). Let \(\Omega_s\) be an \(s\)-dimensional bounded domain in \(R^s\); \(\Omega_{n-s}\) an \((n-s)\)-dimensional bounded domain in \(R^{n-s}\), and \(\Omega=\Omega_s\times\Omega_{n-s}\). The space \(L_{(r_1,r_2)}(\Omega_s,\Omega_{n-s})\) is defined as the set of functions \(f(x)\), defined in \(\Omega\), for which the norm is bounded

\[ \|f\|_{L_{(r_1,r_2)}(\Omega_s,\Omega_{n-s})} = \left\|\,\|f(x_s,x_{n-s})\|_{L_{r_1}(\Omega_s)}\right\|_{L_{r_2}(\Omega_{n-s})}, \]

where

\[ \|\cdot\|_{L_{r_i}(D)} = \begin{cases} \left(\displaystyle\int_D |\cdot|^{r_i}\,d\omega^i\right)^{1/r_i}, & \text{if } 1\le r_i<\infty,\\[6pt] \operatorname*{vrai\,max}\limits_D |\cdot|, & \text{if } r_i=\infty. \end{cases} \]

Lemma 1. If the positive numbers \(r_1,r_2,q_1,q_2\) satisfy at least one of the following three conditions: a) \(r_1<q_1,\ q_2<r_2\); b) \(r_2<q_1,\ q_2<r_1\); c) \((r_1,r_2)\), \((q_1,q_2)\) are distinct solutions of the equation \(2x_1x_2-(n-s)x_1-sx_2-\alpha=0,\ \alpha>0\), then
\[ \bigl(L_{(r_1,r_2)}(\Omega)\cup L_{(q_1,q_2)}(\Omega)\bigr)\setminus L_{(r_1,r_2)}(\Omega)\ne \varnothing, \quad \bigl(L_{(r_1,r_2)}(\Omega)\cup L_{(q_1,q_2)}(\Omega)\bigr)\setminus L_{(q_1,q_2)}(\Omega)\ne \varnothing, \]
\[ \varnothing \text{ is the empty set.} \]

We denote by \(X^\alpha\) the following class of Banach spaces (\(\alpha\) is any positive number):

\[ X^\alpha= \left\{ L_{(r_1,r_2)}(\Omega), \begin{array}{l} 2r_1r_2-(n-s)r_1-sr_2-\alpha=0,\quad \alpha>0,\\[4pt] \infty>r_1> \begin{cases} 1, & \text{if } s=1,\\ s/2, & \text{if } s\ge 2, \end{cases}\\[10pt] \infty>r_2> \begin{cases} 1, & \text{if } n-s=1,\\ (n-s)/2, & \text{if } n-s\ge 2. \end{cases} \end{array} \right\}. \]

We note that any two spaces \(L_{(r_1,r_2)}(\Omega)\), \(L_{(\bar r_1,\bar r_2)}(\Omega)\), belonging to \(X^\alpha\), are distinct if only \(r_1\ne \bar r_1\) and \(r_2\ne \bar r_2\).

By \(D_\alpha\) we denote the set of those points \((r_1,r_2)\) of the plane \(r_1 O r_2\) for which \(L_{(r_1,r_2)}(\Omega)\subset X^\alpha\).

Theorem 1. If \(p_i=2r_i'\) \((i=1,2)\), \(1/r_i+1/r_i'=1\), \((r_1,r_2)\in D_\alpha\), then

\[ \|v\|_{L_2(\Omega)} \leq c_0(\operatorname{mes}\Omega_{n-s})^{1/\rho-1/q}\|\nabla v\|_{L_2(\Omega)},\qquad \forall v\in \dot W_2^1(\Omega); \tag{2} \]

\[ \|u\|_{L_{(p_1,p_2)}(\Omega)} \leq c\|u\|_{W_2^1(\Omega)},\qquad \forall u\in W_2^1(\Omega), \tag{3} \]

\[ \|v\|_{L_{(p_1,p_2)}(\Omega)} \leq c_1\|\nabla v\|_{L_2(\Omega)},\qquad \forall v\in \dot W_2^1(\Omega); \tag{4} \]

\[ \|v\|_{L_{(p_1,p_2)}(\Omega)}^2 \leq \tilde c\,[\varepsilon\beta\|\nabla v\|_{L_2(\Omega)}^2 +\varepsilon^{-1/(1-\beta)}(1-\beta)\|v\|_{L_2(\Omega)}^2], \qquad \forall v\in \dot W_2^1(\Omega), \tag{5} \]

where \(\varepsilon\) is any positive number; the numbers \(\beta,\rho\), and \(q\) are such that \(0<\beta<1\); \(1<\rho<2<q\), \(1/2r_2<1/\rho-1/q<1/2(n-s)\); the constants \(c,c_1\) depend on \(\Omega\), while \(c_0,\tilde c\) do not depend on \(\Omega\); \(\Omega_i\) \((i=s,n-s)\) satisfy the cone condition.

Let \((r_1,r_2)\) be a fixed point belonging to \(D_\alpha\). We shall assume that the coefficients and the free term of equation (1) satisfy the conditions

\[ \nu\xi_i\xi_i \leq a_{ij}\xi_i\xi_j \leq \mu\xi_i\xi_i,\qquad \nu,\mu=\operatorname{const}>0; \tag{6} \]

\[ \|a_i^2,b_i^2,a\|_{L_{(r_1,r_2)}(\Omega)} \leq \mu; \tag{7} \]

\[ \|f\|_{L_{(p_1',p_2')}(\Omega)}<\infty . \tag{8} \]

We denote

\[ a_0=(\operatorname{mes}\Omega)^{-1}\int_\Omega a(x)\,dx,\qquad a(x)=a^+(x)-a^-(x), \tag{9} \]

where

\[ a^+(x)=\max\{a(x)-a_0;0\},\qquad a^-(x)=-a_0+\max\{-a(x)+a_0;0\}, \]

\[ M\equiv \max\{\|(b_i-a_i)^2\|_{L_{(r_1,r_2)}(\Omega)},\ \|a^+\|_{L_{(r_1,r_2)}(\Omega)}\}, \]

\[ c_2=2Mc^2\nu^{-2}(1+2\nu)(\beta^{-1}-1)\beta^{1/(1-\beta)} . \]

A generalized solution from \(W_2^1(\Omega)\) of the first boundary-value problem for equation (1) is a function \(u(x)\) belonging to \(W_2^1(\Omega)\) and satisfying the identity

\[ L(u,\eta)\equiv \int_\Omega \left[(a_{ij}u_{x_j}+a_i u)\eta_{x_i} -(b_i u_{x_i}+au)\eta\right]\,dx = -\int_\Omega f\eta\,dx \]

for every \(\eta(x)\) from \(W_2^1(\Omega)\), and the condition

\[ u(x)-\varphi(x)\in \dot W_2^1(\Omega), \]

where \(\varphi(x)\) is an extension from the boundary \(\Gamma\) to the whole domain \(\Omega\) of the function \(\tilde\varphi(z)\) determining the boundary values of \(u(x)\), i.e.

\[ u|_\Gamma=\varphi|_\Gamma . \tag{10} \]

Theorem 2. Let \((r_1,r_2)\subset D_\alpha\). If conditions (6), (7) and

\[ \left(2c_2+\frac{8}{\nu}a_0\right)c_0^2(\operatorname{mes}\Omega_{n-s})^{2/\rho-2/q}<1, \tag{11} \]

are fulfilled, then problem (1), (10) has at most one generalized solution from \(W_2^1(\Omega)\).

Corollary 1. For any differential operator \(L\), defined in \(\Omega\) and satisfying conditions (6) and (7), the uniqueness theorem for the Dirichlet problem is valid in any subdomain \(\Omega'=\Omega_s'\times\Omega_{n-s}'\) of the domain \(\Omega\), provided only that \(\operatorname{mes}\Omega\) is sufficiently small.

Corollary 2. In domains of arbitrary dimension the uniqueness theorem for the Dirichlet problem holds for the operators \(L-\lambda E\), \(\lambda \geqslant \lambda_0\), if \(L\) satisfies conditions (6) and (7), and \(\lambda_0\) is sufficiently large.

Corollary 3. Theorem 2 is valid for any point \((r_1,r_2)\in D_\alpha\).

Corollary 4. Theorem 2 is valid for any \(\alpha>0\).

Theorem 3. Let \((r_1,r_2)\in D_\alpha\). If conditions (6), (7) are satisfied and, in addition, the estimate

\[ c_2+\frac{4}{\gamma}a_0 \leqslant 0, \tag{12} \]

holds, then problem (1), (10) has a generalized solution from \(W_2^1(\Omega)\) for any \(f\) from \(L_{(p'_2,p'_2)}(\Omega)\) and \(\varphi(x)\) from \(W_2^1(\Omega)\).

Define the sets \(\widehat X^\alpha\) and \(\widetilde X^\alpha\):

\[ \widehat X^\alpha=\bigcup_{(r_1,r_2)\in D_\alpha} L_{r_1,r_2}(\Omega),\qquad \widetilde X^\alpha=\bigcup_{\substack{p_i=2r'_i\\ (r_1,r_2)\in D_\alpha}} L_{(p'_1,p'_2)}(\Omega). \]

Theorem 4. If \(a_i^2,b_i^2,a\in \widehat X^\alpha\) and

\[ \|a_i^2\|_{L_{(r_1^i,r_2^i)}(\Omega)},\qquad \|b_i^2\|_{L_{(\widetilde r_1^i,\widetilde r_2^i)}(\Omega)},\qquad \|a\|_{L_{\simeq(r_1,r_2)}(\Omega)} \leqslant \mu, \]

then, under conditions (6) and (11), problem (1), (10) has at most one generalized solution from \(W_2^1(\Omega)\).

Theorem 5. Under the conditions of the preceding theorem, problem (1), (10) has a generalized solution \(u\) from \(W_2^1(\Omega)\) for any \(f\in \widetilde X^\alpha\) and \(\varphi\) from \(W_2^1(\Omega)\).

Let

\[ \widehat X_\alpha^p=\bigcup_{\substack{p<r_1,r_2\\ (r_1,r_2)\in D_\alpha}} L_{(r_1,r_2)}(\Omega),\qquad \overline X_p^\alpha=\left\{L_{(r_1,r_2)}(\Omega);\ \begin{array}{l} p\leqslant r_1,r_2,\\ (r_1,r_2)\in D_\alpha \end{array} \right\}. \]

Lemma 2. \(\widehat X_p^\alpha\subset L_p(\Omega)\).

From Theorem 4 and Lemma 2 it follows:

Theorem \(4'\). Let \(a_i^2,b_i^2,a\in L_p\), \(p\geqslant 1\). If \(a_i^2,b_i^2,a\in \widehat X_p^\alpha\) and

\[ \|a_i^2\|_{L_{(r_1^i,r_2^i)}(\Omega)},\qquad \|b_i^2\|_{L_{(\widetilde r_1^i,\widetilde r_2^i)}(\Omega)},\qquad \|a\|_{L_{\simeq(r_1,r_2)}(\Omega)}\leqslant \mu, \]

where

\[ L_{(r_1^i,r_2^i)}(\Omega),\qquad L_{(\widetilde r_1^i,\widetilde r_2^i)}(\Omega),\qquad L_{\simeq(r_1,r_2)}(\Omega)\in \overline X_p^\alpha, \]

then, under conditions (6) and (11), problem (1), (10) cannot have more than one generalized solution.

Theorem \(5'\). Under the conditions of the preceding theorem, problem (1), (10) has a generalized solution \(u\) from \(W_2^1(\Omega)\) for any \(f\) from \(\widetilde X^\alpha\) and \(\varphi\) from \(W_2^1(\Omega)\).

Denote

\[ I_1(v,\xi)=\int_\Omega (a_i v\eta_{x_i}-b_i v_{x_i}\eta-a^+v\eta)\,dx,\qquad [u,v]=\int_\Omega (a_{ij}u_{x_i}\overline{v}_{x_j}+\overline a u\overline v)\,dx, \]

\[ l_\lambda(\eta)=-L(\varphi,\eta)-(f,\eta)-\lambda(\varphi,\eta). \tag{13} \]

We define the operators \(A\) and \(B\) by the equalities

\[ [Av,\eta]=I_1(v,\eta),\qquad [Bv,\eta]=(v,\eta). \tag{14} \]

Theorem 6. Suppose that conditions (6), (7), and (12) are satisfied; then the problem

\[ Lu-\lambda u=f, \tag{15} \]

\[ u|_{\Gamma}=\varphi|_{\Gamma} \tag{16} \]

has a unique generalized solution in \(W_2^1(\Omega)\) for arbitrary \(\varphi\) in \(W_2^1(\Omega)\) and \(f\) in \(L_p\), for all \(\lambda\) in the complex plane except for a countable set \(\lambda=\lambda_k,\ k=1,2,\ldots\).

To each \(\lambda=\lambda_k\) there corresponds a finite number of linearly independent solutions in \(W_2^1(\Omega)\) of the equation

\[ Lv-\lambda v=0. \tag{17} \]

For \(\lambda=\lambda_k,\ k=1,2,\ldots\), problem (15), (16) has a solution if and only if the conditions

\[ l_{\lambda_k}(\omega_k^i)=0,\qquad i=1,2,\ldots,N_k, \tag{18} \]

are satisfied, where \(\omega_k^i\) are all the solutions in \(W_2^1\) of the equation

\[ (E+A^*)\omega+\overline{\lambda}_k B\omega=0 \tag{19} \]

adjoint to the equation

\[ v+\lambda(E+A)^{-1}Bv=0. \tag{20} \]

The number of conditions (18) coincides with the number of linearly independent solutions of equation (20) for \(\lambda=\lambda_k\).

Theorem 7. Suppose that for all operators \(L^m,\ m=1,2,\ldots\), of the form (1), the conditions of Theorem 3 are satisfied with the same constants. Suppose that \(a_{ij}^m(x)\), while remaining uniformly bounded, converge almost everywhere to \(a_{ij}(x)\), and that the functions \(a_i^m,\ b_i^m,\ a^m,\ f^m,\ \varphi^m\) converge to \(a_i,\ b_i,\ a,\ f,\ \varphi\) in the norms \(L_{(2r_1,2r_2)}\), \(L_{(2r_1,2r_2)}\), \(L_{(r_1,r_2)}\), \(L_{(p_1',p_2')}\), \(W_2^1\), respectively.

Then the generalized solutions \(u^m\) in \(W_2^1(\Omega)\) of the problems

\[ L^m u=f^m,\qquad u|_{\Gamma}=\varphi^m|_{\Gamma} \tag{21} \]

converge strongly in \(W_2^1(\Omega)\) to the generalized solution in \(W_2^1(\Omega)\) of the limiting problem (1), (10).

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR

Received
15 II 1967

CITED LITERATURE

1. O. A. Ladyzhenskaya, N. N. Ural’tseva, Linear and Quasilinear Equations of Elliptic Type, “Nauka,” 1964.
2. A. Kh. Gudiev, DAN 147, No. 4 (1962).
3. A. Kh. Gudiev, Izv. AN UzSSR, No. 5 (1965).