Abstract
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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}(\Omega_s,\Omega_{n-s})
=
\left|\,|f(x_s,x_{n-s})|{L\right|}(\Omega_s){L,}(\Omega_{n-s})
]
where
[
|\cdot|{L}(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_20), 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,\qquad})^{1/\rho-1/q}|\nabla v|_{L_2(\Omega)
\forall v\in \dot W_2^1(\Omega);
\tag{2}
]
[
|u|{L,\qquad}(\Omega)} \leq c|u|_{W_2^1(\Omega)
\forall u\in W_2^1(\Omega),
\tag{3}
]
[
|v|{L,\qquad}(\Omega)} \leq c_1|\nabla v|_{L_2(\Omega)
\forall v\in \dot W_2^1(\Omega);
\tag{4}
]
[
|v|{L^2}(\Omega)
\leq \tilde c\,[\varepsilon\beta|\nabla v|{L_2(\Omega)}^2
+\varepsilon^{-1/(1-\beta)}(1-\beta)|v|^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 \leq \mu;}(\Omega)
\tag{7}
]
[
|f|{L<\infty .}(\Omega)
\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,\ |a^+|}(\Omega){L},}(\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,\qquad}(\Omega)
|b_i^2|{L,\qquad}(\Omega)
|a|{L \leqslant \mu,}(\Omega)
]
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,\qquad}(\Omega)
|b_i^2|{L,\qquad}(\Omega)
|a|{L\leqslant \mu,}(\Omega)
]
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|
\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|
\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
- O. A. Ladyzhenskaya, N. N. Ural’tseva, Linear and Quasilinear Equations of Elliptic Type, “Nauka,” 1964.
- A. Kh. Gudiev, DAN 147, No. 4 (1962).
- A. Kh. Gudiev, Izv. AN UzSSR, No. 5 (1965).