Abstract Generated abstract
The paper studies the Dirichlet problem for uniformly elliptic second order equations in divergence form whose lower order coefficients may be discontinuous and unbounded, but belong to mixed norm spaces rather than ordinary Lebesgue classes. It develops embedding and interpolation estimates for mixed norm spaces, then applies them to generalized solutions in Sobolev space. The main results give conditions for uniqueness and existence of the Dirichlet problem, including small subdomains and the case of a sufficiently negative zeroth order coefficient. Additional theorems establish local and global boundedness criteria for generalized solutions, with dependence of the bounds specified in terms of ellipticity constants, mixed norm data, domain geometry, and boundary values.
Full Text
UDC 517.946
MATHEMATICS
A. Kh. Gudiev
ON THE DIRICHLET PROBLEM FOR EQUATIONS WITH COEFFICIENTS BELONGING TO SPACES WITH MIXED NORM
(Presented by Academician S. L. Sobolev on 12 I 1968)
Known results \((^3,\ ^4{}^{-}{}^6)\) concern the first boundary-value problem for the uniformly elliptic equation
\[ Lu \equiv \frac{\partial}{\partial x_i}\left(a_{ij}u_{x_j}+a_i u\right)+b_i u_{x_i}+au = \frac{\partial f_i}{\partial x_i}+f \tag{I} \]
with discontinuous unbounded lower-order coefficients. In particular, in \((^2)\) conditions are given for the existence and uniqueness of a generalized solution from \(W_2^1\) of the Dirichlet problem for equation (I). These conditions are, in a certain sense, not only sufficient but also necessary, i.e., in terms of \(L_p\)-spaces these results cannot be improved.
However, equations are known a priori whose coefficients do not satisfy the indicated conditions, but for which existence and uniqueness theorems for the Dirichlet problem hold. The present paper is devoted to the study of generalized solutions of the Dirichlet problem for such equations. Existence and uniqueness theorems are obtained for the solution of the Dirichlet problem for equations of the form (I) with coefficients belonging to a space with mixed norm. In addition, conditions for boundedness of the generalized solution are given. The results obtained cannot be improved in terms of the spaces under consideration. The paper uses the results of \((^2,\ ^3,\ ^6)\).
Let \(s\) be a natural number not exceeding \(n\); \(E^n\) is \(n\)-dimensional Euclidean space. We shall denote each point \(x \in E^n\) in the form of a pair of vectors \((x_s,x_{n-s})\) or \((x^{(1)},x^{(2)})\), where
\[ x_s(x_1,\ldots,x_s)\equiv x^{(1)}(x_1^{(1)},\ldots,x_{n_1}^{(1)}),\quad x_{n-s}(x_{s+1},\ldots,x_n)\equiv x^{(2)}(x_1^{(2)},\ldots,x_{n_2}^{(2)}), \]
\(n_1=s,\ n_2=n-s\). Let \(E^{n_i}\) be the \(n_i\)-dimensional space of vectors \(x^{(i)}\) \((i=1,2)\). By \(|x^{(i)}-y^{(i)}|\) we shall denote the distance between the points \(x^{(i)}\) and \(y^{(i)}\) in \(E^{n_i}\); \(|x-y|\) is the distance between \(x\) and \(y\) in \(E^n\).
Let \(D\) be a bounded domain in \(E^n\) and \(D_1=D\cap (x^{(2)}=\mathrm{const})\); \(D_2=\operatorname{pr}_{E^{n-s}}D\). Denote by \(L_{(p_1,p_2)}(D_1,D_2)\) the set of functions defined in \(D\) and satisfying the condition
\[ \left\|\, \|f(x)\|_{L_{p_1}(D_1)} \,\right\|_{L_{p_2}(D_2)}<\infty, \]
where
\[ \|\cdot\|_{L_{p_i}(D_i)}= \begin{cases} \left(\displaystyle\int_{D_i}|\cdot|^{p_i}\,dx^{(i)}\right)^{1/p_i}, & \text{if } 1\le p_i<\infty,\\[1.2em] \operatorname*{vrai\ max}\limits_{x^{(i)}\in D_i}|\cdot|, & \text{if } p_i=\infty. \end{cases} \]
If in \(L_{(p_1,p_2)}(D_1,D_2)\) one introduces the norm by the equality
\[ \|f\|_{L_{(p_1,p_2)}(D_1,D_2)} = \left\|\, \|f\|_{L_{p_1}(D_1)} \,\right\|_{L_{p_2}(D_2)}, \]
then \(L_{(p_1,p_2)}(D_1,D_2)\) will be a complete normed space.
In the case when we are dealing with only one space \(L_{(p_1,p_2)}(D_1,D_2)\), one may use a more convenient notation for them, namely
\[
D_{(p_1,p_2)}(D_1,D_2)\equiv L_{(p_1,p_2)}(D)\equiv L_p(D).
\]
Let \(a\) be any nonnegative number; \(k\) a positive integer; \(\beta\) a fixed positive number. Denote by \(Y_\beta\) the following class of Banach spaces:
\[
Y_\beta=\{L_{(\beta p,p)}(D_1,D_2);\ \max\{1/\beta,1\}<p<\infty\}.
\]
For \(\beta\ne 1\), \(Y_\beta\) forms a continuous scale, different from the scale of \(L_p\) spaces. Obviously,
\[
Y_1=\{L_p\}
\]
(\(\{L_p\}\) is the scale of \(L_p\)-spaces). By \(Y\) we denote the set of scales \(Y_\beta\) for all possible nonnegative values of \(\beta\), i.e.
\[
Y=\{Y_\beta;\ 0<\beta<\infty\}.
\]
From each scale \(Y\) we consider one representative, and the set of these representatives will be denoted by \(X_k^\alpha\). The class \(X_k^\alpha\) is defined as follows:
\[
X_k^\alpha=
\left\{
\begin{array}{l}
L_{(r_1,r_2)}(D_1,D_2);\quad
kr_1r_2-(n-s)r_1-sr_2-\alpha=0;\\[3pt]
\infty>r_1>
\begin{cases}
1, & \text{if } 1<s<k,\\
s/k, & \text{if } s\ge k;
\end{cases}\\[8pt]
\infty>r_2>
\begin{cases}
1, & \text{if } n-s<k,\\
(n-s)/k, & \text{if } n-s\ge k.
\end{cases}
\end{array}
\right.
\]
Denote by \(\Omega_k^{(\alpha)}\) the set of those points \((r_1,r_2)\) of the plane \(r_1Or_2\) for which
\[
L_{(r_1,r_2)}(D_1,D_2)\in X_k^\alpha .
\]
Let \(p\) be an integer greater than or equal to one. The class \(X_{k;p}^\alpha\) is defined by the equality
\[
X_{k;p}^\alpha=\{L_{(r_1,r_2)}(D_1,D_2);\ (r_1,r_2)\in \Omega_k^{(\alpha)}\cap\{p\le r_1,r_2\}\}.
\]
Put
\[
\Omega_{k;p}^{(\alpha)}=\Omega_k^{(\alpha)}\cap\{p\le r_1,r_2\}.
\]
Theorem 1. If \(D\) is a bounded domain of \(n\)-dimensional Euclidean space and
\[
s/p_1+(n-s)/p_2-1\le s/q_1+(n-s)/q_2;\qquad
1\le p_2\le p_1\le q_1,q_2,
\]
then for every function \(f\in \mathring W_{(p_1,p_2)}^1(D)\) the estimate
\[
\|f\|_{L_{(q_1,q_2)}(D)}
\le C_1\|\nabla f\|_{L_{(p_1,p_2)}(D)}^\sigma
\|f\|_{L_{(p_1,p_2)}(D)}^{1-\sigma},
\]
holds, where \(C_1\) is a constant independent of \(D\), and \(\sigma\) is determined from the equality
\[
\sigma=s(1/p_1-1/q_1)+(n-s)(1/p_2-1/q_2).
\]
Theorem 2. If \(D\) is a bounded domain of \(n\)-dimensions, star-shaped with respect to some ball, and \(f\in W_2^1(D)\), then, for
\[
n/2-1\le s/p_1+(n-s)/p_2,\qquad 2\le p_1,p_2,
\]
the function \(f\) belongs to the space \(L_{(p_1,p_2)}(D)\), and, moreover, the estimate
\[
\|f\|_{L_{(p_1,p_2)}(D)}
\le C_2\|f\|_{W_2^1(D)}.
\]
Theorem 3. Let the positive numbers \(\alpha,p_1,p_2\) be such that
\[
\alpha>\max\{1-2/p_1,\ 1-2/p_2\},\qquad
s/p_1+(n-s)/p_2=n/2-\alpha;
\]
\[
2\le p_1,p_2,
\]
and let the function \(f\in W_2'(D)\) satisfy the condition
\[
\int_D f(x)\,dx=0;
\]
then
\[
\|f\|_{L_{(p_1,p_2)}(D)}
\le C_2\|\nabla f\|_{L_2(D)}^\alpha
\|f\|_{L_2(D)}^{1-\alpha},
\]
where the constant \(C_2\) depends on the domain \(D\), but remains unchanged under a similarity transformation.
Theorem 4. If the numbers \(\alpha, p_1, p_2\) satisfy the conditions
\[ \alpha>\max\{1-2/p_1,\;1-2/p_2\},\qquad s/p_1+(n-s)/p_2=n/2-\alpha, \]
\[ 2\leq p_1,p_2, \]
then for any function \(f\) in \(W_2^1(D)\) the estimate
\[ \|f\|_{L_{(p_1,p_2)}(D)} \leq C_3\bigl(\|f\|_{L_2(D)}+\|\nabla f\|_{L_2(D)}^{\alpha}\|f\|_{L_2(D)}^{1-\alpha}\bigr). \]
holds.
Theorem 5. If \(D\) is a bounded domain of \(n\)-dimensional Euclidean space and \(f\in \dot W_2^1(D)\), then the estimate
\[ \|f\|_{L_2(D)} \leq C_1\bigl[\operatorname{mes}(\operatorname{pr}_{E^{\,n-s}}D)\bigr]^{1/(n-s)}\|\nabla f\|_{L_2(D)} \]
holds, where \(C_1\) is the constant from Theorem 1.
Theorem 6. If the function \(f\in W_2^1(D)\) has a bounded \(\operatorname{vrai\,max}_{S} f(x)\) and, for \(k\geq k_0\geq \operatorname{vrai\,max}_{S} f(x)\), satisfies the inequalities
\[ \int_{A_k}|\nabla f|^2\,dx \leq \gamma\left[ \int_{A_k}(f-k)^2\,dx +k^2\|1\|_{L_{(p_1,p_2)}(A_k)}^2 \right], \tag{1} \]
where
\[ s/p_1+(n-s)/p_2=(n/2-1)(1+\varepsilon),\qquad \varepsilon>0, \tag{2} \]
then \(\operatorname{vrai\,max}_{D} f(x)\) is bounded.
Theorem 7. Let \(D\) be an \(n\)-dimensional bounded domain with piecewise smooth boundary \(S\), and let the function \(f\in W_2^1(D)\), for \(k\geq \widetilde{k}\), satisfy inequalities (1), (2).
Then \(\operatorname{vrai\,max}_{D} f(x)\) is estimated by a constant depending on \(\widetilde{k}, C_1, \gamma, \varepsilon, n, \|f\|_{L_1(A_{\widetilde{k}})}\), and on the boundary \(S\).
Theorem 8. Let \(f\in W_2^1(K_{(1+\delta)R})\), \(K_{(1+\delta)R}\subset D\), \(\delta>0\), and suppose that for any pair of balls \(K_{(1-\sigma)\rho}\) and \(K_\rho\), concentric with \(K_{(1+\delta)R}\), with \(\widetilde R\leq \rho(1-\sigma)<\rho\leq (1+\delta)R\), the following holds:
\[ \int_{A_{k,\rho(1-\sigma)}}|\nabla f|^2\,dx \leq \gamma\left[ (\sigma\rho)^{-2}\int_{A_{k,\rho}}(f-k)^2\,dx +\rho^{-2r}k^2\|1\|_{L_{(p_1,p_2)}(A_{k,\rho})}^2 \right], \]
where \(\delta,\gamma,k,p_1,p_2,r\) are fixed positive numbers, and
\[ s/p_1+(n-s)/p_2=n/2-1+r. \]
Then \(\operatorname{vrai\,max}_{K_R}\) does not exceed a certain number determined only by
\[ k,\delta,n,r,\gamma,p_1,p_2 \quad\text{and}\quad R^{-n}\int_{K_{(1+\delta)R}}|f(x)|^2\,dx; \]
\(A_{k,\rho}\) is the set of points \(x\) in \(K_\rho\) for which \(f(x)>k\).
Let us now consider, in a bounded domain \(D\subset E^n\) with boundary \(S\), the equation
\[ Lu\equiv \frac{\partial}{\partial x_i}(a_{ij}u_{x_j}+a_i u) +b_i u_{x_i}+au = \frac{\partial f_i}{\partial x_i}+f \tag{3} \]
with coefficients satisfying the conditions:
\[ \nu\xi_i\xi_i\leq a_{ij}\xi_i\xi_j\leq \mu\xi_i\xi_i,\qquad \nu,\mu=\mathrm{const}>0; \tag{4} \]
\[ \|a_i^2;\,b_i^2;\,a\|_{L_{(r_1,r_2)}(D_1,D_2)}\leq K; \tag{5} \]
\[ \left\|\sum f_i^2\right\|_{L_1(D)},\qquad \|f\|_{L_{(q_1,q_2)}(D_1,D_2)}<\infty; \tag{6} \]
\[ (r_1,r_2)\in \Omega_2^{(\alpha)},\qquad (q_1,q_2)\in \Omega_{n/2+1}^{(0)}. \tag{7} \]
Theorem 9. The Dirichlet problem in \(D' \subset D\) for equation (3) has no more than one generalized solution from \(W_2^1\), if conditions (4), (5), (7) are satisfied and \(\operatorname{mes}(\operatorname{pr}_{E^n} D')\) is sufficiently small.
Theorem 10. The Dirichlet problem for equation (3) in domains \(D' \subset D\) of arbitrary size has no more than one generalized solution from \(W_2^1\), provided that conditions (4), (5), (7) are satisfied and \(a(x) < -N\), where \(N\) is a sufficiently large positive number.
Theorem 11. Let \((r_1,r_2) \in \Omega_2^{(\alpha)}\), \((q_1,q_2) \in \Omega_{n/2+1}^{(0)}\), and suppose that conditions (4), (5), (6), (7) are satisfied and \(a(x) < -N\); then the Dirichlet problem for equation (3) has a generalized solution \(u(x)\) from \(W_2^1\) for any boundary value \(\varphi(x)\) from \(W_2^1\).
Theorem 12. Let \((r_1,r_2) \in \Omega_2^{(\alpha)}\) and suppose that the conditions
\[ \nu \xi_i\xi_i \leqslant a_{ij}\xi_i\xi_j \leqslant \mu \xi_i\xi_i,\qquad \nu,\mu=\operatorname{const}>0,\qquad \|a_i^2,b_i^2,f_i^2,a,f\|_{L_{(r_1,r_2)}(D)}\leqslant K. \]
are satisfied. Then for any generalized solution \(u(x)\) from \(W_2^1\) of equation (1) in any subdomain \(D'\) of the domain \(D\), the quantity \(\operatorname{vrai\,max}_{D'} u(x)\) is finite, i.e.
\[ \operatorname{vrai\,max}_{D'} |u(x)|<C, \]
where \(C\) is a constant depending only on \(\nu,\mu,K,r_1,r_2,s,n,\|u\|_{L_2(D)}\) and the distance of \(D'\) from the boundary \(S\) of the domain \(D\).
Theorem 13. If the conditions of the preceding theorem are satisfied and
\[ \operatorname{vrai\,max}_{S}|u(x)|<M, \]
then
\[ \operatorname{vrai\,max}_{D}|u(x)| \]
is bounded.
Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR
Received
5 I 1968
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