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MATHEMATICS
A. A. NOVRUZOV
ON PROPERTIES OF SOLUTIONS OF ELLIPTIC EQUATIONS
(Presented by Academician I. G. Petrovskii, 10 IV 1961)
In papers \((^{1-4})\) a number of properties were obtained for solutions of elliptic equations of the form
\[ \sum_{i,k=1}^{n} A_{ik}(x)\frac{\partial^2 u}{\partial x_i \partial x_k} + \sum_{i=1}^{n} B_i(x)\frac{\partial u}{\partial x_i} + C(x)u=0, \tag{1} \]
where \(x=(x_1,x_2,\ldots,x_n)\), which follow from the following lemma:
Let \(K_R\) be the ball of radius \(R\) with center at the origin \(O\), and let \(G \subset K_R\) be a domain containing the center of the ball and having limit points on the boundary of the ball. Let the volume of \(G\) be less than \(R^n/M\), where \(M\) is a sufficiently large constant. Then, for a solution \(u(x)\) of equation (1) which vanishes on that part of the boundary of the domain \(G\) which lies strictly inside \(K_R\), the inequality
\[ 2u(0) \leq \max_{x\in \overline{G}} u(x) \tag{2} \]
holds.
In these papers it is assumed that \(C(x)\leq 0\). The constant \(M\) in this case, generally speaking, increases as \(R\) increases. In those cases when an estimate independent of \(R\) is needed (in the proof of the Phragmén—Lindelöf theorem), it is assumed that \(B_i(x)\equiv 0\).
The present note is devoted to inequality (2). In it an estimate uniform with respect to \(R\) is obtained when \(B_i(x)\not\equiv 0\).
Denote by \(K_R\) the open ball in \(n\)-dimensional space of arbitrary radius \(R\) with center at the origin \(O\), by \(\mu_n E\) the \(n\)-dimensional Lebesgue measure of a set \(E\) lying in \(n\)-dimensional space. Denote by \(G\) a domain lying inside \(K_R\), containing the center of the ball \(K_R\) and having limit points on the boundary of \(K_R\), and by \(S_\rho^*\) the intersection of the domain \(G\) with the sphere of radius \(\rho\) and center at the origin \(O\). Consider in \(G\) the equation
\[ \sum_{i,k=1}^{n} \frac{\partial}{\partial x_i} \left(a_{ik}(x)\frac{\partial u}{\partial x_i}\right) + \sum_{i=1}^{n} b_i(x)\frac{\partial u}{\partial x_i} + c(x)u=0. \tag{3} \]
With respect to its coefficients it is assumed that in \(G\) the inequalities
\[ \sum_{i,k=1}^{n} a_{ik}(x)\xi_i\xi_k > \alpha \sum_{i=1}^{n}\xi_i^2, \qquad \alpha>0 \tag{4} \]
hold for any real \(\xi_i\), \(\sum_{i=1}^{n}\xi_i^2\ne 0\),
\[ (\operatorname{grad} a,\vec{\rho})\big|_G \geq 0, \qquad |a_{ik}(x)|\leq a_0; \tag{5} \]
\[ \operatorname{div}\mathbf b \geq 0; \tag{6} \]
\[ (\mathbf b,\vec\rho)\big|_{G}\leq 0; \tag{7} \]
\[ c(x)\leq 0, \tag{8} \]
where
\[ a=\sum_{j=1}^{n}\sum_{i,k=1}^{n}\frac{\partial a_{ik}}{\partial x_j}\cos\alpha_i\cos\alpha_k,\qquad \mathbf b=(b_1,b_2,\ldots,b_n), \]
and \(\vec\rho\) is the radius vector from the center to the given point.
Lemma. Let \(\Gamma\) be that part of the boundary of the domain \(G\) which lies strictly inside \(K_R\). Suppose that equation (3) is defined in \(G\). There exists a constant \(M\), depending only on the constants \(a\) and \(a_0\) in inequalities (4), (5) and on the dimension \(n\) of the space, such that if, for every \(R\),
\[ \mu_n G<\mu_n K_R/M, \tag{9} \]
then, for any positive solution \(u(x)\) of equation (3) in \(G\), continuous in \(\bar G\) and vanishing on \(\Gamma\), the inequality
\[ 2u_0<\max_{x\in \bar G}u(x),\qquad \text{where } u_0=u(0). \]
holds.
Proof. Suppose the lemma is false. Denote by \(E_t\) the set of points \(x\in G\) for which \(u(x)=t,\ t\in [u_0/2,u_0]\). Let \(E_t^*(r)\) be that part of \(E_t\) which lies inside \(K_r\) \((0<r<R)\). Denote by \(G_t\) the set of points \(x\in G\) at which \(u(x)>t\). Let \(G_r^t=G_t\cdot K_r\). Then, according to (3), there exist \(t_0\) \((u_0/2<t_0<u_0)\) and a set \(E\subset [0,R]\) of positive measure such that, for all \(r\in E\),
\[ \int_{E_{t_0}^*(r)}\left|\frac{\partial u}{\partial \tau}\right|\,dl > M u_0\,\bigl(\mu_n G_r^t\bigr)^{(n-2)/n}, \]
where \(\tau\) is the exterior normal to the surface \(E_{t_0}^*(r)\). Denote by \(G_1\) the set of points \(x\in G\) for which \(u(x)>t_0\). Let \(G_r=G_1\cdot K_r,\ r\in[0,R]\). Consider the function \(f(r)=\mu_n G_r\) on the interval \([0,R]\). By inequality (9), the function \(f(r)\) satisfies all the conditions of Lemma 6 of the paper \((^3)\). Consequently, there exist nonintersecting intervals \([\alpha_1,\beta_1]\), \([\alpha_2,\beta_2],\ldots,[\alpha_s,\beta_s]\), belonging to the interval \([0,R]\), such that
\[ \tfrac12 f(\alpha_j)\leq f(\beta_j)\leq 2f(\alpha_j)\qquad (j=1,2,\ldots,s); \]
\[ f(r)\geq \min [f(\alpha_j),f(\beta_j)]\qquad \text{for } \alpha_j<r<\beta_j; \]
\[ \beta_j-\alpha_j>\frac{R^{1/2n}M^{1/2n}}{C_n}\,f^{1/2n}(r); \qquad \sum_{j=1}^{s}(\beta_j-\alpha_j)>\frac{R}{4}, \]
where \(C_n\) is a constant.
By the same lemma there exists such a \(\lambda_j\in[\alpha_j,\beta_j]\), \(j=1,2,\ldots,s\), that
\[ \int_{E_{t_0}^*(\lambda_j)}\left|\frac{\partial u}{\partial \tau}\right|\,dl > M u_0\,(\mu_n G_{\lambda_j})^{(n-2)/n}; \tag{10} \]
\[ f(\beta_j)\leq 2f(\lambda_j); \tag{11} \]
\[ \beta_j-\lambda_j>\frac{R^{1/2n}M^{1/2n}}{C_n}\,f^{1/2n}(r). \tag{12} \]
Further, by inequality (11), there exist \(r_0\) \((\lambda_j<r_0<(3\lambda_j+\beta_j)/4)\), \(r_1\) \(((\lambda_j+\beta_j)/2<r_1<\beta_j)\), such that
\[ \mu_{n-1}S_{r_0}^*<\frac{\mu_n G_{\lambda_j}}{\beta_j-\lambda_j}; \tag{13} \]
\[ \mu_{n-1}S_{r_1}^{*}<\frac{2\mu_n G_{\lambda_j}}{\beta_j-\lambda_j}. \tag{14} \]
Consider the integral
\[ \int_{S_r^*}\sum_{i,k=1}^{n} a_{ik}(x)\cos\nu_i\,\frac{\partial u}{\partial x_k}\,ds, \]
where \(\cos\nu_i\) are the direction cosines of the normal to \(S_r^*\). In view of the inequalities (5), (13), and (14), there will be an \(r'\) \((r_0<r'<r_1)\) such that
\[ \int_{S_{r'}^*}\sum_{i,k=1}^{n} a_{ik}(x)\cos\nu_i\,\frac{\partial u}{\partial x_k}\,ds \le \frac{24\mu_0 a_0 n^2\mu_n G_{\lambda_j}}{(\beta_j-\lambda_j)^2}. \tag{15} \]
Denote by \(S_{r'}\) the boundary of the domain \(G_{r'}=G_1\cdot K_{r'}\). Clearly, \(S_{r'}\) consists of points belonging to \(E_{t_0}^*\) and \(S_{r'}^*\). Applying Green’s formula in the domain \(G_{r'}\), from equation (3) we obtain
\[ \int_{E_{t_0}^*(r')}\sum_{i,k=1}^{n} a_{ik}\cos\nu_i\cos\nu_k\,\frac{\partial u}{\partial \tau}\,dl + \int_{S_{r'}^*}\sum_{k,i=1}^{n} a_{ik}\cos\nu_i\,\frac{\partial u}{\partial x_k}\,ds + \]
\[ + \int_{S_{r'}^*}(u-t_0)\sum_{i=1}^{n} b_i\cos\nu_i\,ds - \iint_{G_{r'}}(u-t_0)\operatorname{div}\mathbf b\,d\omega + \iint_{G_{r'}}cu\,d\omega =0. \tag{16} \]
By virtue of inequalities (4) and (10),
\[ \int_{E_{t_0}^*(r')}\sum_{k=1}^{n} a_{ik}\cos\nu_i\cos\nu_k\,\frac{\partial u}{\partial \tau}\,dl < -M\alpha u_0\,(\mu_n G_{\lambda_j})^{(n-2)/n}. \tag{17} \]
Further, by virtue of inequalities (6), (7), and (8) we have
\[ \int_{S_{r'}^*}\sum b_i\cos\nu_i\,(u-t_0)\,ds\le 0, \qquad - \iint_{G_{r'}}(u-t_0)\operatorname{div}\mathbf b\,d\omega\le 0, \]
\[ \iint_{G_{r'}}cu\,d\omega\le 0. \tag{18} \]
Combining inequalities (15), (17), and (18) with equality (16), we obtain
\[ -M\alpha u_0(\mu_n G_{\lambda_j})^{(n-2)/n} + \frac{24a_0\mu_0 n^2\mu_n G_{\lambda_j}}{(\beta_j-\lambda_j)^2}\,C_n >0, \]
whence, taking \(M=25n^2a_0C_n/\alpha\), by virtue of inequality (12), we have \(u_0<0\), which is impossible. The lemma is proved. From this lemma the following theorems are obtained:
Theorem 1. Let the domain \(G\) lie inside the ball \(K_R\) (\(R\) arbitrary), contain the center of the ball \(K_R\), and have limit points on the boundary of the ball \(K_R\). Let in \(G\) there be defined a solution \(u(x)\) of equation (3), vanishing on that part of the boundary of the domain \(G\) which is situated strictly inside \(K_R\). Let the inequality \(\mu_n G<\mu_n K_R/M\) hold, where by \(M\) here and everywhere below is denoted the constant of the lemma. Let \(u_0>0\). Then
\[ \sup u(x)>u_0\exp(\mu_n K_R/M_1\mu_n G)^{\frac{1}{n-1}}, \]
where \(M_1\) is a positive constant depending only on the constants \(\alpha\), \(a_0\) of inequalities (4), (5), and on the dimension \(n\) of the space.
Theorem 2. Let equation (3) be defined in the ball \(K_R\). Let \(u(x)\) be some solution of this equation in the ball \(K_R\). Let \(|u(x)|<1\)
in \(K_R^n\). Let \(G_1, G_2, \ldots\) be the maximal connected domains in which the function \(u(x)\) preserves a constant sign. Let \(a\) \((0<a<1)\) be some number. Let \(G_{i_1}, G_{i_2}, \ldots, G_{i_m}\) be those among the domains \(G_1, G_2, \ldots\) which possess the following two properties:
- The intersection \(G_{i_k}\cap K_{R/2}\) is nonempty.
- \(\displaystyle \max_{x\in G_{i_k}K_{R/2}} |u(x)| \ge a.\)
Then the number \(m\) of these domains satisfies the inequality
\[ m<M_2\left(\lg \frac{1}{a}\right)^{1/(n-1)}, \]
where \(M_2\) is a constant depending only on the constants \(\alpha\), \(a_0\) in inequalities (4), (5), and on the dimension \(n\) of the space.
Theorem 3. Let \(G\) be an unbounded domain. Let equation (3) be given in \(G\). Let \(R\) be an arbitrary positive number. Let \(\sigma\) be some other number satisfying the inequality
\[ \sigma<\mu_n K_R/M. \tag{19} \]
Suppose the domain \(G\) has the property that for every ball \(K_R\) of radius \(R\) the inequality \(\mu_n(G\cdot K_R)<\sigma\) holds. Let the domain \(G\) contain the origin \(O\). Let there be defined in the domain \(G\) a solution \(u(x)>0\) of equation (3), which vanishes on the boundary of the domain. Put
\[ M(r)=\max_{\sum_{i=1}^{n}x_i^2=r^2} u(x). \]
Then, for all \(r>R\), the inequality
\[ M(r)>u_0 \exp \left(R^{\,n-1}r/M_3\sigma\right)^{1/(n-1)} \]
holds, where \(M_3\) is a positive constant depending only on the constants \(\alpha\), \(a_0\) in inequalities (4), (5), and on the dimension \(n\) of the space.
Theorem 4 (Phragmén—Lindelöf). Let \(G\) be an unbounded domain having the following property. There exists a number \(\eta<1/M\) such that, if for a natural number \(m\) one denotes by \(G_m\) the intersection of \(G\) with \(K_{2^m}\) (\(K_{2^m}\) is the ball of radius \(2^m\) with center at the origin \(O\)), then
\[ -\mu_n G_m' /\mu_n K_{2^m}<\eta \]
for all \(m\), beginning with some \(m_0\). Let there be defined in \(G\) a solution \(u(x)\) of equation (3), continuous in \(\overline G\) and nonpositive on the boundary of the domain \(G\). Then: 1) either \(u(x)\le 0\) everywhere in \(G\); 2) or, if one puts
\[ M(R)=\sup_{\sum_{i=1}^{n}x_i^2=R^2} u(x), \]
then
\[ \liminf_{R\to\infty}\frac{M(R)}{R^{(1/M_4\eta)^{1/(n-1)}}}>0, \]
where \(M_4\) is a positive constant depending only on the constants \(\alpha\), \(a_0\) in inequalities (4), (5), and on the dimension \(n\) of the space.
The author expresses deep gratitude to E. M. Landis for assistance and guidance.
Moscow State University
named after M. V. Lomonosov
Institute of Mathematics and Mechanics
Academy of Sciences of the Azerbaijan SSR
Received
7 IV 1961
References
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- E. M. Landis, Doctoral dissertation, MSU, 1956.
- E. M. Landis, UMN, 14, issue 1 (85) (1959).
- A. A. Novruzov, Izv. AN AzerbSSR, No. 5 (1960).