Mathematics

E. M. LANDIS

ON THE DEPENDENCE BETWEEN THE NUMBER OF SIGN CHANGES OF A SOLUTION OF AN ELLIPTIC EQUATION AND THE GROWTH OF THE SOLUTION

(Presented by Academician I. G. Petrovskii, 8 VII 1958)

Let there be given a linear elliptic equation of the second order

\[ Lu \equiv \sum_{i,k=1}^{n} a_{ik}(x_1,\ldots,x_n)\, \frac{\partial^2 u}{\partial x_i \partial x_k} + \sum_{i=1}^{n} b_i(x_1,\ldots,x_n)\, \frac{\partial u}{\partial x_i} + c(x_1,\ldots,x_n)u=0, \tag{1} \]

defined in the spherical layer \(D=\{r_1<|x|<r_2\}\), where \(x=(x_1,\ldots,x_n)\). Let \(c(x)\le 0\). Suppose, further, that the coefficients \(a_{ik}\), \(i,k=1,\ldots,n\), of equation (1) are twice continuously differentiable in \(D\), and the coefficients \(b_i\), \(i=1,\ldots,n\), are continuously differentiable in \(D\). Suppose that in \(D\) the inequalities

\[ |a_{ik}|,\ |\partial a_{ik}/\partial x_j|,\ |\partial^2 a_{ik}/\partial x_j\partial x_l|,\ |b_i|,\ |\partial b_i/\partial x_j|,\ |c|<M; \]
\[ i,j,k,l=1,\ldots,n \tag{2} \]

hold.

Suppose that in \(D\) the condition of uniform ellipticity is satisfied,

\[ \sum_{i,k=1}^{n} a_{ik}\xi_i\xi_k \Big/ \sum_{i=1}^{n} \xi_i^2 > a>0. \tag{3} \]

Let \(u(x)\) be a solution of equation (1) in \(D\), continuous in \(\overline D\). Denote by \(D^+\) and \(D^-\), respectively, the sets of points \(x\in D\) at which \(u(x)>0\) and \(u(x)<0\). We shall call a component of the set \(D^+\) or \(D^-\) essential if it has limit points both on the inner and on the outer spheres bounding the spherical layer \(D\). The total number of all essential components of the sets \(D^+\) and \(D^-\) will be called the number of sign changes of the function \(u(x)\) in the spherical layer \(D\).

Theorem 1. There exists a constant \(C\), depending on the constant \(M\) in inequality (2) and on the constant \(a\) in inequality (3), such that for every solution \(u(x)\) of equation (1) in the spherical layer \(D=\{r_1<|x|<r_2\}\), where \(r_2<1\), continuous in \(\overline D\) and having \(N\) sign changes, at least one of the following two inequalities holds:

\[ \max_{|x|=r_2}|u(x)| \Big/ \max_{|x|=\sqrt{r_1r_2}} |u(x)| > \left(\frac{r_2}{r_1}\right)^{N^{\frac{1}{n-1}}/C} \]

or

\[ \max_{|x|=r_1}|u(x)| \Big/ \max_{|x|=\sqrt{r_1r_2}} |u(x)| > \left(\frac{r_2}{r_1}\right)^{N^{\frac{1}{n-1}}/C}. \]

The proof of this theorem is based on the following lemmas.

Lemma 1. Let \(K\subset D\) be a ball with center at the point \(O\). Let \(G\subset K\) be a domain containing the point \(O\) and having limit points on the boundary of the ball \(K\). There exists a constant \(C_1\), depending on the constan-

of \(M\) in inequality (2) and on the constant \(a\) of inequality (3), such that, whenever for the domain \(G\) the inequality

\[ \mu_n G < \mu_n K/C_1 \tag{4} \]

is satisfied (by \(\mu_k E\) we shall denote the \(k\)-dimensional measure of the set \(E\)), and \(u(x)\) is a solution of equation (1), defined in \(G\) and vanishing on that part of the boundary of \(G\) which lies strictly inside \(K\), then

\[ \sup_{x\in G} u(x) \geq 2u(0). \]

Proof. Without loss of generality, one may assume that \(u(x)>0\) for \(x\in G\). Let \(R\) be the radius of the ball \(K\). For every \(r\), \(0<r<R\), denote by \(K_r\) the ball of radius \(r\) with center at the point \(O\), and by \(S_r\) the surface of the ball \(K_r\). Put \(G_r=G\cap K_r\) and \(\Gamma_r=G\cap S_r\). From inequality (4) it follows that there is an \(r_0\), \(0<r_0<R\), such that

\[ \mu_{n-1}\Gamma_{r_0}<\mu_{n-1}S_{r_0}/C_1. \tag{5} \]

By Serrin’s theorem \({}^{(1)}\) there exists a function \(K(x,x')\), \(x\in K_{r_0}\), \(x'\in S_{r_0}\), such that for any continuous function \(\varphi(x')\), defined on \(S_{r_0}\), the function \(v(x)\), defined by the equality

\[ v(x)=\int_{S_{r_0}}\frac{K(x,x')\varphi(x')}{\mu_{n-1}S_{r_0}}\,ds, \tag{6} \]

has the properties: 1) \(Lv\leq 0\) and 2) \(v/S_{r_0}=\varphi\). Moreover

\[ 0<K(0,x')<C^*, \tag{7} \]

where \(C^*\) is a constant depending on the constants \(M\) and \(a\) of inequalities (2) and (3). Put \(C_1=2C^*\). Further put: \(\varphi(x')=u(x')\) for \(x'\in\Gamma_{r_0}\); \(\varphi(x')=0\) for \(x'\in S_{r_0}\setminus\Gamma_{r_0}\). Since on the boundary of \(G_{r_0}\) we have the inequality \(u(x)\leq v(x)\), by virtue of property 1) of the function \(v\) we obtain \(u(x)\leq v(x)\) in \(G_{r_0}\). From (5), (6), and (7) we find \(v(0)<\sup_{x'\in S_{r_0}}\varphi(x')\,C^*/C_1\), and hence \(v(0)<\sup_{x\in S_{r_0}}\varphi(x')/2\), whence \(u(0)<\sup_{x\in G}u(x)/2\), as was required to prove.

Lemma 2. Let \(P\) be the cylinder \(\sum_{k=2}^{n}x_k^2<h^2<1\). Let \(\Gamma_1\) and \(\Gamma_2\) be \((n-1)\)-dimensional manifolds lying inside the cylinder \(P\), with boundary on the boundary of \(P\), each of which separates, in the cylinder \(P\), points with sufficiently large in absolute value negative coordinates \(x_1\). Let \(G\) be the part of \(P\) lying between \(\Gamma_1\) and \(\Gamma_2\). Let in \(G\) equation (1) be defined, satisfying in \(G\) conditions (2) and (3). Let \(u(x)\) be a solution of the equation, defined in \(G\) and continuously differentiable in \(\overline G\). Suppose that the function \(u(x)\) satisfies the conditions: 1) \(u|_{\Gamma_1}=u_0>0\); 2) \(u|_{\Gamma_2}=0\); 3) \(\partial u/\partial n|_{\Gamma_2}\leq0\), where \(\partial/\partial n\) is differentiation with respect to the inner normal, and 4) \(u(x)\geq -u_0\) for \(x\in G\). Denote by \(G_1\) the aggregate of points \(x\in G\) for which \(u_0/2<u(x)<u_0\).

Then \(\mu_n G_1>h^n/C_2\), where \(C_2\) is a constant depending on the constant \(M\) of inequalities (2) and on the constant \(a\) of inequality (3).

Here we shall indicate the way to prove this lemma. Put \(h^n/\mu_nG_1=C_2^*\). Denote by \(E_t\) the aggregate of points \(x\in G\) for which \(u(x)=t\), and by \(E_t^*\) the intersection of \(E_t\) with the cylinder \(\sum_{k=2}^{n}x_k^2<h^2/4\). There exists a \(u^*\), \(u_0/2<u^*<u_0\), such that: a) the level set \(E_{u^*}^*\) contains no points \(x\), where

\(\operatorname{grad} u(x)=0\), and, consequently, consists of smooth \((n-1)\)-dimensional manifolds; b) the inequality

\[ \int_{E_{u^*}^*}\left|\frac{\partial u}{\partial n}\right|\,d\sigma> \frac{C_2^* u_0 h^{\,n-2}}{L_1}, \tag{8} \]

holds, where \(L_1\) is an absolute constant.

Denote by \(G_{u^*}\) the set of points \(x\in G\) for which \(u(x)<u^*\). There is an \(h^*\), \(h/2<h^*<h\), such that, if by \(S_{h^*}\) we denote the set of points \(x\in G_{u^*}\) for which

\[ \sum_{k=2}^{n} x_k^2 = h^{*2}, \]

then the inequality

\[ \int_{S_{h^*}}\left|\frac{\partial u}{\partial n}\right|\,d\sigma < L_2 u_0 h^{\,n-2}, \tag{9} \]

holds, where \(L_2\) is an absolute constant. By condition 3), imposed on the solution \(u(x)\), we have

\[ \int_{\Gamma_2}\frac{\partial u}{\partial n}\,d\sigma<0. \tag{10} \]

Consider the equality \(\int_{G_{h^*}} Lu\,d\omega=0\) and apply Green’s formula to its left-hand side. From inequalities (8), (9), (10), (2), (3) and condition 4), taking into account that

\[ \left.\frac{\partial u}{\partial n}\right|_{E_{u^*}}<0 \]

(the normal here is internal with respect to \(G_{u^*}\)), we obtain

\[ (aC_2^*/L_1-Mn^2L_2)h^{\,n-2}u_0 \leq L_3h^{\,n-1}u_0+L_4h^n u_0, \]

where \(L_3\) and \(L_4\) are absolute constants; and since \(0<h<1\), we arrive at the required estimate of \(C_2^*\) in terms of \(M\) and \(a\).

With the aid of Lemmas 1 and 2 one proves:

Lemma 3. Let in the spherical layer \(D=\{h/4<|x|<h\}\), \(0<h<1\), a solution \(u(x)\) of equation (1), continuous on \(\overline D\), be defined. Let \(u(x)\) have \(N\) changes of sign in \(D\). Let \(g_1,g_2,\ldots,g_N\) be the essential components of the sets \(D^+\) and \(D^-\). Put

\[ m'=\max_{|x|=h/4}|u(x)|,\qquad m''=\max_{|x|=h/2}|u(x)|,\qquad m'''=\max_{|x|=h}|u(x)|; \]

\[ m_i'=\max_{\substack{|x|=h/4,\; x\in \overline g_i}} |u(x)|,\qquad m_i''=\max_{\substack{|x|=h/2,\; x\in \overline g_i}} |u(x)|,\qquad m_i'''=\max_{\substack{|x|=h,\; x\in \overline g_i}} |u(x)|. \]

There exists a constant \(C_3\), depending on the constant \(M\) in inequalities (2) and on the constant \(a\) in inequality (3), such that from the fact that

\[ m'''/m'' \leq 2^{N^{\frac1{n-1}}/C_3},\qquad m'/m'' \leq 2^{N^{\frac1{n-1}}/C_3} \tag{11} \]

it follows that

\[ \min_i m_i'' > 2^{N^{\frac1{n-1}}/2^{\frac{3n+1}{\,n-1}}+2}\, C_1^{\frac1{n-1}}\,m'', \tag{12} \]

where \(C_1\) is the constant of Lemma 1.

From Lemmas 1 and 3 it follows:

Lemma 4. In the notation of the preceding lemma, the following assertion holds: there exists a constant \(C_4\), depending on the constant \(M\) in inequalities (2) and on the constant \(a\) in inequality (3), such that

\[ m'''/m'' > 2^{N^{\frac1{n-1}}/C_4} \quad\text{or}\quad m'/m'' > 2^{N^{\frac1{n-1}}/C_4}. \]

Proof. Suppose that inequalities (11) are satisfied and, consequently, by Lemma 3, inequality (12) is satisfied. We have the obvious inequality \(\sum_{i=1}^{N}\mu_n g_i < \omega_n h^n\), where \(\omega_n\) is the volume of the unit \(n\)-dimensional ball, and therefore there is an \(i_0\) such that

\[ \mu_n g_{i_0} < \omega_n h^n/N . \tag{13} \]

Let

\[ N_1=\left[ N^{\frac{1}{n-1}} \bigg/ 2^{\frac{3n+1}{\,n-1\,}} C_1^{\frac{1}{n-1}} \right]. \]

Let \(N\) be so large that \(N_1>3\) (the case when \(N_1 \leqslant 3\) is considered separately; in this case the assertion of the lemma is obtained by an argument analogous to the proof of Lemma 1). Put
\(t_i' = h/2 - h(i-1/2)/4N_1,\ i=1,\ldots,N;\)
\(t_i'' = h/2 + h(i-1/2)/2N_1,\ i=1,2,\ldots,N;\)
\(m_{k,i_0}'=\max_{|x|=t_k',\,x\in g_{i_0}} |u(x)|\) and
\(m_{k,i_0}''=\max_{|x|=t_k'',\,x\in g_{i_0}} |u(x)|\).
Suppose these maxima are attained at the points \(P_{k,i_0}'\) and \(P_{k,i_0}''\). By the maximum principle, at least one of the following two alternatives holds:
\(m_{k,i_0}' \leqslant m_{k+1,i_0}',\ k=1,2,\ldots,N-1,\)
or
\(m_{k,i_0}'' \leqslant m_{k+1,i_0}'',\ k=1,2,\ldots,N-1\).
For definiteness, suppose the first case occurs (the second case is analogous). Denote by \(K_{k,i_0}'\) the ball with center at the point \(P_{k,i_0}'\) and radius \(h/8N_1\), and by \(g_{k,i_0}'\) the component of the intersection of \(g_{i_0}\) with this ball which contains the point \(P_{k,i_0}'\). Consider those \(k\) \((k=1,2,\ldots,N_1)\) for which
\(\mu_n g_{k,i_0}' < \omega_n h^n / C_1 2^{3n} N_1^n\).
The number of such \(k\) does not exceed \(N_1/2\), since otherwise we would have

\[ \mu_n g_{i_0} \geqslant \sum_{k=1}^{N_1} \mu_n g_{k,i_0}' > \frac{N_1}{2}\,\omega_n \frac{h^n}{C_1 2^{3n} N_1^n} \geqslant \omega_n \frac{h^n}{N}, \]

which contradicts (13). Consequently, there are at least

\[ N_1/2-1 \geqslant N^{\frac{1}{n-1}} \bigg/ 2^{\frac{3n+1}{\,n-1\,}} C_1^{\frac{1}{n-1}} \]

distinct \(k\) for which
\(\mu_n g_{k,i_0}' \geqslant \omega_n h^n / C_1 2^{3n} N_1^n\), i.e. such that Lemma 1 is applicable to \(g_{k,i_0}'\). Hence we obtain

\[ m' \geqslant m_{N_1,i_0}' \geqslant 2^{\,N^{\frac{1}{n-1}} \big/ 2^{\frac{3n+1}{\,n-1\,}+1} C_1^{\frac{1}{n-1}}} m_{1,i_0}' > \]

\[ > 2^{\,N^{\frac{1}{n-1}} \big/ 2^{\frac{3n+1}{\,n-1\,}} C_1^{\frac{1}{n-1}}} \,2^{-N^{\frac{1}{n-1}} \big/ 2^{\frac{3n+1}{\,n-1\,}+2} C_1^{\frac{1}{n-1}}} m'' = 2^{\,N^{\frac{1}{n-1}} \big/ 2^{\frac{3n+1}{\,n-1\,}+2} C_1^{\frac{1}{n-1}}} m'' . \]

Putting

\[ C_4=\max\left(C_3,\ C_1^{\frac{1}{n-1}}2^{\frac{3n+1}{\,n-1\,}+2}\right), \]

we obtain the constant needed.

Theorem 1 is easily obtained from Lemma 4 if one divides the spherical layer
\(D=\{r_1<|x|<r_2\}\) into \([\log_4(r_2/r_1)]\) spherical layers with ratio of radii equal to 4.

Received
4 VII 1958

REFERENCES

¹ J. Serrin, J. Analyse Math., 4, No. 2, 292 (1955–1956).