Mathematics

R. Ya. Glagoleva

THE THREE-CYLINDER THEOREM AND ITS APPLICATIONS

(Presented by Academician I. G. Petrovskii, February 11, 1965)

This note considers a three-cylinder theorem for a parabolic equation of second order, analogous to Hadamard’s three-circle theorem for analytic functions of a complex variable \((^1)\).

Denote by \(\Pi_r^{[t_1,t_2]}\) the cylinder
\[ (x_1^2+\cdots+x_n^2)^{1/2}=|x|\le r,\qquad t_1\le t\le t_2, \]
and, for a function \(f(x,t)\) continuous in the cylinder \(\Pi_r^{[t_1,t_2]}\), put
\[ M_f(r,t_1,t_2)=\max_{(x,t)\in \Pi_r^{[t_1,t_2]}} |f(x,t)|. \]
Suppose that in the cylinder \(\Pi_R^{[0,T]}\), \(R\le 1\), the equation
\[ Hu\equiv \sum_{i,k=1}^{n} a_{ik}(x,t)\frac{\partial^2 u}{\partial x_i\partial x_k} +\sum_{i=1}^{n} b_i(x,t)\frac{\partial u}{\partial x_i} +c(x,t)u-\frac{\partial u}{\partial t}=0 \tag{1} \]
is defined such that: 1) \(a_{ik}(x,t)\) are three times continuously differentiable with respect to \(x_i\) and continuously differentiable with respect to \(t\); 2)
\[ \sum_{i,k=1}^{n} a_{ik}(x,t)\xi_i\xi_k \ge \mu \sum_{i=1}^{n}\xi_i^2,\qquad \mu>0; \]
3) the coefficients \(a_{ik}(x,t)\) and their derivatives up to the indicated order, as well as the coefficients \(b_i(x,t)\) and \(c(x,t)\), are bounded in modulus by the constant \(K\).

Theorem 1 (on three cylinders). Let \(u(x,t)\) be a solution of equation (1) satisfying the conditions
\[ M_u(r_0,0,T)=\Delta;\qquad M_u(R,0,T)=M. \tag{2} \]
Then for any \(r\), \(r_0<r\le R/2C\), and any positive \(a\), the inequality
\[ \ln M_u(r,a,T-a)\le \ln M_u(r_0,0,T)\frac{\ln(Cr)}{\ln(Br_0)} + \]
\[ +\ln M_u(R,0,T)\frac{\ln(Br_0/Cr)}{\ln(Br_0)}-\ln(Cr), \tag{3} \]
holds, where \(B,C\) are constants depending on the equation, \(a\), and \(n\).

Proof. According to the results of Li and Dž-huan \((^2)\), there exists a transformation of coordinates \(x=x(y)\) that maps \(\Pi_R^{[0,T]}\) into a domain containing the cylinder \(\Pi_s^{*[0,T]}\), \(|y|=\rho\le s,\ 0\le t\le T\), the cylinder \(\Pi_{\rho_0}^{*[0,T]}\) into a domain containing the cylinder \(\Pi_{r_0}^{[0,T]}\), and transforms the operator \(H\) into an operator \(H^*\) of the form
\[ H^* \equiv \frac{\partial^2}{\partial \rho^2} +\frac{n-1}{\rho}\frac{\partial}{\partial \rho} +\frac{1}{\rho^2}M_\rho -b(\rho)\frac{\partial}{\partial t} +\sum_{i=1}^{n} p_i(y,t)\frac{\partial}{\partial y_i} +p_0, \]
where \(M_\rho\) is an operator containing only differentiation on the unit sphere. In this case, for functions \(w\) with the properties: 1) \(w(y,t)\) have continuous derivatives of second order with respect to \(y_i\) and of first order with respect to \(t\); 2) \(w\equiv0\) in \(\Pi_\varepsilon^{*[0,T]}\) for some positive \(\varepsilon\); 3) \(w|_{\rho=s}= \partial w/\partial \rho|_{\rho=s}=0\); 4) \(w|_{t=0}=w|_{t=T}=0\), the inequality can be obtained
\[ \int_{\Pi_s^{*[0,T]}} \frac{w^2}{\rho^{2\beta+n-2}}\,dy\,dt \le \frac{2}{\beta^2(\beta+n-3)} \int_{\Pi_s^{*[0,T]}} \frac{(H^*w)^2}{\rho^{2\beta+n-4}}\,dy\,dt. \tag{4} \]

Let \(u^*(y,t)\equiv u[x(y),t]\) be a solution of the equation \(H^*u^*=0\), satisfying the equalities (2), where \(M=1\). We multiply \(u^*(y,t)\) by the cutoff functions \(f(\rho)\) and \(\varphi(t)\) so that the function \(w\equiv u^*(y,t)f(\rho)\varphi(t)\) has properties 1)—4), and \(w\equiv u^*(y,t)\) in the domain \(G:\rho_0\le |y|\le s'/2,\ a/2\le t\le T-a/2\). Taking into account that, for sufficiently large \(\beta\), \((\rho_0/2)^\beta\le \Delta\), from inequality (4) one can obtain the inequality

\[ \int_G \left(\frac{u^{*2}}{\rho^{2\beta+n-2}}\right)\,dy\,dt \le \left(\frac{2}{s}\right)^{2\beta-1}. \tag{5} \]

Hence we have

\[ \frac{1}{\tilde\rho^{\,n-1}} \int_{\Pi_\rho^*[a/2,T-a/2]} u^{*2}\,dy\,dt \le (C_1\tilde\rho)^{2\beta-1} \]

for any \(\tilde\rho\), \(\rho_0<\tilde\rho<1/2C_1\). Returning, by means of the inverse transformation, to the variables \((x_1,\ldots,x_n,t)\), we obtain

\[ \frac{1}{\tilde r^{\,n-1}} \int_{\Pi_{\tilde r}^{[a/2,T-a/2]}} u^2\,dx\,dt \le (C_2\tilde r)^{2\beta-1} \tag{6} \]

for all \(\tilde r\), \(r_0<\tilde r\le R/2C_2\). Let now \(\tilde r\) be such that \(2r_0<\tilde r<R/2C_2\). By the mean-value theorem for the integrals

\[ \int_{a/2}^{3a/4}\int_{D_t} u^2\,dx\,dt, \]

\[ \int_{3\tilde r/4}^{\tilde r}\int_{K_1}\int_0^T u^2 r^{n-1}\,dr\,dO\,dt \]

(\(K_1\) is the unit sphere and \(dO\) is the element of the surface of \(K_1\)), there exist \(r_1\) and \(t_1\) such that \(3\tilde r/4\le r_1\le \tilde r\), \(a/2\le t_1\le 3a/4\), and

\[ \int_{K_1}\int_0^T u^2\big|_{r=r_1}\,r_1^{n-1}\,dO\,dt \le \frac{4}{\tilde r}(C_2\tilde r)^{2\beta-1}(\tilde r^{\,n-1}); \tag{7} \]

\[ \int_0^r\int_{K_1} u^2\big|_{t=t_1}\,r^{n-1}\,dr\,dO \le \frac{4}{a}(C_2\tilde r)^{2\beta-1}(\tilde r^{\,n-1}). \tag{8} \]

For any point \((x,t)\in \Pi_{\tilde r/2}^{[a,T-a]}\), the solution \(u(x,t)\) is representable in the form

\[ u(x,t)=\int_{\Pi_1}\Gamma_1(x,t,\xi,\tau)u(\xi,\tau)r_1^{n-1}\,dO\,d\tau +\int_{\Pi_2}\Gamma_2(x,t,\xi,t_1)u(\xi,t_1)\,d\xi, \tag{9} \]

where

\[ \Pi_1=\{(x,t)\mid |x|=r_1,\ t_1\le t\le T-a\}; \]

\[ \Pi_2=\{(x,t)\mid 0\le |x|\le r_1,\ t=t_1\}. \]

\(\Gamma_1(x,t,\xi,\tau)\) and \(\Gamma_2(x,t,\xi,\tau)\) are Green’s functions. Using Schwarz’s inequality, (7) and (8), as well as the known estimates for Green’s functions (see (3, 4)), we obtain \(|u(x,t)|\le (Cr)^{\beta-1}\) for all \(\tilde r\), \(r_0\le \tilde r\le R/2C\), and \(t\), \(a\le t\le T-a\); \(C\) is a constant depending on \(a,n\), and the equation; \(\beta=\ln\Delta/\ln(Br_0)\); \(2B=\max_{\rho_0/2\le\tilde\rho\le1/2C_1}(\tilde\rho/\tilde r)\).

Let \(M(R,0,T)=M\ne1\). Introduce the function \(v=u/M\). By what has been proved earlier, \(M_v(r,a,T-a)\le (Cr)^{\beta-1}\), where \(\beta=\ln(\Delta/M)/\ln(Br_0)\). Taking into account that \(M_u(r,a,T-a)=M\cdot M_v(r,a,T-a)\) and taking logarithms, we obtain the assertion of the theorem.

Corollary. Let \(r_0=R/4C,\ r=R/2C\), where \(C\) is the constant of Theorem 1, \(M_u(r_0,0,T)=r_0^\gamma\); \(M_u(R,0,T)=M\). Then the inequality \(M(r,a,T-a)<r^{\gamma'}\) is valid, where \(\gamma'=\gamma\alpha_1,\ \alpha_1=\ln(R/2)/2\ln(BR/4C)\).

Lemma 1. Let \(u(x,t)\) be a solution of equation (1), satisfying the conditions

\[ M_u(r_0,0,T)=\Delta,\qquad M_u(R,0,T)=M. \tag{10} \]

For any \(\varepsilon>0\) the inequality
\[ M_u(R-\varepsilon,t_1,t_2)<(R/4C)^{\gamma\omega^N}, \tag{11} \]
holds, where
\[ N=\left[\ln {2C\varepsilon \over R(2C-1)} \bigg/ \ln {2(2C-1)\over 4C-1}\right]; \qquad t_1=a\sum_{p=0}^{N}\left[{2(2C-1)\over 4C-1}\right]^{2p}; \]
\[ t_2=T-t_1; \]
\[ \gamma={\ln\Delta\cdot \ln(R/4)\over 2\ln(Br_0)\ln(R/4C)},\qquad \omega={\ln(R/4C)\over \ln(R/2C)}\cdot {\ln(R/2)\over 2\ln(Br/4C)}; \]
\(B,C\) are the constants of Theorem 1.

Proof. By Theorem 1, for \(\tilde r=R/4C\) we have
\[ M_u(R/4C,a,T-a)\le (R/4)^{(\ln\Delta/\ln(Br_0))-1+\ln M\cdot\{1/\ln(R/4)-1/\ln(Br_0)\}\gamma} <(R/4C)^{\tilde\gamma}, \]
where
\[ \gamma={\ln\Delta\over 2\ln(Br_0)}{\ln(R/4)\over \ln(R/4C)}. \]
By the corollary to Theorem 1, for \(r=R/2C\) we have
\[ M_u(R/2C,2a,T-2a)\le (R/2C)^{\gamma\alpha_1}. \]

Denote by \(K_{\bar O}^{\bar r}\) the ball in the space \(\{x\}\) with center at \(\bar O\) and radius \(\bar r\), and set
\[ M_u^{\bar O}(\bar r,\bar t_1,\bar t_2)= \max_{(x,t)\in K_{\bar O}^{\bar r}\times[\bar t_1,\bar t_2]} |u(x,t)|. \]

In \(K_O^{R/2C}\) inscribe a ball \(K_{O_1}^{r_1}\) tangent to it from inside, and in \(K_O^R\) a ball \(K_{O_1}^{R_1}\) tangent to it from inside, so that \(r_1=R_1/4C\).

Make the change of coordinates
\[ x_j'=k_1[x_j-(x_j)_{O_1}],\qquad t'=k_1^2t, \]
where
\[ k_1=R/R_1=(4C-1)/2(2C-1). \]
Under this transformation the equation \(Hu=0\) becomes \(H_1u_1\), where
\[ H_1\equiv \sum_{i,k=1}^{n} a_{ik}{\partial^2\over \partial x_i'\partial x_k'} +\sum_{i=1}^{n}{b_i\over k_1}{\partial\over \partial x_i'} +{c\over k_1^2}-{\partial\over \partial t'}, \]
with coefficients satisfying the same conditions as the coefficients of the operator \(H\), while
\[ u_1(x',t')\equiv u[x'(x),t'(t)] \]
will satisfy the conditions
\[ M_{u_1}^{O_1}(R/4C;k_1^2\cdot 2a;k_1^2(T-2a)) = M_u^{O_1}(R_1/4C;2a;T-2a) \le M_u^O(R/2C;2a;T-2a)<(R/4C)^{\gamma\alpha_1\alpha_2}, \]
\[ M_{u_1}^{O_1}(R;k_1^2 2a;k_1^2(T-2a)) = M_u^{O_1}(R_1;2a;T-2a) \le M_u^O(R;2a;T-2a)\le M, \]
where
\[ \alpha_2={\ln(R/4C)\over \ln(R/2C)}. \]
Then the corollary to Theorem 1 is applicable to \(u_1\). We have
\[ M_{u_1}^{O_1}(R/2C;k_1^2\cdot 2a+2a;k_1^2(T-2a)-2a) = M_u^{O_1}(R/2C;t_1^{(1)};t_2^{(1)}) \le (R/2C)^{\gamma\alpha_1^2\alpha_2} = (R/4C)^{\gamma\omega^2}, \]
where
\[ \omega=\alpha_1\alpha_2,\qquad t_1^{(1)}=2a(1+1/k_1^2),\qquad t_2^{(1)}=T-t_1^{(1)}. \]

Carrying out an analogous construction for the balls \(K_{O_1}^{R/2C}\) and \(K_{O_1}^{R_1}\), by the same method we obtain the estimate
\[ M_u^{O_2}(R_2/2C;t_1^{(2)};t_2^{(2)})\le (R/4C)^{\gamma\omega^3}. \]

Let \(N\) be the number of steps of this process for which
\[ (1-1/2C)R_{N-1}>\varepsilon\ge (1-1/2C)R_N, \]
i.e.
\[ N=\left[\ln {2C\varepsilon\over R(2C-1)}\bigg/ \ln {2(2C-1)\over 4C-1}\right]. \]
Then the inequality
\[ M_u^O(R-\varepsilon,t_1,t_2)<(R/4C)^{\gamma\omega^N} \]
is valid, where
\[ t_1=2a\sum_{p=0}^{N}\left[{2(2C-1)\over 4C-1}\right]^{2p},\qquad t_2=T-t_1, \]
which was required to be proved.

Lemma 2. Let \(u(x,t)\) be a solution of equation (1) in \(\Ц_R^{[0,T]}\), satisfying the conditions:
\(M_u(R,0,T)=M;\quad M_u(R-\varepsilon,t_1,T-t_1)<\delta;\)
\(\left.u\right|_{|x|=R}<\delta.\) Let
\[ m(t^*)=\max_{R-\varepsilon\le |x|\le R,\;t=t^*}|u(x,t)|. \]
Then for any \(t\in(t_1,T-t_1)\)
\[ m(t)\le 2\max\left\{\delta\exp Kt;\; M\sqrt{2}\exp\left[(K-\mu/\varepsilon^2)(t-t_1)\right]\right\}, \]
where \(K,\mu\) are constants from conditions 2) and 3) imposed on the coefficients of equation (1).

The proof is carried out by means of constructing a barrier.

Theorem 2 (on two cylinders). Let \(u(x,t)\) be a solution of equation (1) in \(\Ц_R^{[0,T]}\), satisfying the conditions:
\(M_u(R,0,T)=M;\; M_u(r_0,0,T)=\Delta;\)
\(\left.u\right|_{|x|=R}=0.\) Then for any \(r\), \(r_0<r<R\),
\[ \ln M_u(r;T/4;T)\le C_1\left[\ln\Delta/\ln(Br_0)\right]^\theta, \]
where
\[ 1/\theta=1+2C\ln\left[\ln(4C/B)/\ln\sqrt{2}\right]; \]
\(C,B\) are the constants of Theorem 1; \(C_1\) is a constant depending on \(a,n\), and the equation.

Proof. By Lemma 1, for any \(\varepsilon>0\) inequality (11) is valid. Choosing \(a\) so small that \(t_1<T/8\), and setting \(\delta=(R/4C)^{\gamma\omega^N}\), by Lemma 2 we obtain
\[ M_u(r;T/4;T)\le \min_{\varepsilon>0}\left(2\max\left\{(R/4C)^{\gamma\omega^N}\exp(3KT/4);\; M\sqrt{2}\exp\left[(K-\mu/\varepsilon^2)(T/4)\right]\right\}\right) \tag{12} \]
for any \(r\le R\).

From monotonicity considerations for the quantities in the braces in inequality (12), it is clear that the optimal value \(\varepsilon=\varepsilon_0\) is obtained in the case of equality of these quantities. Taking a value close to \(\varepsilon_0\),
\[ \varepsilon=\frac{(\ln(Br_0))\mu TR}{(2C-1)\,16C\,\ln\Delta\cdot \ln(R/4C)}, \]
we obtain the estimate
\[ M_u(r;T/4;T)<(\theta_1)^{[\ln\Delta/\ln(Br_0)]^\theta}, \]
where \(\theta_1<1\) depends on \(a,n\), and the equation,
\[ 1/\theta=1+2C\ln\left[\ln(4C/B)/\ln\sqrt{2}\right]; \]
\(C\) and \(B\) are the constants of Theorem 1. The assertion of Theorem 2 follows from this.

We give two applications of Theorems 1 and 2 proved above.

Theorem 3 (an analogue of Ito’s problem). Let \(u(x,t)\) be a solution of equation (1) in \(\Ц_R^{[-T,0]}\), satisfying the conditions
\[ M_u(r_0,-\tau,0)=\Delta;\qquad M_u(R,-T,0)=M;\quad \left.u\right|_{|x|=R}=0. \]
Then
\[ M(R,-T+1,0)\le A_1\theta_1^{A_2[\ln\Delta/\ln(Br_0)]^\theta}, \]
where \(\theta\) is the constant of Theorem 2. \(\theta_1<1\) is a constant depending on \(\tau,n\), and the equation.

The proof is obtained from Theorem 2 of the present article and Theorem 2 of paper (5).

Theorem 4. (Possible rate of decrease of a solution of equation (1) in a neighborhood of an irregular boundary point.) Let \(\Ц=D\times[0,T]\), where \(D\) is some domain in the space of \(x\). Let \(x^0\) be a boundary point of the domain \(D\) such that there exists a circular cone with vertex at \(x^0\), lying entirely in the domain \(D\). Let \(\varphi\) be the angle made by the generatrix with the axis. Let \(u(x,t)\) be a solution of equation (1) in \(\Ц\). Put
\[ M(r)=\max_{|x-x_0|\le r,\;(x,t)\in\Ц}|u(x,t)|. \]
If
\[ \varlimsup_{r\to 0}\left[M(r)\exp\left(C_1/r^{C_2/\varphi}\right)\right], \]
where \(C_1,C_2\) are constants depending on the equation, then \(u\equiv0\) in \(\Ц\).

This assertion is a consequence of Theorem 1 (on three cylinders). Its proof is similar to the proof of Lemma 1.

Moscow Aviation Institute
named after Sergo Ordzhonikidze

Received
25 I 1965

REFERENCES

1. A. I. Markushevich, Theory of Analytic Functions, Moscow–Leningrad, 1950.
2. L. I. Danilyan, DAN, 129, No. 5, 979 (1959).
3. W. Pogorzelski, Ann. Polon. Math., 4, No. 1, 61 (1957); 4, No. 1, 110 (1957).
4. W. Pogorzelski, Ann. Polon. Math., 4, No. 3, 288 (1958).
5. R. Ya. Glagoleva, DAN, 148, No. 1, 20 (1963).