UDC 517.946

MATHEMATICS

E. M. LANDIS

THEOREMS OF PHRAGMÉN–LINDELÖF TYPE FOR SOLUTIONS OF ELLIPTIC EQUATIONS OF HIGH ORDER

(Presented by Academician I. G. Petrovskii on 31 XII 1969)

Two types of generalizations of the classical Phragmén–Lindelöf theorem for analytic functions of a complex variable to harmonic functions are possible. In the first of them, with an analytic function \(f(z)\) one associates the harmonic function \(\ln |f(z)|\). This yields theorems of the following type: on the boundary of a certain unbounded domain the Dirichlet data of a harmonic function are nonpositive; then either this function is nonpositive everywhere in the domain, or at infinity it grows no more slowly than at a certain rate depending on the shape of the domain (for a strip this is an exponential). In the second generalization, the function \(f(z)\) is associated with the harmonic function \(\operatorname{Re} f(z)\). Here theorems of the following kind arise: if on the boundary of an unbounded domain the Cauchy data of a harmonic function are bounded in modulus, then this function either is uniformly bounded in modulus everywhere in this domain, or at infinity grows no more slowly than at a certain rate depending on the domain (for a strip this is an iterated exponential).

In the present note we shall deal with a generalization of the second type. For harmonic functions of many independent variables, as well as for solutions of a number of particular types of elliptic equations, this question has been studied by M. A. Evgrafov, I. S. Arshon, M. A. Pak, G. A. Dzhafarli \((^{1-7})\) and others.

In the note the case of a general linear elliptic equation with variable coefficients of arbitrary order without multiple characteristics is considered (more precisely, subject to Hörmander’s condition, which will be formulated below).

Let
\[ P(x,D)=\sum_{|\alpha|\le m} a_\alpha(x)D^\alpha \]
be a uniformly elliptic operator defined in a domain \(\Omega \subset \mathbf{R}^n\). The coefficients (generally speaking, complex) of the derivatives of highest order are from \(C_1\), while the remaining ones satisfy a Hölder condition and are bounded in \(\Omega\).

We shall say that the equation
\[ P(x,D)u=0 \tag{1} \]
satisfies Hörmander’s condition \((^8)\) if, for any real vector \(N\ne 0\) and any real vector \(\xi\) not proportional to \(N\), the equation \(P_m(x,\xi+i\tau N)=0\), where
\[ P_m(x,\xi)=\sum_{|\alpha|=m} a_\alpha(x)\xi^\alpha, \]
has no double real roots \(\tau\).

We shall assume that equation (1) satisfies this Hörmander condition.

Theorem 1. Let \(\Omega\) be a cylinder defined by the inequalities
\[ \sum_{i=2}^{n} x_i^2 \le h^2,\qquad x_1 \le x_1^0, \]
and let \(S\) be its lateral surface. Let \(u(x)\) be a solution in \(\Omega\) of equation (1), satisfying the conditions
\[ \left.\left|\partial^k u/\partial n^k\right|\right|_S < 1,\qquad k=0,\ldots,m-1 \]
(\(\partial/\partial n\) is differentiation along the normal to the surface \(S\)).

Then either \(u(x)\) is uniformly bounded in \(\Omega\), or

\[ \overline{\lim_{x_1\to\infty}}\,[|u(x)|/\exp(\exp Cx_1)]>0, \tag{2} \]

where \(C>0\) is a constant depending on the equation and on \(h\).

Proof. For an arbitrary \(a\ge x_1^0+1\), denote by \(\Pi_a\) and \(\Pi_a'\) the cylinders defined, respectively, by the inequalities

\[ \sum_{i=2}^{n}x_i^2\le h^2,\quad a-1\le x_1\le a+2 \quad\text{and}\quad \sum_{i=2}^{n}x_i^2\le h^2,\quad a\le x_1\le a+1. \]

Let \(S_a\) be the lateral surface of the cylinder \(\Pi_a\).

From the results of M. M. el Borai \((^{10})\) it follows that there exist constants \(\varepsilon_0>0\) and \(\mu\), \(0<\mu<1\), depending on the equation and on \(h\), such that, if \(\varepsilon\), \(0<\varepsilon<\varepsilon_0\), is an arbitrary number and \(v(x)\) is a solution of equation (1) in \(\Pi_a\) satisfying the conditions

\[ |v|\le 1\ \text{in }\Pi_a,\qquad \left|\partial^k v/\partial n^k\right|\big|_{S_a}<\varepsilon,\quad k=0,\ldots,m-1, \tag{3} \]

then

\[ |v|<\varepsilon^\mu\ \text{in }\Pi_a'. \tag{4} \]

Denote

\[ M_a=\max_{x\in\Pi_a}|u(x)|\quad\text{and}\quad M_a'=\max_{x\in\Pi_a'}|u(x)|. \]

Put \(M=\max(1/\varepsilon_0,1)\) and suppose that \(u(x)\) is unbounded in the cylinder \(\Omega\). Then one can find a number \(a\ge x_1^0+2\) such that

\[ M_a'>M,\qquad M_a'>M_{a-1}'. \tag{5} \]

Consider in the cylinder \(\Pi_a\) the function \(v(x)=u(x)/M_a\). It is a solution of equation (1), and for it we have \(|v(x)|\le 1\) in \(\Pi_a\),

\[ \left|\partial^k v/\partial n^k\right|\big|_{S_a}<1/M_a,\quad k=0,\ldots,m-1. \]

Consequently, by (3) and (4),

\[ |v|<(1/M_a)^\mu\ \text{in }\Pi_a', \]

i.e.

\[ M_a>(M_a')^{1/(1-\mu)}. \tag{6} \]

Since \(M_a=\max(M_{a-1}',\,M_a',\,M_{a+1}')\), it follows from (5) and (6) that \(M_a=M_{a+1}'\), and thus

\[ M_{a+1}'>(M_a')^{1/(1-\mu)}. \]

Now we may repeat the argument, replacing \(a\) by \(a+1\), then by \(a+2\), and so on. We obtain

\[ M_{a+k}'>(M_a')^{(1/(1-\mu))^k} =\exp\left[\ln M_a'\exp\left(k\ln\frac{1}{1-\mu}\right)\right], \]

whence inequality (2) follows, where as \(C\) one may take any number less than \(\ln[1/(1-\mu)]\).

Theorem 2. Let \(\Omega\) be the same cylinder as in the preceding theorem. Let \(u(x)\) be a solution of equation (1) in \(\Omega\). Let \(\alpha>0\) be an arbitrary constant, and suppose the inequalities

\[ \overline{\lim_{x_1\to\infty}} \left[\left|\frac{\partial^k u(x)}{\partial n^k}\right|\bigg|_{S}\big/ \exp(-\exp \alpha x_1)\right]<\infty, \qquad k=0,\ldots,m-1. \]

Then either

\[ \overline{\lim_{x_1\to\infty}}\,[|u(x)|/\exp(-\exp \alpha x_1)]<\infty, \]

or

\[ \overline{\lim_{x_1\to\infty}}\,[|u(x)|/\exp(C\exp \alpha x_1)]>0, \]

where \(C>0\) is a constant depending on the equation, on \(\alpha\), and on \(h\).

The proof is similar to the proof of the preceding theorem.

Theorem 3. Let \(\Omega\) be a ball with its center removed,
\[ \Omega=\{x\mid 0<|x|\leq R\}. \]
Let \(K\) be the cone defined by the inequalities
\[ \sum_{i=2}^{n} x_i^2<a^2x_1^2,\qquad a\ne 0,\qquad x_1>0. \]

Let \(u(x)\) be a solution of equation (1) in \(\Omega\), bounded in \(\Omega\cap K\). Then either \(u(x)\) is bounded everywhere in \(\Omega\), or
\[ \overline{\lim_{|x|\to 0}}\,[|u(x)|/\exp(1/|x|^C)]>0, \]
where \(C>0\) is a constant depending on the equation.

Proof. We exclude from \(\Omega\) the narrower cone \(K'\)
\[ K'=\left\{x\left|\sum_{i=2}^{n}x_i^2<\frac{a^2}{4}x_1^2,\ x_1>0\right.\right\} \]
and denote the remaining set by \(\Omega'\). We map \(\Omega'\) onto the cylinder
\[ \hat{\Omega}=\left\{y\left|\sum_{i=2}^{n}y_i^2\leq 1,\ y_1\geq \ln\frac1R\right.\right\} \]
in Euclidean space \((y_1,\ldots,y_n)\) by means of the following transformation. Let \(\hat K_{n-1}\) be the ball
\[ \sum_{i=2}^{n}y_i^2\leq 1 \]
on the hyperplane \(y_1=0\). Let \(S_n\) be the unit \(n\)-dimensional sphere \(|x|=1\). Put \(S_n'=S_n\setminus K'\). Let \(f:S_n'\to \hat K_{n-1}\) be some fixed \(m\)-times continuously differentiable diffeomorphism such that it maps \(S_n\setminus K\) into the ball concentric with \(\hat K_{n-1}\) of half the radius.

Let \(x\in \Omega'\). Put \(y_1=\ln 1/|x|\) and \((y_2,\ldots,y_n)=f(x/|x|)\). This transformation carries equation (1) into the equation \(\hat P(y,D_y)u=0\), uniformly elliptic in the cylinder \(\hat{\Omega}\) and with the same properties of the coefficients as equation (1). Its solution \(\hat u(y)=u(x)\) will be uniformly bounded in the cylindrical layer
\[ \frac14<\sum_{i=2}^{n}y_i^2<1, \]
and therefore \((^9)\) there exists a constant \(\hat M>0\) such that on the lateral surface \(\hat S\) of the cylinder
\[ \hat{\Omega}=\left\{y\left|\sum_{i=2}^{n}y_i^2\leq \frac12,\ y_1\geq \ln\frac1R\right.\right\} \]
one has \(|\partial^k\hat u/\partial n|<\hat M\), \(k=0,\ldots,m-1\). Applying Theorem 1 to \(u\) in \(\hat{\Omega}\) and then making the inverse transformation to the variables \(x\), we obtain inequality (7).

Similarly, from Theorem 2, using the uniqueness theorem from \((^{11})\), the following theorem is obtained:

Theorem 4. Let \(\Omega\) and \(K\) have the same meaning as in the preceding theorem. Let \(\alpha>0\) be an arbitrary number. Let \(u(x)\not\equiv 0\) be a solution of equation (1) in \(\Omega\), and let the equation itself be defined in \(\overline{\Omega}\). Suppose
\[ \overline{\lim_{\substack{x\in K\\ |x|\to 0}}}\,[|u(x)|/\exp(-1/|x|^\alpha)]<\infty. \]
Then
\[ \overline{\lim_{\substack{x\in \Omega\\ |x|\to 0}}}\,[|u(x)|/\exp(C/|x|^\alpha)]>0, \]
where \(C>0\) is a constant depending on the equation and on \(\alpha\).

In a similar way one proves a theorem generalizing Theorem 3:

Theorem 5. Let \(\Omega\) and \(K\) be the same as above. Let \(u(x)\) be a solution of equation (1) in \(\Omega\), and suppose that for some number \(C_1\) the inequality

\[ \overline{\lim_{\substack{x\in K\\ |x|\to 0}}}\,[\,|u(x)|/|x|^{C_1}\,]<\infty . \]

is satisfied. Then either

\[ \overline{\lim_{\substack{x\in \Omega\\ |x|\to 0}}}\,[\,|u(x)|/|x|^{C_1}\,]<\infty, \]

or

\[ \overline{\lim_{\substack{x\in \Omega\\ |x|\to 0}}}\,[\,|u(x)|/\exp(1/|x|^{C})\,]>0, \]

where \(C>0\) is a constant depending on the equation.

Instead of estimating \(u\) by the maximum of its modulus, one can pass to an estimate in \(L_2\). Then it is not necessary to require Hölder continuity of the coefficients of the lower-order terms. Their measurability and boundedness are sufficient. In this form, for example, Theorem 5 may be reformulated as follows:

Theorem 6. Let \(\Omega\) and \(K\) be the same as in Theorem 5. Put

\[ M(r)=\left(\int_{|x|<r} u^2\,dx\right)^{1/2}; \qquad M_K(r)=\left(\int_{\substack{|x|<r\\ x\in K}} u^2\,dx\right)^{1/2}. \]

Let \(u(x)\in H_m^{\mathrm{loc}}\) be a solution of equation (1) with the new conditions on the lower-order coefficients in \(\Omega\). Suppose that for some number \(C_1>0\) the inequality

\[ \overline{\lim_{|x|\to 0}}\,[\,M_k(|x|)/|x|^{C_1}\,]<\infty . \]

is satisfied. Then either

\[ \overline{\lim_{|x|\to 0}}\,[\,M(|x|)/|x|^{C_2}\,]<\infty, \]

or

\[ \overline{\lim_{|x|\to 0}}\,[\,M(|x|)/\exp(1/|x|^{C})\,]>0, \]

where \(C>0\) is a constant depending on the equation, and \(C_2>0\), moreover, also on \(C_1\).

Moscow State University
named after M. V. Lomonosov

Received
25 XII 1969

CITED LITERATURE

\(^{1}\) M. A. Evgrafov, DAN, 126, No. 3, 478 (1959).
\(^{2}\) M. A. Evgrafov, Izv. AN SSSR, ser. matem., 27, No. 4, 843 (1963).
\(^{3}\) I. S. Arshon, M. A. Evgrafov, DAN, 134, No. 3, 507 (1960).
\(^{4}\) I. S. Arshon, M. A. Evgrafov, DAN, 143, No. 1, 9 (1962).
\(^{5}\) I. S. Arshon, M. A. Evgrafov, DAN, 147, No. 4, 755 (1962).
\(^{6}\) I. S. Arshon, G. A. Difforl, Diff. Equations, journal, 2, No. 4 (1966).
\(^{7}\) I. S. Arshon, M. A. Ilin, DAN, 177, No. 1 (1967).
\(^{8}\) L. Hörmander, Linear Partial Differential Operators, Moscow, 1965.
\(^{9}\) S. Agmon, A. Douglis, L. Nirenberg, Estimates of Solutions of Elliptic Equations near the Boundary, IL, 1962.
\(^{10}\) M. M. El Borai, Vestn. MGU, No. 4, 15 (1968).
\(^{11}\) E. G. Sitnikova, Matem. sborn., 81 (123), No. 3 (1970).