UDC 517.9

MATHEMATICS

A. S. BRATUS’

A PRIORI ESTIMATES FOR EQUATIONS WITH A PARAMETER

(Presented by Academician I. G. Petrovskii, 16 XII 1969)

Consider differential operators depending on a parameter, with smooth complex-valued coefficients in a bounded domain (\Omega \subset R_n), of the form

[
P(x,D,q)=\sum_{|\alpha|+j\le m} a_{\alpha j}(x) q^j D^\alpha,\qquad x\in\Omega,
\tag{1}
]

[
D^\alpha=\left(\frac{1}{i}\right)^{|\alpha|}
\frac{\partial^{|\alpha|}}{\partial x_1^{\alpha_1}\cdots \partial x_n^{\alpha_n}},
\qquad
|\alpha|=\alpha_1+\alpha_2+\cdots+\alpha_n.
]

Here (q) is a parameter taking values on the half-line (\Theta) issuing from the origin of the complex plane. By (P_m(x,D,q)) we denote the principal part of the operator (P) of order (m).

In paper ((^1)), boundary-value problems for operators of this type were investigated under the condition that the characteristic polynomial (P_m(x,\xi,q)\ne 0) for (|\xi|+|q|\ne 0), (\xi\in R_n), (q\in\Theta). In the present note the question is considered of the existence and regularity of solutions of the differential equation (P(x,D,q)u=f) in the case when this condition is violated. The central point is the proof of estimates of the form

[
|q|\sum_{|\alpha|\le m-1} |q|^{2(m-1-|\alpha|)}
\iint |D^\alpha u|^2\,dx
\le
K\iint |P(x,D,q)u|^2\,dx,
\tag{2}
]

[
u\in C_0^\infty(\Omega),\qquad |q|\ge d_0>0.
]

Estimates of this type for operators not depending on a parameter, in the spaces (L_2(\Omega)) with a certain weight, were considered in works ((^{2-5})).

Theorem 1. Let (P) be an operator of the form (1) with coefficients from (C^1(\overline{\Omega})). Suppose that there exist a positive constant (K) and a number (d_0>0) such that estimate (2) holds.

Then, for those ((x,\xi_0,q_0)\in \overline{\Omega}\times R_n\times\Theta) for which (\xi_0\ne 0), (q_0\ne 0), and (P_m(x,\xi_0,q_0)=0), the inequality holds

[
|\xi_0|^{2(m-1)}
\le
2K|q_0|^{-1}\operatorname{Im}
\sum_{j=1}^n
P_{m(j)}(x,\xi_0,q_0)\,\overline{P_m^{(j)}(x,\xi_0,q_0)}.
\tag{3}
]

Here and below

[
P_{m(j)}(x,\xi,q)=\frac{\partial}{\partial x_j}P_m(x,\xi,q),
\qquad
P_m^{(j)}(x,\xi,q)=\frac{\partial}{\partial \xi_j}P_m(x,\xi,q).
]

Proof. Choose (w(x)\in C^\infty) so that (w(x)=\langle x,\xi_0\rangle+O(|x|^2)) as (x\to 0), and set (q=\lambda q_0), where (\lambda>0). Consider the function
(u^\lambda(x)=\exp[i\lambda w(x)]\psi(x\sqrt{\lambda})), where (\psi\in C_0^\infty(R_n)). Applying the Leibniz formula, introducing the new variable (x\sqrt{\lambda}), and passing to the limit as (\lambda\to\infty), we obtain that (2) is equivalent to

[
|\xi_0|^{2(m-1)}|q_0|\iint |\psi|^2\,dx
\le
K\iint \left|
\sum_{j=1}^n \bigl(P_{m(j)}(0,\xi_0,q_0)\cdot x_j\bigr)\psi
+
\sum_{j=0}^n P_m^{(j)}(0,\xi_0,q_0)D^j\psi
\right|^2 dx;
]

using the result of papers ((^2)), § 8, 1 and ((^3)), § 1, 2, we obtain condition (3). The theorem is proved.

We now show that a condition of type (3) is also sufficient for the validity of estimate (2). Let us first consider the case of operators with real coefficients.

Theorem 2. Let (P) be an operator of the form (1) with coefficients of the principal part from (C^2(\overline{\Omega})), and let (\operatorname{Im} q \ne 0). Suppose that at those points (x \in \overline{\Omega}) where there exist such (0 \ne \xi \in R_n), (0 \ne q \in \Theta), that (P_m(x,\xi,q)=0), the condition

[
\operatorname{Im} \sum_{j=1}^{n} P_{m(j)}(x,\xi,q)\,\overline{P}^{(j)}_{m}(x,\xi,q)>0
\tag{4}
]

is satisfied. Further, wherever (x \in \overline{\Omega}) and there exists such (0 \ne \xi \in R_n) that (P_m(x,\xi,0)=0), the condition

[
\lim_{q \to 0}\operatorname{Im}\frac{q}{|q|}
\sum_{j=1}^{n}
\left(
P^{(q)}{m(j)}(x,\xi,0)\,P^{(j)}(x,\xi,0)
-
P_{m(j)}(x,\xi,0)\,P^{(j)(q)}_{m}(x,\xi,0)
\right)>0
\tag{5}
]

is satisfied.

Here and below
[
P^{(q)}_{m}(x,\xi,0)=\frac{d}{dq}P_m(x,\xi,q)\quad \text{at } q=0.
]

Then there exist a constant (K), independent of the function (u), and a positive number (d_0) such that estimate (2) is satisfied.

Proof. By carrying out a finite partition of unity in (\overline{\Omega}), one can show that estimate (2) is local in character; we also note that, by choosing the number (d_0) sufficiently large, it is easy to obtain that (2) does not depend on adding operators of order less than (m). Consider (x=0 \in \overline{\Omega}) and (U_\delta={x;\ x\in\Omega,\ |x|<\delta}). Integrating by parts for (u\in C_0^\infty(U_\delta)), one can obtain that

[
\int |P_m(x,D,q)u|^2\,dx
\ge
\int ([P^*,P]u)\,\overline{u}\,dx.
\tag{6}
]

Here (P^=P_m(x,D,\overline{q})+P_{m-1}), where (P_{m-1}) is an operator of order not greater than (m-1), and ([P^,P]=P^P-PP^). Put (G_q(x,D)u=[P^*,P]u); the principal part of the symbol of the operator (G_q(x,D)) is determined by the formula

[
G_q(x,\xi)=2\operatorname{Im}\sum_{j=1}^{n}
P_{m(j)}(x,\xi,q)\,\overline{P}^{(j)}_{m}(x,\xi,q),
\qquad \xi\in R_n.
\tag{7}
]

Represent
[
G_q(x,\xi)=\sum_{j=0}^{2m-1} q^j G^{(j)}(x,\xi);
]
in view of the reality of the coefficients, it follows from (7) that (G^{(0)}(x,\xi)=0); therefore the expansion of (G_q) in powers of (q) has the form
[
\sum_{j=1}^{2m-1} q^j G^{(j)}(x,\xi),
]
where (G^{(j)}(x,\xi)) are polynomials with continuous coefficients of order (2m-1-j) in (\xi). It can be shown, using conditions (4) and (5), that there exist constants (C_1) and (C_2) such that for all (0\ne \xi\in R_n) and (q\in\Theta) the estimate

[
\left(|\xi|^2+|q|^2\right)^{m-1}
\le
C_1 |q|^{-1}G_q(0,\xi)
+
C_2 |P_m(0,\xi,q)|^2\left(|\xi|^2+|q|^2\right)^{-1};
\tag{8}
]

is satisfied. Multiplying (8) by (|\hat{u}(\xi)|^2), where

[
\hat{u}(\xi)=\int e^{-i\langle x,\xi\rangle}u(x)\,dx,
]

and integrating, we obtain

[
(2\pi)^{-n}\int |\hat{u}|^2\left(|\xi|^2+|q|^2\right)^{m-1}\,d\xi
\le
C_1 |q|^{-1}\int (G_q(0,D)u)\,\overline{u}\,dx
+
C_2|P_m(0,D,q)u|_q^2,
]

where by (\left|!\left|!\left|P_m(0,D,q)u\right|!\right|!\right|_q^2) we denote the expression

[
\int |P_m(0,\xi,q)u|^2 (|\xi|^2+|q|^2)^{-1}\,d\xi .
]

Integrating by parts, using the Cauchy—Bunyakovsky inequality and taking (7) into account, one can show that, if (\delta) is sufficiently small, then the estimate

[
\frac12 |q|(2\pi)^{-n}\int |\hat u|^2 (|\xi|^2+|q|^2)^{m-1}\,d\xi
\le
(C_1+2C_2|q|^{-1})\int |P'_m(x,D,q)u|^2\,dx,
]

holds, whence (2) follows. The proof is complete.

To derive the estimates (2) for operators with complex-valued coefficients, we introduce an additional assumption.

Definition. We shall say that an operator (P) of the form (1) is normal in the principal part in (\Omega) if its coefficients belong to (C^2(\overline{\Omega})) and there exists a differential operator (Q(x,D,0)) of order (m-1) with coefficients in (C^1(\overline{\Omega})) such that

[
C_{2m-1}(x,\xi,0)=2\operatorname{Im}\sum_{j=1}^{n}
P_{m(j)}(x,\xi,0)\overline{P}_m^{(j)}(x,\xi,0)
=
]

[
=2\operatorname{Re} P_m(x,\xi,0)\overline{Q}(x,\xi,0).
\tag{9}
]

Theorem 3. Let (P) be an operator of the form (1), normal in the principal part. Suppose that at those points (x\in \Omega) where there exist (0\ne \xi\in R_n) and (0\ne q\in\Theta) such that (P_m(x,\xi,q)=0), condition (4) is satisfied. Further, where (x\in\overline{\Omega}) and (P_m(x,\xi,0)=0), (0\ne \xi\in R_n), the condition

[
\operatorname{Im}\left{\left(\lim_{q\to 0}\frac{q}{|q|}\right)
\sum_{j=1}^{n}\left(P_{m(j)}^{(q)}(x,\xi,0)\overline{P}m^{(j)}(x,\xi,0)
-
P_m^{(j)(q)}(x,\xi,0)\overline{P}(x,\xi,0)\right)\right}
-
]

[
-2\operatorname{Re}\left{\lim_{q\to 0}\frac{q}{|q|}
\left(P_m^{(q)}(x,\xi,0)\overline{Q}(x,\xi,0)\right)\right}>0.
\tag{10}
]

is satisfied. Then there exists a constant (K) such that the estimate (2) holds for functions in (C_0^\infty(\Omega)).

The proof of Theorem 3 is, on the whole, analogous to the proof of Theorem 2.

Remark 1. If the coefficients of (P_m(x,D,q)) are real and (\operatorname{Im} q\ne 0), then for (Q) one may take an arbitrary operator with purely imaginary coefficients. In this case we can always satisfy inequality (10) by choosing (Q(x,\xi,0)=-iM P_m^{(q)}(x,\xi,0)), where (M) is a positive or negative constant, depending on the sign of (\displaystyle \lim_{q\to 0}\operatorname{Im}\frac{q}{|q|}).

Remark 2. By virtue of the results of [2], p. 249, conditions (4) and (10) are invariant under changes of variables. Therefore all considerations of this note remain valid also when (\Omega) is a precompact open set on a manifold.

Example. Let

[
P=-x\partial^2/\partial x_1^2-\partial^2/\partial x_2^2+iq\,\partial/\partial x_1-q^2,\qquad x=(x_1,x_2)
]

with (q=i\tau), (\tau\ge 0), and let (\Omega) be some neighborhood of the origin. Then

[
P(x,\xi,q)=x\xi_1^2+\xi_2^2-i\tau\xi_1+\tau^2,\qquad \tau\ge 0.
]

At the point (x=0), (P(x,\xi,0)=0) for (\xi_2=0) and (\xi_1\ne 0). Note that at this same point (P^{(1)}=0) and (P^{(2)}=0) for (\xi_2=0), (\xi_1\ne 0), and (q=0); therefore the operator (P) is not an operator of principal type in the sense of the definitions of [2]. Let us verify condition (5):

[
\left(\lim_{\tau\to 0}\operatorname{Im}\frac{i\tau}{\tau}\right)
\left(-P^{(1)(q)}P_{(1)}\right)=\xi_1^2>0.
]

Next, (P(x,\xi,q)\ne 0) for (q\ne 0); consequently, for the operator (P) in (\Omega) there exists an estimate of type (2). Using the averaging technique (see (2), § 2.4), we can pass from estimates in the space (L_2) to estimates in the space (\mathscr H_{(s)}).

Theorem 4. Let (P) be an operator of the form (1) with coefficients of the principal part in (C^\infty(\Omega)), normal in the principal part. Suppose that conditions (4) and (10) of Theorem 3 are satisfied.

Then for any pair of real numbers (s) and (t) there exists a positive number (d_0) such that if, for at least one (q) such that (|q|\ge d_0), one has
(P(x,D,q)u(x,q)=f(x,q)\in\mathscr H_{(s)}), (u(x,q)\in\mathscr H_{(t)}), then in fact (u\in\mathscr H_{(s+m-1)}) for each such (q), and for any such (s) and (t) and compact (K\subset\Omega) there exists a positive constant (C_{s,t,k}), independent of the function (u) and of the parameter (q\in\Theta), such that the estimate holds
[
|u|{(s+m-1)}\le C u\subset K.}|P(x,D,q)u|_{(s)},\quad \text{if } \operatorname{supp
]

Theorem 5. Let the assumptions of Theorem 4 be satisfied, and let (\Omega') be an open set (\Subset\Omega). Suppose that (f(x,q)\in\mathscr H_{(s)}) for all (q) such that (|q|\ge d_0), for some positive number (d_0). Denote by ({}^{t}P) the operator adjoint to the operator (P), i.e., such that
[
\int (P(x,D,q)u)v\,dx=\int u({}^{t}P(x,D,q)v)\,dx,\quad
u,v\in C_0^\infty(\Omega).
]

Then one can find a function (v(x,q)\in\mathscr H_{(s+m-1)}) for which ({}^{t}P(x,D,q)v=f) in (\Omega') for all such (q).

Theorem 6. Let the conditions of Theorem 4 be satisfied, and let (\Omega') be an open set (\Subset\Omega). Suppose that the equation ({}^{t}P(x,D,q)v=0) has no nonzero solutions with support in (\Omega') for all (q) such that (|q|\ge d_0), where (d_0) is some positive number.

Then there exists a linear mapping (\mathscr E) of the space (L_2(\Omega')) into itself such that
[
P(x,D,q)\mathscr E f=f \text{ in } \Omega',\quad \text{if } f\in L_2(\Omega'),
]
[
\mathscr E P(x,D,q)u=u \text{ in } \Omega',\quad \text{if } u\in C_0^\infty(\Omega'),
]
(D^\alpha\mathscr E) is a bounded operator in (L_2(\Omega')), provided (|\alpha|