UDC 517.944 + 517.946

MATHEMATICS

M. D. RAMAZANOV

A PRIORI ESTIMATES OF SOLUTIONS OF THE DIRICHLET PROBLEM FOR SOME HYPOELLIPTIC EQUATIONS

(Presented by Academician S. L. Sobolev on 29 V 1967)

Let

[
P(x,D_x)u \equiv \sum_{|\alpha|\le m} a_\alpha(x)D_x^\alpha u \equiv
]

[
\equiv \sum_{\alpha_1+\ldots+\alpha_n\le m}
a_{\alpha_1\ldots\alpha_n}(x_1,\ldots,x_n)
\frac{\partial^{\alpha_1+\cdots+\alpha_n}}
{(ix_1)^{\alpha_1}\ldots(ix_n)^{\alpha_n}}\,u
\tag{1}
]

be a hypoelliptic operator of constant strength ((^1)) of order (m), defined on functions of (n) real variables (u(x)=u(x_1,\ldots,x_n)), given in a bounded domain (\Omega) with continuously differentiable boundary (\Gamma). The coefficients of the operator (a_\alpha(x)) are continuous functions in (\Omega+\Gamma).

By a solution of the Dirichlet problem we mean a sufficiently smooth solution (u(x)) of the equation

[
P(x,D_x)u=f,\qquad x\in\Omega,
\tag{2}
]

for which, at each boundary point (y\in\Gamma), all derivatives up to an order less than a certain number (m_+), depending on the operator (P) and the point (y), vanish. If (A_y) is the matrix of such a rotation of the coordinate system (A_yx=\tilde x) under which the directions of the axis (\tilde x_n) and of the inward normal at the point (y\in\Gamma) coincide, and (A_y') is the transposed matrix, then (m_+(y)) is equal to the number of roots with positive imaginary part of the polynomial (P(y,A_y'\xi)), solved with respect to (\xi_n), for sufficiently large values of (|\xi'|=(\xi_1^2+\cdots+\xi_{n-1}^2)^{1/2}) with real (\xi_1,\ldots,\xi_{n-1}) ((0\le m_+(y)\le m)).

Suppose that there exist such mutually orthogonal directions (l_1,\ldots,l_s) that, after a rotation (B'x=\tilde x) aligning the directions of the axes (\tilde x_1,\ldots,\tilde x_s) with (l_1,\ldots,l_s), the polynomial (P(y,B'\xi)) has order (m) in each of the variables (\xi_1,\ldots,\xi_s), and for some constants (C_1,C_2) the conditions

[
\xi_j^m/|P(y,B'\xi)|C_2\text{ and }j=1,\ldots,s.
\tag{3}
]

are satisfied.

Let (n(y)) be the unit inward normal at the point (y\in\Gamma), and let (N(y)) be the projection of (n(y)) onto the subspace spanned by the vectors (l_j),

[
N(y)=\sum_{j=1}^{s}(n(y),l_j)l_j.
]

We require that:

[
\begin{gathered}
\text{the function } |N(y)| \text{ defined on } \Gamma \text{ have zeros}\
\text{of order not higher than the first.}
\end{gathered}
\tag{4}
]

Except for the set ({y\in\Gamma,\ N(y)=0}), (\Gamma) is assumed to be (m) times continuously differentiable, with uniform estimates of the derivatives.

Theorem. Under the imposed restrictions (1), (3), (4), the solution of the Dirichlet problem for equation (2) admits the estimate

[
|Q(x,D_x)u|{\mathscr L_2(\Omega)}
\le K\bigl(|f|\bigr),}+|u|_{\mathscr L_2(\Omega)
\tag{5}
]

where the constant (K) does not depend on (u), and (Q(x,D_x)) is any operator weaker than (P(x,D_x)), i.e., with some constants (C_1) and (C_2),

[
|Q(x,\xi)/P(x,\xi)|C_2 .
\tag{6}
]

The method of proof is a generalization of the usual method of local estimates, and the main attention is directed to the choice of a special cut-off function that makes it possible to pass to local considerations.

Let us note, first, that the function (u) can be represented as the sum of two functions, one of which is concentrated in an (\varepsilon_1)-neighborhood of (\Gamma) with sufficiently small (\varepsilon_1), while the support of the other lies inside (\Omega). For the function concentrated in (\Omega), the estimate we need is already known ((^1)) (Theorem 7.4.2), and therefore it is enough to consider functions concentrated in an (\varepsilon_1)-neighborhood of (\Gamma).

In a neighborhood of a point (y\in\Gamma) introduce local coordinates (\tilde{x}) with origin at the point (y) and with the axis (\tilde{x}_n) directed along the interior normal, (\tilde{x}=A_y(x-y)). Let (u(x)=u(\tilde{x},y)). Then equation (2) can be written in the form

[
P(y,A_y'D_{\tilde{x}})u(\tilde{x},y)=f_1\equiv f+[P(y,D_x)-P(x,D_x)]u .
\tag{2′}
]

Let (\zeta(\tilde{x},y)), for fixed (y\in\Gamma), be a function of (\tilde{x}), smooth in (\Omega). We shall impose several restrictions on the form of this function, and for the moment we require that the following property hold.

Property 1. As a function of (\tilde{x}), (\zeta(\tilde{x},y)) is concentrated in a (2\varepsilon_1)-neighborhood of the point (y\in\Gamma), and the intersection (\nu(y)) of its support with the boundary (\Gamma) lies in an (\varepsilon(y)=\varepsilon_2|N(y)|)-neighborhood of the point (y). The constant (\varepsilon_2) must be so small that in the (\varepsilon(y))-neighborhood of the point (y) the transformation defined for each (y) in the (\varepsilon(y))-neighborhood of (y) as a displacement in the direction (N(y)), under which (\gamma(y)) passes into the plane tangent to (\Gamma) at the point (y), is uniformly bounded in the (C^m) norm with respect to (y\in\Gamma). Such a choice (0<\varepsilon_2\le \varepsilon_1) is possible owing to the smoothness of the boundary (\Gamma).

For each (y), the displacement defined in the (\varepsilon(y))-neighborhood can be extended by a displacement with preservation of the uniform boundedness of the (C^m)-norm to the whole space and, in particular, to the support of the function (\zeta(\tilde{x},y)).

Applying this change of variables (\tilde{x}=R(z)), we transform the operator (P(y,A_y'D_{\tilde{x}})) to the form

[
P(y,A_y'D_z)+\sum_{j=1}^{r} c_j(y,z)P_j(y,D_z)
]

with some finite (r), continuous coefficients (c_j(y,z)), (c_j(y,0)=0), and operators (P_j(y,D_z)), each of which has coefficients constant with respect to (z) and is weaker than (P(y,A_y'D_z)) (this is ensured by condition (3)).

Considering the function (v(z,y)=u(R(z),y)\zeta(R(z),y)), we can then write equation (2′) as

[
P(y,A_y'D_z)v=f_2,\qquad z_n>0,
\tag{7′}
]

where

[
f_2\equiv f_1\zeta+\sum_{0<|\alpha|\le m}\frac{1}{\alpha!}
\bigl[D_\xi^\alpha P(y,A_y'\xi)\bigr]{\xi=D_x}u(\tilde{x},y)\,D_x^\alpha \zeta(\tilde{x},y)
-
\sum c_j(y,z)P_j(y,D_z)v .}^{r
\tag{7″}
]

The function (v) satisfies the boundary conditions

[
D_{z_n}^{j}v\big|{z_n=0}=0,\qquad j=0,\ldots,m+-1.
]

The function (v), solving problem ((7′),(7‴)), can be written through

[
\widetilde{f}{2+}(\xi)=\int\,dz,} f_2(z)e^{-i\xi z
]

[
v(z,y)=\frac{1}{(2\pi)^n}\int_{-\infty}^{\infty} d\xi\, e^{i\xi z_n}\tilde f_{2+}(\xi)/P(y,A'_y\xi).
]

Let us now note that the variables ((y,z'=0,z_n)=(y,z_n)) define a coordinate system in a sufficiently small (\varepsilon_1)-neighborhood of the boundary (\Gamma), and the transformation ((y,z_n)\leftrightarrow x) is one-to-one, smooth, with Jacobian separated from zero and infinity.

Property 2 of the function (\zeta(\tilde x,y)): in the variables (z), in an (\varepsilon_1)-neighborhood of the boundary, for (z'=0) all derivatives of the function (\zeta) vanish, while the function itself is equal to 1.

If (Q(x,D_x)) is an operator with bounded coefficients which is weaker than (P(x,D_x)), then, after replacing the variables (x) by (z,y) according to the formula (z=R(A_y(x-y))), (Q) is transformed into (Q_1(z,y,D_z)) with bounded coefficients and weaker than (P(y,A'_yD_z)). Therefore, with certain bounded functions (b_j(y,z)) ((j=1,\ldots,r)),

[
Q_1(z,y,D_z)=\sum_{j=1}^{r} b_j(z,y)P_j(y,D_z) \; (^{1}).
]

By property 2 of the function (\zeta),

[
\sum_{j=1}^{r} b_j(z,y)P_j(y,D_z)v(z,y)\big|{z'=0}
=
Q(x,D_x)u\big|.
]

Therefore, instead of estimating
(|Q(x,D_x)u|{\mathscr L_2(\Omega)}), it is enough for us to be able to estimate
(|P_j(y,D_z)v|) ((j=1,\ldots,r)).}|_{\mathscr L_2(y,z_n)

It is not difficult to obtain the estimate

[
|P_j(y,D_z)v|{z'=0}|
\leq K_1|f|_{\mathscr L_2(y,z_n)} .
\tag{8}
]

In order to pass from inequality (8) to the required

[
|P_j(y,D_z)v|{z'=0}|
\leq K_2|f|{\mathscr L_2(y,z_n)}
\leq K|f|,
]

we apply the usual arguments using the small variation of continuous coefficients in a small domain and the possibility of obtaining the estimate (8) for any operators weaker than (P(x,D_x)).

Here the main role is played by the estimate

[
\left|\left{[D_\xi^\alpha P(y,A'y\xi)]\,u(\tilde x,y)\right}
D_x^\alpha \zeta(\tilde x,y)\Big|{\substack{\tilde x\to z\ z'=0}}\right|
\leq K_3|f|_{\mathscr L_2(y,z_n)},
]

which we derived from the inequality

[
\sup_{y,\xi_n}
\left|
\frac{D_\xi^\alpha P(y,A'y\xi)}{P(y,A'_y\xi)}
\intD_z^\alpha\zeta}^{\infty} e^{-iz\xi
\prod_{j=1}^{n}\frac{\sin L_1 z_j}{z_j}\,dz
\right|_{\mathscr L_2(|\xi'|>L_2-|\xi_n|)}
\leq L_3<\infty
\tag{9}
]

with certain constants (L_1,L_2,L_3).

Inequality (9) may be regarded as property 3 of the function (\zeta).
The requirement (4) of the theorem is a rather rough sufficient condition for (9) to hold.

Boundary-value problems for certain special types of hypoelliptic equations are the subject of the works ((^{2-5})).

Let us note here that for parabolic equations ((^{3,4})) and for the special type of boundary considered in these works, it is also possible to construct a function (\zeta) with properties 1, 2, 3 and, consequently, to obtain the estimate (5).

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR

Received
13 V 1967

REFERENCES

1. L. Hörmander, Linear Partial Differential Operators, Moscow, 1965.
2. S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic equations in partial derivatives under general boundary conditions, 1, Moscow, 1962.
3. V. P. Mikhailov, Matem. sbornik, 64 (106), no. 1, 10 (1964); 63 (105), no. 2, 238 (1964).
4. V. A. Kondrat’ev, DAN, 163, no. 2, 285 (1965).
5. B. Pini, Ann. di mat. pura ed appl., ser. 4, 73, 33 (1966).