UDC 517.946.9

MATHEMATICS

I. A. KIPRIYANOV

BOUNDARY-VALUE PROBLEMS FOR SINGULAR ELLIPTIC PARTIAL DIFFERENTIAL OPERATORS

(Presented by Academician P. S. Novikov on 20 III 1970)

At present, boundary-value problems for elliptic equations of higher order that degenerate on the boundary of the domain are being intensively studied ((^{3-5})).

In the present note one more class of elliptic operators of higher order is studied, which degenerate on the corresponding hyperplane. The case when such degeneration occurs in a tangent plane was studied by the author earlier ((^{2})).

1. Let (R_n^+) denote the half-space (x_n=y>0) of the Euclidean (n)-dimensional space (R_n) of points (x=(x',y)), where (x'=(x_1,\ldots,x_{n-1})). We introduce for consideration a class of function spaces (H_\gamma^l), a detailed description of which is given in ((^{1})).

Let (L) be a linear differential operator of the form

[
L=\sum_{|l'|+2l_n\le r} a_{l',l_n}(x)D_{x'}^{l'}B_y^{l_n},
\tag{1}
]

where the coefficients (a_{l',l_n}) are functions sufficiently smooth in (\overline{R_n^+}), constant for sufficiently large values of (x) in modulus. Here and below (B_y) denotes the singular Bessel operator

[
\frac{\partial^2}{\partial y^2}+\frac{2\gamma}{y}\frac{\partial}{\partial y}
\qquad (\gamma>0).
]

On the operator (L) at all points (x=x_0) belonging to the domain (y\ge 0) we impose the condition

[
\sum_{|l'|+2l_n=r} a_{l',l_n}(x)\xi^{l'}\xi_n^{2l_n}\ne 0
\qquad \text{for } |\xi|\ne 0.
\tag{2}
]

Such operators were called (B)-elliptic operators by the author ((^{2})). We note that a condition of type (2) was first formulated by the author in ((^{2})). An operator (T) is called a smoothing operator if it is a bounded operator from (H_\gamma^s) to (H_\gamma^{s+1}). Let first the operator (L) have constant coefficients. Consider in all of (R_n^+) the equation (Lu=L(D_{x'},B_y)u(x)+f(x)). Let (L_0) be the principal part of the operator (L).

Theorem 1. Let (u\in H_\gamma^s), (s\ge 2m), be a solution of the equation (L_0u(x)=f(x)). Then for (\gamma>0):

1) the a priori estimate is valid

[
|u|{H\gamma^s}\le C\left(|L_0u|{H\gamma^{s-2m}}+|u|{H\gamma^0}\right),
\tag{3}
]

2) there exists a bounded operator (R_0), acting from (H_\gamma^{s-2m}) to (H_\gamma^s), such that

[
L_0R_0=I+T.
\tag{4}
]

With the aid of the preceding theorem the following is proved.

Theorem 2. Let (u \in H_\gamma^s) and (s \geq 2m). Let

[
L(x;D_{x'},B_y)=\sum_{|l|\leq 2m} a_l(x)D_{x'}^{\,l'}B_y^{\,l_n}
]

be a (B)-elliptic operator with infinitely differentiable coefficients, bounded in all of (R_n^+), and with higher coefficients satisfying the condition: there exists an (\varepsilon>0) such that for all (x\in \overline{R}_n^+),

[
|a_l(x)-a_l(0)|<\varepsilon .
]

Let, moreover, (a_l(x)) satisfy the condition

[
|D_{x',y}^{\,l'+k} a_l(x)|\leq C y^{2r-i-k},
\tag{5}
]

where (k+i\leq 2r,\ 1\leq i\leq 2r-1,\ l'+2r\leq s-2m,\ |l|\leq 2m-1). Then for (\gamma>0), for a solution of the equation (Lu=f), the a priori estimate

[
|u|{H\gamma^s}\leq C\left(|Lu|{H\gamma^{s-2m}}+|u|{H\gamma^0}\right)
\tag{6}
]

is valid, and there exists a bounded operator (R_B), acting from (H_\gamma^{s-2m}) into (H_\gamma^s), such that

[
LR_B=I+T.
]

We now consider a linear differential operator of the form

[
L(x;D_{x'},D_y^2)=\sum_{|l|\leq 2m} a_l(x)D_{x'}^{\,l'}D_y^{2l_n}.
\tag{7}
]

The conditions on the coefficients (a_l) of the operator (L) will be formulated below. We shall call the operator (7) (B)-elliptic if condition (2) is fulfilled for it in the domain (y\geq 0). With the aid of Theorem 2 the following assertion is proved.

Theorem 3. Let (u\in H_\gamma^s) and (s\geq 2m). Let

[
L(x;D_{x'},D_y^2)=\sum_{|l|\leq 2m} a_l(x)D_{x'}^{\,l'}D_y^{2l_n}
\tag{8}
]

be a (B)-elliptic operator with infinitely differentiable coefficients, bounded in (R_n^+), and with higher coefficients satisfying the condition:

[
|a_l(x)-a_l(0)|<\varepsilon .
]

Let, moreover, (a_l(x)) satisfy the conditions

[
\left|D_{x'}^{\,l'}\frac{\partial^k a_l(x)}{\partial y^k}\right|\leq C y^{2r-i-k},
\tag{9}
]

[
k\leq 2r-i,\qquad 1\leq i\leq 2r-1,\qquad l'+2r\leq s-2m,\qquad |l|\leq 2m-1;
]

[
|a_l(x)|\leq C_0 y^r,\qquad 1\leq r\leq 2m-1,\qquad |l|\leq 2m-1;
\tag{10}
]

[
\left|D_{x'}^{\,l'}\frac{\partial^{\,i-k-r}a_l(x)}{(y\partial y)^{\,i-k-r}}\,y^{\,i-k-\tau}\right|\leq C_1 y^{j+\tau},
\tag{11}
]

[
l'+i\leq s-2m,\qquad 1\leq j\leq 2m-1,\qquad |l|\leq 2m,\qquad l_n\neq 0.
]

Then for (\gamma<0), for a solution of the equation (Lu=f) in (R_n^+), the estimate

[
|u|{H\gamma^s}\leq C\left(|Lu|{H\gamma^{s-2m}}+|u|{H\gamma^0}\right)
\tag{12}
]

is valid.

and there exists a bounded operator (R) such that

[
LR=I+T,
\tag{13}
]

where (R) acts from (H_\gamma^{s-2m}) into (H_\gamma^s), and (T) is a smoothing operator.

The results obtained in this part allow us to pass to a bounded domain.

1. Consider a bounded domain (\Omega) in the space (R_n). Let the boundary (\partial\Omega) of this domain be a smooth manifold. Denote by ({V_{\nu_1}}) a covering of the boundary (\partial\Omega) by a finite system of domains (V_1,\ldots,V_r) in (R_n) such that in each (V_k) there is defined an infinitely smooth change of coordinates (x\to y=y^{(k)}) with positive Jacobian, under which the boundary (\partial\Omega) is transformed locally into the hyperplane (y_n=0), while the part of the domain adjacent to (\partial\Omega) is transformed into (y_n>0). In addition, it is assumed that under these changes of coordinates the normals to (\partial\Omega) are transformed into normals to (y_n=0).

Let ({V_{\nu_2}}) be such a finite system of open sets that

[
\Omega\setminus \bigcup_{\nu_1} V_{\nu_1}\subset \bigcup_{\nu_2} V_{\nu_2},
]

and, moreover,

[
\overline{\bigcup_{\nu_2} V_{\nu_2}}\cap \partial\Omega=\varnothing .
]

The space (H_\gamma^l(\Omega)) is the set of functions defined in (\Omega) for which the norm

[
|u|{H\gamma^l(\Omega)}
=\sum_{\nu_1}|\varphi_{\nu_1}u|{H\gamma^l(R_n^+)}
+\sum_{\nu_2}|\varphi_{\nu_2}u|_{H^l},
\tag{14}
]

is finite, where the norm (|\varphi_{\nu_1}u|{H\gamma^l}) is computed in the local coordinates (y^{(\nu_1)}) as the norm of a function defined in the corresponding half-space, and (|\varphi_{\nu_2}u|_{H^l}) is the usual norm in the Sobolev space.

Let the operator (L) inside the domain (\Omega) have the form

[
L(x,D_x)=\sum_{|\alpha|\le r} a_\alpha(x)D_x^\alpha,
\tag{15}
]

where the coefficients (a_\alpha) are sufficiently smooth functions inside the domain (\Omega). We now define the operator (L) near the boundary (\partial\Omega) of the domain (\Omega), i.e. at points

[
x\in \Omega\cap \left(\bigcup_{\nu_1}V_{\nu_1}\right).
]

We require of the operator (L) that, for any (\nu_1), the operator (\widetilde L_{\nu_1}) be representable in the following form:

[
\widetilde L_{\nu_1}
=\sum_{|\alpha'|+2\alpha_n\le r}
a_{\alpha',\alpha_n}(y)D_y^{\alpha'}B_{y_n}^{\alpha_n},
\tag{16}
]

where (a_{\alpha',\alpha_n}) are sufficiently smooth functions defined in (\overline{R_n^+}\cap \widetilde V_{\nu_1}), and (\widetilde V_{\nu_1}) is the image of the set (V_{\nu_1}) under the transformation (x\to y^{(\nu_1)}). We impose the following restriction on the operator (L). At every point (x\in\Omega) the operator (L) is elliptic, i.e.

[
\sum_{|\alpha|=r} a_\alpha(x)\xi^\alpha \ne 0
\quad\text{for }|\xi|\ne 0,\quad \xi\in R_n,\quad x\in\Omega .
\tag{17}
]

For any (\nu_1), the operator (\widetilde L_{\nu_1}) satisfies, for (|\xi|\ne 0), (\xi\in R_n), (y\in \widetilde V_{\nu_1}\cap{y_n\ge 0}), the condition

[
\sum_{|\alpha'|+2\alpha_n=r}
\widetilde a_{\alpha',\alpha_n}(y)\xi^{\alpha'}\xi_n^{2\alpha_n}\ne 0 .
\tag{18}
]

An operator (L) satisfying these conditions will be called a (B)-elliptic operator in (\Omega).

Theorem 4. Let the operator (L) be (B)-elliptic in the domain (\Omega). Suppose its coefficients satisfy the conditions of Theorem 2. Then, for (\gamma>0), the estimate

[
|u|{H\gamma^{s}(\Omega)}
\leq
C\left(|Lu|{H\gamma^{s-2m}(\Omega)}
+
|u|{H\gamma^{0}(\Omega)}\right)
\tag{19}
]

holds, and there exists a bounded operator (R), acting from (H_\gamma^{s-2m}(\Omega)) into (H_\gamma^s(\Omega)), such that

[
LR=I+T,
\tag{20}
]

where (I) is the identity operator and (T) is a smoothing operator.

An analogous assertion holds for the operator (\bar L), defined inside the domain (\Omega) by means of the operator (15), and near the boundary by means of an operator of the form

[
\widetilde L_{\nu_1}
=
\sum_{|\alpha'|+2\alpha_n=r}
\tilde a_{\alpha',\alpha_n}(y)D_y^{\alpha'}D_y^{2\alpha_n},
\tag{21}
]

whose coefficients satisfy, near the boundary, the conditions of Theorem 3. The operators (\bar L) and (\widetilde L_{\nu_1}) themselves satisfy conditions (17) and (18), respectively.

Using the Fourier–Bessel transform, one can show that the smoothing operators (T) thus obtained are completely continuous for a bounded domain (\Omega). Hence the operators (L) and (\bar L), acting respectively from (H_\gamma^{2m+k}) into (H_\gamma^k), are Noetherian.

Voronezh State University
named after the Lenin Komsomol

Received
19 III 1970

CITED LITERATURE

(^{1}) I. A. Kipriyanov, Trudy Mat. Inst. AN SSSR, 89 (1967).
(^{2}) A. I. Kipriyanov, DAN, 158, No. 2 (1964).
(^{3}) M. I. Vishik, V. V. Grushin, Mat. sborn., 79 (121), 1 (1969).
(^{4}) M. I. Vishik, V. V. Grushin, Mat. sborn., 80 (122), 4 (1969).
(^{5}) L. V. Fursikov, Mat. sborn., 79 (121), 3 (1969).