UDC 517.9

MATHEMATICS

I. A. KIPRIYANOV

ON GÅRDING’S INEQUALITY FOR DEGENERATE ELLIPTIC OPERATORS

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

Let (\Omega) be a bounded domain of the ((n+1))-dimensional Euclidean space of points (x=(x_1,\ldots,x_n,x_{n+1})). Let (m) be a fixed natural number. Suppose that for every (z\in\Omega) there is given a real homogeneous form (a(z,\xi)) of the real variables (\xi_1,\ldots,\xi_n,\xi_{n+1}) of degree (2m), with: 1) the coefficients of the form sufficiently smooth and uniformly continuous; 2) (\inf_{|\xi|=1} a(z,\xi)) having a positive lower bound in (\Omega). With every such form one can associate the Dirichlet form

[
\sum_{|\alpha|=|\beta|=m} a_{\alpha\beta}(z)D^\alpha uD^\beta u,
\tag{1}
]

where the coefficients (a_{\alpha\beta}(z)) are real, symmetric, satisfy condition 1), and, moreover,

[
\sum_{|\alpha|=|\beta|=m} a_{\alpha\beta}(z)\delta^\alpha\delta^\beta=a(z,\xi).
\tag{2}
]

As is known, such forms exist. If (m=1), then there exists only one Dirichlet form, and this form is positive definite. For (m>1) it may happen that no Dirichlet form is positive definite. From some Dirichlet form belonging to the form (a(z,\xi)) we construct the Dirichlet integral

[
I(u,u)=\int_{\Omega}\sum_{|\alpha|=|\beta|=m} a_{\alpha\beta}(z)D^\alpha uD^\beta u\,dz.
\tag{3}
]

The following assertion is valid.

Gårding’s inequality ((^{1})). There exist positive constants (C_1) and (C_2) such that

[
I(u,u)\geq C_1|u|{W_2^m}^2-C_2|u|^2.
\tag{4}
]

This inequality, as is known, plays an important role in the use of functional methods in the theory of partial differential equations.

In the present note we indicate a class of degenerate elliptic operators for which Gårding’s inequality ((^{1})) retains its validity. Let (E_{n+1}^+) denote the half-space (y>0) ((x_{n+1}=y)) of the Euclidean ((n+1))-dimensional space of points (z=(x,y)) ((x=(x_1,\ldots,x_n))). We shall consider a bounded domain (\Omega^+) lying in the half-space (y>0) and adjacent to the hyperplane (y=0). Let (C_0^\infty(\Omega^+)) denote the set of infinitely differentiable functions having compact support contained in (\Omega^+). On this set we consider the differential operator

[
\widetilde D_y^k u
=
y^k\frac{\partial^k u}{(y\partial y)^k}
=
\frac{\partial^k u}{\partial y^k}
+
\sum_{i=1}^{k} C_i^{(k)}\frac{\partial^{k-i}u}{y^i\partial y^{k-i}}.
\tag{5}
]

For every point (z\in\Omega^+) we prescribe a real homogeneous form (a(z;\xi,\eta)) of the real variables (\xi_1,\ldots,\xi_n,\eta) of degree (2m). In this case

we shall assume that the coefficients satisfy conditions 1) and 2) and, as (y \to 0), behave in a definite manner (see below).

To each such form we associate a Dirichlet form of the type

[
\sum_{|\alpha|+k=m}\sum_{|\beta|+l=m}
a_{\alpha\beta}^{kl}(z)\,\widetilde D^{\alpha+k}u\,\widetilde D^{\beta+l}u,
\tag{6}
]

where (\widetilde D^{\alpha+k}=D_x^\alpha \widetilde D_y^k), (\widetilde D^{\beta+l}=D_x^\beta \widetilde D_y^l); the coefficients (a_{\alpha\beta}^{kl}) are real, symmetric, satisfy the conditions indicated above, and possess the property that

[
\sum_{|\alpha|+k=m}\sum_{|\beta|+l=m}
a_{\alpha\beta}^{kl}(z)\,\xi^\alpha \tau^k \xi^\beta \tau^l
=
a(z;\xi,\tau).
\tag{7}
]

Let us take one of these Dirichlet forms and, from it, form an integral of Dirichlet type

[
\widetilde I_\gamma(u,u)=
\int_{\Omega^+}\sum a_{\alpha\beta}^{kl}(z)\,
\widetilde D^{\alpha+k}u(z)\,\widetilde D^{\beta+l}u(z)\,y^{2\gamma}\,dz.
\tag{8}
]

If the coefficients are constant, then in the Fourier—Bessel images the form (8) is written as follows:

[
\widetilde I_\gamma=
C_\gamma^{(m)}
\int_{E_{n+1}^+}
a(s,t)\,\left|\widetilde f_{\gamma-\frac12}(s,t)\right|^2 t^{2\gamma}\,ds\,dt,
\tag{9}
]

where (a(s,t)) is a homogeneous form of degree (2m) with constant coefficients, and (\widetilde f_{\gamma-\frac12}) is the mixed Fourier—Bessel transform (see (4)). To obtain a lower estimate for the forms just indicated, we introduce the corresponding functional space, considered by the author earlier in papers ((^2!-!^4)). The space (\widetilde W_{y,2,\gamma}^{\,k}(\Omega^+)) is defined as the closure of (C_0^\infty(\Omega^+)) in the norm

[
|u|^2_{\widetilde W_{y,2,\gamma}^{\,k}}
=
\int_{\Omega^+}|u|^2y^{2\gamma}\,dz
+
\int_{\Omega^+}|\widetilde D_y^{\,k}u|^2y^{2\gamma}\,dz.
\tag{10}
]

We shall characterize differential properties in all the remaining directions as follows. Let (l_i) be an integer nonnegative number. The functional spaces (W_{x_i,2,\gamma}^{\,l_i}(\Omega^+)) ((i=1,2,\ldots,n)) are defined as the closure of (C_0^\infty(\Omega^+)) in the usual S. L. Sobolev norm with the weight factor (y^{2\gamma}). The space (\widetilde W_{x,y,2,\gamma}^{\,l,k}(\Omega^+)) is defined as the intersection of the corresponding spaces with the norm determined by the formula

[
\sum_{i=1}^{n}|u|^2_{W_{x_i,2,\gamma}^{\,l_i}(\Omega^+)}
+
|u|^2_{\widetilde W_{y,2,\gamma}^{\,k}(\Omega^+)}.
\tag{11}
]

This space has many remarkable properties, and for it, in terms of Fourier—Bessel images, there is an equivalent norm (see (4)).

If on (C_0^\infty(\Omega^+)) we introduce the scalar product

[
(u,v)j^\gamma
=
\int}\sum_{|\beta|+k=j
\widetilde D^{\beta+k}u(z)\,\widetilde D^{\beta+k}v(z)\,y^{2\gamma}\,dz
\quad (j=0,1,2,\ldots,m)
\tag{12}
]

and complete the set (C_0^\infty(\Omega^+)) in the norm

[
|u|_j^2=(u,u)_j^\gamma+(u,u)_0^\gamma,
\tag{13}
]

then we obtain a Hilbert space (\overset{\circ}{H}{}{j}^{\gamma}) with norm equivalent to the norm of (\widetilde W). Using the apparatus of the mixed Fourier—Bessel transform, we prove the following assertion:}^{\,m,m

Theorem 1. There exist positive constants (c_1) and (c_2) such that

[
\tilde I_\gamma(u,u)\geq C_1|u|^2_{W^m_{2,\gamma}}-C_2|u|^2_{\mathcal L_{2,\gamma}} .
\tag{14}
]

Let us now consider the ordinary Dirichlet form of the type

[
\sum_{|\alpha|+k=m}\sum_{|\beta|+l=m}
a_{\alpha\beta}^{kl}(z)D^{\alpha+k}u(z)D^{\beta+l}u(z)
\tag{15}
]

with all the preceding restrictions on the coefficients (a_{\alpha\beta}^{kl}), and, in addition, assume that they satisfy the inequality

[
\left|a_{\alpha\beta}^{kl}(z)\right|\leq C y^{k+l-2}\qquad (k,l\geq 1).
\tag{16}
]

For this form we construct the Dirichlet integral

[
\tilde I_\gamma(u,u)=
\int_{\Omega^+}\sum a_{\alpha\beta}^{kl}(z)D^{k+\alpha}u(z)D^{\beta+l}u(z)y^{2\gamma}\,dz .
\tag{17}
]

Then the following assertion holds.

Theorem 2. There exist positive constants (C_1') and (C_2') such that

[
\tilde I_\gamma(u,u)\geq C_1'|u|^2_{W^m_{2,\gamma}}-C_2'|u|^2_{\mathcal L_{2,\gamma}} .
\tag{18}
]

Upper estimates in inequalities (14) and (18) are obtained quite simply. Let us now consider the following problem. Let (\mathcal L) be a formally self-adjoint operator of order (2m),

[
\mathcal L=(-1)^m
\sum_{|\alpha|+k=|\beta|+l\leq m}
D^{\alpha+k}\left(y^{2\gamma}a_{\alpha\beta}^{kl}(z)D^{\beta+l}u(z)\right)
\tag{19}
]

with coefficients satisfying the restrictions indicated above. The non-self-adjoint case can be reduced to this one.

Dirichlet problem. In the domain (\Omega^+) a function (g\in H_m^\gamma(\Omega^+)) is given, and an infinitely differentiable function (h) such that (|h|{\mathcal L). Find a solution (u) of the equation}

[
\mathcal L u=h
]

such that (g-u\in \dot H_m^\gamma). The Dirichlet problem is solvable for arbitrary (g) and (h), if the homogeneous problem has only the zero solution. The pair of equations (Qu=h) and (Q^v=g), where (Q^) is the operator formally adjoint to (Q), forms a Fredholm pair. One can study the existence and properties of solutions of the equation

[
Qu-\lambda u=0,
]

belonging to (H_m^\gamma), etc.

Voronezh State
University

Received
12 XI 1967

CITED LITERATURE

(^{1}) L. Gårding, Math. Scand., 1, 55 (1953).
(^{2}) I. A. Kipriyanov, DAN, 147, No. 3 (1962).
(^{3}) I. A. Kipriyanov, DAN, 158, No. 2 (1964).
(^{4}) I. A. Kipriyanov, Tr. Matem. Inst. im. V. A. Steklova AN SSSR, 89, 2 (1967).