UDC 517.946

MATHEMATICS

I. A. KIPRIYANOV

ON SELF-ADJOINT EXTENSIONS OF CERTAIN SINGULAR PARTIAL DIFFERENTIAL OPERATORS

(Presented by Academician M. V. Keldysh on 20 XII 1966)

One of the methods of proving existence theorems for elliptic formally adjoint operators in bounded domains is the method based on self-adjoint extensions (see, for example, \((^1)\)). In the present article a class of singular partial differential operators is indicated for which existence theorems can also be obtained by the method of self-adjoint extensions.

Consider, in the \((n+1)\)-dimensional Euclidean space of points \(z=(x,y)\) \((x=(x_1,\ldots,x_n))\), a domain \(\Omega^+\) situated in the half-space \(y>0\) and adjacent to the hyperplane \(y=0\). The boundary \(\partial\Omega^+\) of the domain \(\Omega^+\) is divided into a part \(\Gamma^0\), lying in the hyperplane \(y=0\), and a part \(\Gamma^+\), lying in the half-space \(y>0\). Let \(C_0^\infty(\Omega^+)\) denote the set of functions each of which is infinitely differentiable and has compact support contained in \(\Omega^+\).

On the indicated set of functions we introduce for consideration a linear differential operator of order \(2m\), having the form

\[ \begin{aligned} \mathcal{L}u &= (-1)^m \sum_{|\alpha|=m} C_m^{(\alpha)} D_B^\alpha \left( \sum_{|\beta|=m} C_m^{(\beta)} a_{\alpha\beta}(z) D_B^\beta u(z) \right) + \\ &\quad + (-1)^m \sum_{|\bar\alpha|=m} C_m^{(\bar\alpha)} D_B^\alpha \left[ \frac{\partial}{\partial y} \left( \sum_{|\bar\beta|=m} C_m^{(\bar\beta)} a_{\bar\alpha\bar\beta}(z) \frac{\partial}{\partial y} D_B^\beta u(z) \right) \right. \\ &\qquad\qquad\qquad\qquad\left. + \sum_{|\bar\beta|=m} C_m^{(\bar\beta)} \frac{k}{y} a_{\bar\alpha\bar\beta}(z) \frac{\partial}{\partial y} D_B^\beta u(z) \right]. \end{aligned} \tag{1} \]

Here we put

\[ \alpha=(\alpha',2\alpha_{n+1}),\qquad \alpha'=(\alpha_1,\ldots,\alpha_n), \]

\[ |\alpha|=|\alpha'|+2\alpha_{n+1} =\alpha_1+\cdots+\alpha_n+2\alpha_{n+1},\qquad D_B^\alpha = D_B^{\alpha'} = D_x^{\alpha'} B_y^{2\alpha_{n+1}}, \tag{2} \]

\[ D_x^{\alpha'}=\partial^{|\alpha'|}/\partial x_1^{\alpha_1}\cdots\partial x_n^{\alpha_n},\qquad \bar\alpha=(\alpha',2\alpha_{n+1}+1),\qquad |\bar\alpha|=\alpha_1+\cdots+\alpha_n+2\alpha_{n+1}+1, \]

\[ C_m^{(\alpha)}=\frac{m!}{\alpha_1!\cdots\alpha_n!(2\alpha_{n+1})!},\qquad C_m^{(\bar\alpha)}=\frac{m!}{\alpha_1!\cdots\alpha_n!(2\alpha_{n+1}+1)!}. \]

An analogous meaning is assigned to other symbols similar to them. The symbol \(B_y\), as usual, denotes the Bessel operator
\[ \frac{\partial^2}{\partial y^2}+\frac{k}{y}\frac{\partial}{\partial y} \quad (k>0,\ y\ge 0). \]

As regards the coefficients \(a_{\alpha\beta}(z)\) and \(a_{\bar\alpha\bar\beta}(z)\), it is assumed that they are bounded and are continuously differentiable a sufficient number of times in \(\overline{\Omega^+}\), in the sense of applying the corresponding number of times the operators occurring in (1).

In addition, it is assumed that they are real and symmetric. The real homogeneous form

\[ a(z,\xi)= \sum_{|\alpha|=|\beta|=m} a_{\alpha\beta}(z) C_m^{(\alpha)} C_m^{(\beta)} \xi^\alpha \xi^\beta + \sum_{|\bar\alpha|=|\bar\beta|=m} a_{\bar\alpha\bar\beta}(z) C_m^{(\bar\alpha)} C_m^{(\bar\beta)} \xi^{\bar\alpha}\xi^{\bar\beta} \tag{3} \]

of the real variables \(\xi_1,\ldots,\xi_n,\xi_{n+1}\) is positive definite.

Moreover, it is assumed that for any position of the point \(z\) in the domain \(\Omega^{+}\) the inequality

\[ a(z,\xi)\geq \mu\left(\sum_{|\alpha|=m} C_m^{(\alpha)}|\xi^\alpha|^2+ \sum_{|\bar\alpha|=m} C_m^{(\bar\alpha)}|\xi^{\bar\alpha}|^2\right) =\mu\sum_{|\gamma|=m} C_m^{(\gamma)}|\xi^\gamma|^2, \tag{4} \]

holds, where \(\mu\) is a positive constant,

\[ \xi=(\xi_1,\ldots,\xi_n,\xi_{n+1}),\qquad \xi^\alpha=\xi_1^{\alpha_1}\cdots \xi_n^{\alpha_n}\xi_{n+1}^{2\alpha_{n+1}}, \qquad C_m^{(\gamma)}=\frac{m!}{\gamma_1!\cdots \gamma_n!\gamma_{n+1}!}, \]

\[ \xi^{\bar\alpha}=\xi_1^{\alpha_1}\cdots \xi_n^{\alpha_n}\xi_{n+1}^{2\alpha_{n+1}+1}. \]

Condition (4) is called the condition of \(B\)-ellipticity of the operator \(\mathcal L\) (2). We shall consider the operator \(\mathcal L\) as an operator acting in the space \(\mathcal L_{2,k}(\Omega^+)\). Here \(\mathcal L_{2,k}(\Omega^+)\) denotes the space of square-summable functions with weight \(y^k\) \((k>0)\) in the domain \(\Omega^+\). Form the scalar product

\[ (\mathcal Lu,u)_k=\int_{\Omega^+} u\,\mathcal Lu\,y^k\,dz . \tag{5} \]

Integrating \(m\) times by parts and taking into account that \(u\in C_0^\infty(\Omega^+)\), we obtain

\[ (\mathcal Lu,u)_k= \int_{\Omega^+}\sum_{|\alpha|=|\beta|=m} a_{\alpha\beta}(z) C_m^{(\alpha)}C_m^{(\beta)} D_B^\alpha u\,D_B^\beta u\,y^k\,dz+ \]

\[ +\int_{\Omega^+}\sum_{|\bar\alpha|=|\bar\beta|=m} a_{\bar\alpha\bar\beta}(z) C_m^{(\bar\alpha)}C_m^{(\bar\beta)} \frac{\partial}{\partial y}D_B^\alpha u\, \frac{\partial}{\partial y}D_B^\beta u\,y^k\,dz . \tag{6} \]

Hence, by virtue of inequality (4), we have

\[ (\mathcal Lu,u)_k\geq \mu\left( \int_{\Omega^+}\sum_{|\alpha|=m} C_m^{(\alpha)} |D_B^\alpha u|^2 y^k\,dz + \int_{\Omega^+}\sum_{|\bar\alpha|=m} C_m^{(\bar\alpha)} \left|\frac{\partial}{\partial y}D_B^\alpha u\right|^2 y^k\,dz \right). \tag{7} \]

Put

\[ r^2=\sum_{i=1}^{n}x_i^2+y^2, \]

and let \(\Delta_B\) denote the singular Beltrami operator

\[ \sum_{i=1}^{n}\frac{\partial^2}{\partial x_i^2}+B_y . \]

In [3] it was proved that the fundamental solution of the operator \(\Delta_B^m\) with singularity at the origin has the form

\[ u(x,y)= \begin{cases} C_1 r^{2m-\gamma}\ln r, & \text{if } 2m\geq \gamma \text{ and } \gamma \text{ is even},\\ C_2 r^{2m-\gamma}, & \text{in all other cases}, \end{cases} \tag{8} \]

where \(\gamma=n+k+1\).

To obtain a fundamental solution with singularity at an arbitrary point, one must apply to the function \(u(x,y)\) the generalized shift operator \(T_{s,t}^{x,y}\) (see [3]). Then the fundamental solution will have the form

\[ E(x,y;s,t)=C_3\int_{0}^{\pi} \left[\sum_{i=1}^{n}(x_i-s_i)^2+y^2+t^2-2yt\cos\alpha\right]^{(2m-\gamma)/2} \times \]

\[ \times \ln\left[\sum_{i=1}^{n}(x_i-s_i)^2+y^2+t^2-2yt\cos\alpha\right]^{1/2} \sin^{k-1}\alpha\,d\alpha, \]

if \(2m\geq \gamma\) and \(\gamma\) is an even number;

\[ E(x,y;s,t)=C_4\int_{0}^{\pi} \left[\sum_{i=1}^{n}(x_i-s_i)^2+y^2+t^2-2yt\cos\alpha\right]^{\frac{2m-\gamma}{2}} \sin^{k-1}\alpha\,d\alpha \tag{9} \]

in all other cases.

It is proved that for any \(u\in C_0^\infty(\Omega^+)\) the integral representation

\[ u(x,y)=\frac{1}{C_{m,n}^{(k)}}\int_{\Omega^+}\sum_{|\alpha|=m} C_m^{(\alpha)}D_B^\alpha u D_B^\alpha E t^k\,ds\,dt+ \]
\[ +\frac{1}{C_{m,n}^{(k)}}\int_{\Omega^+}\sum_{|\bar\alpha|=m} C_m^{(\bar\alpha)} \frac{\partial}{\partial t}D_B^\alpha u\, \frac{\partial}{\partial t}D_B^\alpha E t^k\,ds\,dt . \tag{10} \]

Here \(s=(s_1,\ldots,s_n)\), \(t\geqslant0\), and \((s,t)\) is a point of \((n+1)\)-dimensional Euclidean space.

Estimates of the fundamental solution and its derivatives, when the singularity of the fundamental solution lies inside the domain (see (4)), show that the kernels of the integral operators standing on the right in representation (10) are either bounded or have a weak singularity. If, however, the point \(z\) approaches the part of the boundary \(\Gamma^0\), then the singularity of the kernels increases. The presence of the weight \(t^k\) in the integral representation (10) compensates for the excessive singularity arising in this case. But then from representation (10) we find that

\[ \|u\|_{\mathscr L_{2,k}(\Omega^+)}^2 \leqslant C\sum_{|\alpha|=m} C_m^{(\alpha)} \int_{\Omega^+}|D_B^\alpha u|^2 t^k\,ds\,dt+ \]
\[ + C\sum_{|\bar\alpha|=m} C_m^{(\bar\alpha)} \int_{\Omega^+}\left|\frac{\partial}{\partial t}D_B^\alpha u\right|^2 t^k\,ds\,dt . \tag{11} \]

The latter inequality, together with inequality (7), shows that

\[ (\mathscr Lu,u)_k\geqslant \frac{\mu}{C}\|u\|_k^2 . \tag{12} \]

It is not hard to verify that the operator \(\mathscr L\) is a symmetric operator. Inequality (12) shows that it is also positive definite.

Let now \(\mathscr L\) be a formally self-adjoint operator of \(B\)-elliptic type such that its restriction \(\mathscr L_0(D(\mathscr L_0)=C_0^\infty(\Omega^+))\) is a positive definite operator, i.e. \((\mathscr L_0u,u)_k\geqslant \mu(u,u)_k\) for any \(u\in C_0^\infty(\Omega^+)\) with a positive constant \(\mu\) independent of \(u\). We now consider the self-adjoint positive definite Friedrichs extension of the operator \(\mathscr L_0\): \((\widetilde{\mathscr L})^*=\widetilde{\mathscr L}\supset \mathscr L_0\).

It is known that the range of the operator \(\widetilde{\mathscr L}\) coincides with the whole space (in our case with \(\mathscr L_{2,k}(\Omega^+)\)). Consequently, the equation

\[ \widetilde{\mathscr L}u=f \tag{13} \]

is uniquely solvable. As for functions from the domain of the operator \(\widetilde{\mathscr L}\), i.e. from the set \(D(\widetilde{\mathscr L})\), we shall say that they satisfy the generalized Dirichlet condition, without specifying here for the time being whether this should mean the fulfillment of boundary conditions on the entire boundary \(\partial\Omega^+\), or on a part of it. The solution of equation (13) is an element of the space \(\mathscr L_{2,k}(\Omega^+)\). Above we showed that the restriction \(\mathscr L_0\) of the operator \(\mathscr L\) to infinitely differentiable functions with compact supports lying in \(\Omega^+\) is positive definite.

Let \(H_{\mathscr L_0}\) be the completion of the set \(D(\mathscr L_0)=C_0^\infty(\Omega^+)\) in the metric \(\|u\|_{\mathscr L}^2=(\mathscr L_0u,u)_k\). Then the domain of definition of the Friedrichs extension \(\widetilde{\mathscr L}\) of the operator \(\mathscr L_0\) is described by the relation

\[ D(\widetilde{\mathscr L})=D(\mathscr L_0^*)\cap H_{\mathscr L_0}. \tag{14} \]

Apparently, the norm \(\|u\|_{\mathscr L}\) will be equivalent to the norm

\[ \|u\|_1=\int_{\Omega^+}\sum_{|\alpha|=m}|D_B^\alpha u|^2 y^k\,dz+ \int_{\Omega^+}\sum_{|\bar\alpha|=m}\left|\frac{\partial}{\partial y}D_B^\alpha u\right|^2 y^k\,dz . \tag{15} \]

We note that a description of the domain of definition of the fractional power \(\widetilde{\mathscr L}^{\alpha}\) of the extended operator \(\widetilde{\mathscr L}\) is given in terms of spaces that were introduced by the author earlier in [5].

Voronezh Technological
Institute

Received
19 XII 1966

REFERENCES

\(^{1}\) S. G. Mikhlin, The Problem of the Minimum of a Quadratic Functional, Moscow—Leningrad, 1952.
\(^{2}\) I. A. Kipriyanov, DAN, 158, No. 2 (1964).
\(^{3}\) I. A. Kipriyanov, V. I. Kononenko, DAN, 170, No. 2 (1966).
\(^{4}\) V. I. Kondrashov, DAN, 172, No. 2 (1967).
\(^{5}\) I. A. Kipriyanov, Fourier—Bessel Transforms and Embedding Theorems for Weighted Classes, Dissertation, V. A. Steklov Mathematical Institute, Academy of Sciences of the USSR, 1964.