UDC 517.94

MATHEMATICS

I. A. KIPRIYANOV, M. I. KLYUCHANTSEV

ON GREEN’S FUNCTIONS FOR PROBLEMS WITH THE BESSEL DIFFERENTIAL OPERATOR

(Presented by Academician L. S. Pontryagin, 22 IV 1968)

Let \(E_{n+2}\) denote the \((n+2)\)-dimensional Euclidean space of points
\(z=(x,y,t)\), where \(x=(x_1,\ldots,x_n)\), \(x_{n+1}=y\), \(x_{n+2}=t\). Consider in this space the domain \(y>0,\ t>0\). In this domain consider the following boundary-value problem. Find a solution of the equation

\[ \mathscr{L}(D_x,B_y,D_t) = \sum_{|k|=2m} a_k D_x^{k'} B_y^{k_{n+1}} D_t^{k_{n+2}} u=0, \tag{1} \]

where \(|k|=|k'|+2k_{n+1}+k_{n+2}\),

\[ D_x^{k'} = \left(\frac{1}{i}\frac{\partial}{\partial x_1}\right)^{k_1} \cdots \left(\frac{1}{i}\frac{\partial}{\partial x_n}\right)^{k_n}, \qquad D_t^{k_{n+2}} = \left(\frac{1}{i}\frac{\partial}{\partial t}\right)^{k_{n+2}}, \]

\[ B_y^{k_{n+2}} = \left[ \left(\frac{1}{i}\right)^2 \left( \frac{\partial^2}{\partial y^2} + \frac{2\nu+1}{y}\frac{\partial}{\partial y} \right) \right]^{k_{n+1}} \]

in the domain \(y>0,\ t>0\), satisfying the following boundary conditions:

\[ H_\mu(D_x,B_y,D_t)u\big|_{t=0} = g_\mu(x,y) \qquad (\mu=1,2,\ldots,m). \tag{2} \]

It is assumed that the coefficients \(a_k\) of the operator \(\mathscr{L}\) are constant and that the boundary operators \(H_\mu\) are homogeneous with constant coefficients. In addition, it is assumed that the operator \(\mathscr{L}\) is a \(B\)-elliptic operator \((^2)\) and that the operators \(\mathscr{L}\) and \(H_\mu\) satisfy the Lopatinskii condition (see, for example, \((^1)\)). We shall seek classical solutions of this problem by means of the Fourier–Bessel transform, i.e. we shall seek solutions in the form

\[ u(x,y,t) = \int_{-\infty}^{\infty}\int_{0}^{\infty} v(\xi,\sigma,t)e^{ix\xi}j_\nu(\sigma y)\sigma^{2\nu+1}\,d\xi\,d\sigma . \tag{3} \]

Then for \(v\) one obtains the boundary-value problem

\[ \mathscr{L}(\xi,\sigma,D_t)v=0, \qquad H_\mu(\xi,\sigma,D_t)v\big|_{t=0}=g_\mu(\xi,\sigma) \tag{4} \]

on the half-line \(t>0\) for an ordinary differential equation with constant coefficients, for fixed \(\xi,\sigma\). In order to single out the solutions of problem (4) that decrease exponentially as \(t\to\infty\), factorization is carried out, as usual. Since the Lopatinskii condition is satisfied, problem (4) is solvable, and its solution is written in the form

\[ v(\xi,\sigma,t) = \sum_{k=1}^{m} \tilde v_k(\xi,\sigma,t)\,\hat g_k(\xi,\sigma), \tag{5} \]

where

\[ v_k(\xi,\sigma,t) = \frac{1}{2\pi i} \int_C \frac{N_k(\xi,\sigma,\tau)}{M^+(\xi,\sigma,\tau)} e^{i\tau t}\,d\tau . \tag{6} \]

is a Poisson basis; \(C\) is a contour in the upper half-plane of the complex plane, enclosing all zeros of the polynomial \(M^+\) \((\mathscr L=a_0M^+M^-)\). If in the formula for the solution of our problem we pass from the Fourier–Bessel images to the functions themselves, we obtain

\[ u(x,y,t)=\sum_{j=1}^{m}\int_{-\infty}^{\infty}\int_{0}^{\infty} T_{x,y}^{\alpha,\gamma}K_j(x,y,t)\,g_j(\alpha,\gamma)\,\gamma^{2\nu+1}\,d\alpha d\gamma, \tag{7} \]

where the function \(K_j(x,y,t)\), called the Poisson kernel, is defined by the relation

\[ K_j(x,y,t)=C_\nu\int_{-\infty}^{\infty}\int_{0}^{\infty} \widetilde v_j(\xi,\sigma,t)e^{ix\xi}j_\nu(\sigma y)\sigma^{2\nu+1}\,d\xi d\sigma . \tag{8} \]

Here and below the constants \(C_\nu\) have a quite definite value and their particular form is important. In view of their cumbersomeness we do not write them out.

Taking into account the homogeneity of \(\widetilde v_k\) and introducing polar coordinates in the appropriate way, we obtain the following explicit formulas for the Poisson kernels. If \(n+1+2\nu+1>m_j\), where \(m_j\) is the order of the operator \(H_j\), then

\[ K_j(x,y,t)= \]

\[ =C_j\int_{0}^{\pi}\sin^{2\nu}\varphi\,d\varphi \int_{C_{R,\delta}}d\beta \int_{|\eta|=1} \frac{\dot{\sigma}^{\,2\nu+1}} {(x\dot{\xi}+y\dot{\sigma}\cos\varphi+t\beta)^{\,n+1+2\nu+1-m_j}} \frac{N_j(\eta,\beta)}{M^+(\eta,\beta)}\,d\omega_\eta, \]

\[ \eta=(\dot{\xi},\dot{\sigma}). \tag{9} \]

If, however, \(m_j\ge n+1+2\nu+1\), then the corresponding integral is divergent. Applying the well-known regularization device \((^1)\), in this case we obtain for the Poisson kernels the following two formulas. If \(2\nu+1\) is an integer, then

\[ K_j(x,y,t)= \]

\[ =C_j\int_{0}^{\pi}\sin^{2\nu}\varphi\,d\varphi \int_{C_{R,\delta}}d\beta \int_{|\eta|=1} (x\dot{\xi}+y\dot{\sigma}+t\beta)^{m_j-n-1-2\nu-1}\times \]

\[ \times \ln\frac{x\dot{\xi}+y\dot{\sigma}+t\beta}{i}\, \frac{N_j}{M^+}\dot{\sigma}^{\,2\nu+1}\,d\omega_\eta . \tag{10} \]

If, however, \(2\nu+1\) is a fractional number, then the formula for the Poisson kernel has the form

\[ K_j(x,y,t)= \tag{11} \]

\[ =C_j\int_{0}^{\pi}\sin^{2\nu}\varphi\,d\varphi \int_{C_{R,\delta}}d\beta \int_{|\eta|=1} (x\dot{\xi}+y\dot{\sigma}\cos\varphi+t\beta)^{m_j-n-1-2\nu-1} \frac{N_j}{M^+}\dot{\sigma}^{\,2\nu+1}\,d\omega_\eta . \]

The Poisson kernels obtained above can be differentiated, for \(t>0\), arbitrarily many times. When boundary operators are applied, singularities may arise. We need to have kernels such that both they themselves and all their derivatives up to a certain order are continuous in the closed half-space. This is achieved, as is known \((^1)\), by constructing the so-called adjoint kernels.

Taking into account the formal self-adjointness of the generalized translation operator, with the aid of the adjoint kernels the solution is written in the form

\[ u(x,y,t)= \]

\[ =C\sum_{j=1}^{m}\int_{-\infty}^{\infty}\int_{0}^{\infty} \Delta_{B}^{(\gamma+q)/2}K_{j,q}(s,\tau,t)\, T_{x,y}^{s,\tau}g_j(x,y)\,\tau^{2\nu+1}\,ds\,d\tau . \tag{12} \]

Integrating by parts in formula (12), we obtain, taking into account the commutativity of \(\Delta_B^r\) and \(T_{x,y}^{s,t}\),

\[ u(x,y,t)= \]

\[ = C\sum_{j=1}^{m}\int_{-\infty}^{\infty}\int_{0}^{\infty} K_{j,q}(s,\tau,t)\,T_{x,y}^{s,t}\Delta_B^{(\gamma+q)/2}g_j(x,y)\,\tau^{2\nu+1}\,ds\,d\tau, \tag{13} \]

where the operator \(\Delta_B=\sum \partial^2/\partial x_i^2+B_y\).

If \(2\nu+1\) is an integer (and consequently \(q\) is an integer), then the associated kernels have the form

\[ K_{j,q}(x,y,t)=\int_{0}^{\pi}\sin^{2\nu}\varphi \times \]

\[ \times\, d\varphi \int_{C_{R,\delta}} d\beta \int_{|\eta|=1} (x\dot{\xi}+y\dot{\sigma}\cos\varphi+t\beta)^{m_j+q}\times \]

\[ \times\left(\ln\frac{(x\dot{\xi}+y\dot{\sigma}\cos\varphi+t\beta)}{i}+C\right) \frac{N_j}{M^+}\,\dot{\sigma}^{\,2\nu+1}\,d\omega_\eta. \tag{14} \]

If, however, \(2\nu+1\) is a fractional number (hence \(q\) is fractional), then the associated kernels have the form

\[ K_{j,q}(x,y,t)= \tag{15} \]

\[ = C\int_{0}^{\pi}\sin^{2\nu}\varphi\,d\varphi \int_{C_{R,\delta}} d\beta \int_{|\eta|=1} (x\dot{\xi}+y\dot{\sigma}\cos\varphi+t\beta)^{m_j+q} \frac{N_j(\eta,\beta)}{M^+(\eta,\beta)}\,\dot{\sigma}^{\,2\nu+1}\,d\omega_\eta. \]

Let \(g_\mu\) be infinitely differentiable finite functions in the half-space \(E_{n+1}\). Using the results of V. I. Kononenko \({}^{(3)}\) on the expansion of finite functions into weighted plane waves, it is not difficult to show that formula (13) gives a solution of problem (1)—(2). Each of the kernels \(K_{j,q}\) is analytic at all points of the half-space \(t\ge 0\), with the exception of the origin. As for the origin, it is a pole of a certain order.

Theorem. The kernels \(K_{j,q}(x,y,t)\), for \(t\ge 0\) and \((x,y)\ne 0\), are infinitely differentiable functions, and for their derivatives the following estimates hold:

1. If \(2\nu+1\) is an integer and \(0\le s+2r\le m_j+q\), then the inequality

\[ \left|D_x^s B_y^r K_{j,q}\right| \le C\left(|x|^2+y^2+t^2\right)^{(m_j+q-s-2r)/2}\times \]

\[ \times\left(1+\ln\left(|x|^2+y^2+t^2\right)^{1/2}\right) \tag{16} \]

holds.

If \(s+2r\ge m_j+q+1\), then \(D_x^s B_y^r K_{j,q}\) is homogeneous of degree \(m_j+q-s-2r\), and the logarithmic term in (16) may be omitted.

1. If \(2\nu+1\) is a fractional number (in this case \(q\) is also fractional), then

\[ \left|D_x^s B_y^r K_{j,q}\right| \le C\left(|x|^2+y^2+t^2\right)^{(m_j+q-s-2r)/2}. \tag{17} \]

Analogous assertions are also valid for the Poisson kernels \(K_j\).

Voronezh State University

Received
2 IV 1968

REFERENCES

\({}^{1}\) S. Agmon, A. Douglis, L. Nirenberg, Estimates of solutions of elliptic equations near the boundary, Moscow, 1962.
\({}^{2}\) I. A. Kipriyanov, DAN, 158, No. 2 (1964).
\({}^{3}\) V. I. Kononenko, DAN, 172, No. 2 (1967).