MATHEMATICS

M. V. LOMONOSOV

ON FUNDAMENTAL SOLUTIONS OF ELLIPTIC EQUATIONS

(Presented by Academician S. L. Sobolev on 26 III 1963)

Let the operator

\[ Lu \equiv \sum_{i,j=1}^{N} \frac{\partial}{\partial x_i} \left(a_{ij}(x)\frac{\partial u}{\partial x_j}\right) + \sum_{i=1}^{N} b_i(x)\frac{\partial u}{\partial x_i} + \bigl(c(x)+\lambda^2 g(x)\bigr)u \tag{1} \]

be given and elliptic in the whole \(N\)-dimensional space \(E_N\), i.e. \(a_{ij}(x)=a_{ji}(x)\),

\[ \sum_{i,j=1}^{N} a_{ij}(x)\xi_i\xi_j \geq a \sum_{i=1}^{N}\xi_i^2 \quad (a=\mathrm{const}>0) \]

for arbitrary real \(\xi_1,\ldots,\xi_N\) at any point \(x\in E_N\). We assume that: a) \(a_{ij}(x), b_i(x)\in C^{(1,\mu)}(E_N)\); \(c(x),g(x)\in C^{(0,\mu)}(E_N)\) \((\mu>0)\); b) \(g(x)\geq g_0>0\); c) outside a prescribed bounded domain \(T_0\), \(a_{ij}(x)\equiv \delta_{ij}\), \(b(x),c(x)\equiv 0\), \(g(x)\equiv 1\); d) the numerical parameter \(\lambda\) takes arbitrary complex values.

The note is devoted to two problems:

A. In the case \(b_i(x)\equiv 0\), for all \(\lambda\) in a certain neighborhood of the upper \(\lambda\)-half-plane, the so-called principal fundamental solution of the self-adjoint operator \(L\) is constructed—an analogue of the solution \(\exp(i\lambda|x-y|)/4\pi|x-y|\) of the three-dimensional equation \(\Delta u+\lambda^2u=0\).

B. In the case of real coefficients \((\lambda=0)\), the existence of a fundamental solution of a general non-self-adjoint operator \(L\) in any bounded domain is proved.

The principal fundamental solution is known for the three-dimensional Schrödinger equation \(\Delta u+(q(x)+\lambda^2)u=0\) (see, for example, \(\left({}^{1}\right)\)). With regard to problem B, the last result belongs to Yu. I. Lyubich \(\left({}^{2}\right)\), who proved the existence of a fundamental solution under more stringent conditions \(a_{ij}\in C^{(4)}\), \(b_i\in C^{(3)}\), \(c\in C^{(2)}\). We note that below no assumptions whatever are made on the sign of the coefficient \(c(x)\).

1. We shall need twice the following two basic propositions.

Theorem of E. M. Landis \(\left({}^{3}\right)\). Let \(\lambda^2\) be real; suppose, furthermore, that a function \(v(x)\) belongs to the class \(C^{(1)}\) in a bounded closed domain \(\Omega\) with smooth boundary \(\Sigma\), satisfies in \(\Omega\) the equation \(Lv=0\), and on the boundary

\[ v\big|_{\Sigma}=\frac{\partial v}{\partial n}\bigg|_{\Sigma}=0 \quad \left(\frac{\partial}{\partial n}\text{ is the derivative along the normal to }\Sigma\right). \]

Then \(v(x)\equiv 0\) in \(\Omega\).

In \(\left({}^{3}\right)\) this theorem is proved in a different formulation. However, the changes in the proof adapting the theorem to conditions a) are trivial; they were indicated to us by E. M. Landis.

Lemma 1. There exists a sufficiently small \(\rho>0\) such that in every ball \(\Omega_\rho\) of radius \(\rho\) the following assertion is true. Let a function \(\chi(x)\) of class \(C^{(0,\mu)}(\Omega_\rho+\Sigma_\rho)\) \((\Sigma_\rho\) is the boundary of \(\Omega_\rho)\) be equal to zero on \(\Sigma_\rho\) and satisfy the equality

\[ \int_{\Omega_\rho} \chi(x)Lu(x)\,dx=0 \]

for every finite in \(\Omega_\rho\) function \(u(x)\) of class \(C^{(2)}(\Omega_\rho)\). Then \(\chi(x)\equiv 0\) in \(\Omega_\rho\).

1. We turn to the construction of the principal fundamental solution.

Definition 1. An \(S\)-kernel is a function \(A(x,y)\) of a pair of points \(x,y\in E_N\), satisfying the following requirements:
a) \(A(x,y)\) depends analytically on the parameter \(\lambda\) for all \(\lambda\ne 0\);
b) \(A(x,y)\) is a Levi function (see \((4)\)) for \(x,y\in E_N\);
c) for any \(y\in E_N\), \(L_xA(x,y)=0\), when \(x\in E_N-T_0\), and

\[ L_{x'}A(x',y)-L_{x''}A(x'',y) = O\left(|x'-x''|^\mu \rho^{\nu-\mu-N}\right) \qquad (\mu\le \nu) \]

uniformly for \(x',x'',y\in E_N\) (\(\rho\) is the distance from the point \(y\) to the segment \(x'x''\));
d) for \(\arg\lambda=0\) the Sommerfeld condition is satisfied

\[ \frac{\partial A}{\partial r_{xy}}-i\lambda A=o\left(r_{xy}^{(1-N)/2}\right). \tag{2} \]

Definition 2. A principal fundamental solution of the operator \(L\) is an \(S\)-kernel \(G(x,y)\) for which \(L_xG(x,y)\equiv 0\).

It is easy to give an example showing that the class of \(S\)-kernels is nonempty. Following E. E. Levi \((5)\), represent \(G\) by means of an arbitrary \(S\)-kernel \(A(x,y)\) in the form

\[ G(x,y)=A(x,y)+\int_T A(x,t)R(t,y)\,dt, \]

where \(T\) is a ball containing the domain \(T_0\). The function \(R\) is found from an integral equation which defines it as the resolvent of the kernel \(L_xA(x,y)\). The function \(R\) exists for sufficiently large negative \(\lambda^2\) in modulus; therefore it exists everywhere except at points that are isolated poles.

It can be shown that if \(\lambda\) is a pole of \(G\), then there exist solutions of the equation

\[ \varphi(y)=\int_T \overline{L_xA(x,y)}\,\varphi(x)\,dx \tag{3} \]

which are the same for any \(S\)-kernel \(A\). Hence, with the aid of Lemma 1, we find the following properties of these solutions:
a) \(\varphi(x)\in C^{(2)}(E_N)\);
b) \(L\varphi(x)\equiv 0,\ x\in E_N\);
c) \(\partial\varphi/\partial r-i\lambda\varphi=O\left(r^{-(1+N)/2}\right)\) \((r=|x|)\).

For \(\operatorname{Im}\lambda\ge 0\), it follows from b), c) that \(\varphi\in L_2(E_N)\). Since the operator \(L\) is self-adjoint, the functions \(\varphi\) can be nonzero only when \(\operatorname{Im}\lambda^2=0\).

If \(\lambda^2\le -\max c(x)/\min g(x)\), then \(\varphi\equiv 0\) by the maximum principle. If \(\lambda^2>0\), then, as A. Ya. Povzner \((6)\) showed, from \(\varphi\in L_2(E_N)\) and the equality \(\Delta\varphi+\lambda^2\varphi=0\), which holds outside \(T_0\), it follows that \(\varphi(x)=0\) outside \(T_0\). By Landis’ theorem, \(\varphi(x)\equiv 0\).

Thus, we have shown that the principal fundamental solution exists in the entire \(\lambda\)-plane, with the exception of the interval \(\left(+\operatorname{Re}\sqrt{\max c(x)/\min g(x)},0\right)\) of the imaginary axis, where there may be at most a finite number of poles of the function \(G(x,y;\lambda)\).

1. For \(N=2\) the operator \(L\) (with \(g(x)\equiv 1\)) is reduced to the form

\[ Lu=\Delta u+a_1(x)\frac{\partial u}{\partial x_1} +a_2(x)\frac{\partial u}{\partial x_2} +\left(c(x)+\lambda^2\right)u. \tag{1'} \]

By a method similar to that used by L. D. Faddeev in \((1)\) for estimates, for large \(|\lambda|\), of the resolvent kernel of the three-dimensional Schrödinger operator, we found for the operator \((1')\) \((|\lambda|>\Lambda)\):

\[ G(x,y)=-\frac{i}{\sqrt{2\pi}}H_0^1(r_{xy}\lambda)\left(1+\lambda^{-1/2}g(x,y;\lambda)\right), \]

\[ \frac{\partial}{\partial x_i}G(x,y) = -\frac{i\lambda(x_i-y_i)}{\sqrt{2\pi}\,r_{xy}} H_1^1(r_{xy}\lambda) \left(1+\lambda^{-1/2}g'(x,y;\lambda)\right), \tag{4} \]

where \(g,g'\) are uniformly bounded for all \(x,y,\lambda\) and are zero for \(x=y\). Here we require that \(a_i\in C^{(2)}\), \(c\in C^{(1)}\).

1. We turn to the question of the existence of a fundamental solution in a bounded domain \(\Omega\) (\(\lambda = 0\)). As is known ((4), pp. 221–222), the question reduces to the existence in \(\Omega\) of a solution of the equation \(Lu=\varphi\) for an arbitrary function \(\varphi \in C^{(0,\mu)}\). Adding the boundary condition \(u|_{\Sigma}=\zeta\), we reduce the resulting Dirichlet problem to an equivalent system of integral equations. In doing so one uses the principal fundamental solution of the equation

\[ \sum_{i,j=1}^{N}\frac{\partial}{\partial x_i} \left(a_{ij}(x)\frac{\partial u}{\partial x_j}\right)-u=0. \]

The boundary-value problem is solvable when conditions of the form

\[ \int_{\Omega} v_k\varphi\,dx+\int_{\Sigma}\omega_k\zeta\,d\sigma=0 \qquad (k=1,\ldots,q) \tag{5} \]

are satisfied, or

\[ \int_{\Sigma}\omega_k\zeta\,d\sigma=\alpha_k. \tag{5'} \]

We prove, with the aid of Lemma 1 and Landis’ theorem, that the functions \(\omega_k\) are linearly independent; in this case the requirements (5′) are easily satisfied. Choosing \(\zeta\) in an appropriate way, we solve the boundary-value problem, and hence also the equation \(Lu=\varphi\). This proves the existence of a fundamental solution.

In conclusion I express my warm gratitude to my scientific adviser V. A. Il’in and to I. A. Shishmarev for formulating the problem and for their attention to the work; I am also grateful to E. M. Landis for useful consultations.

REFERENCES CITED

\(^{1}\) L. D. Faddeev, Vestn. LGU, No. 7, issue 2, 164 (1957).
\(^{2}\) Yu. I. Lyubich, Matem. sborn., 57 (99), No. 1, 45 (1962).
\(^{3}\) E. M. Landis, DAN, 107, No. 5, 640 (1956).
\(^{4}\) C. Miranda, Equations with Partial Derivatives of Elliptic Type, IL, 1957.
\(^{5}\) E. E. Levi, UMN, issue 8, 249 (1940).
\(^{6}\) A. Ya. Povzner, Matem. sborn., 32 (74), 109 (1953).