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Mathematics
L. G. MIKHAILOV
ELLIPTIC EQUATIONS WITH SINGULAR COEFFICIENTS
(Presented by Academician I. N. Vekua, March 4, 1961)
1. Notation. \(E\) is \(n\)-dimensional Euclidean space; \(x=(x_1,x_2,\ldots,x_n)\); \(f(x)=f(x_1,x_2,\ldots,x_n)\), \(dx=dx_1,dx_2,\ldots,dx_n\), \(r(x,t)=\left(\sum_{i=1}^n (x_i-t_i)^2\right)^{1/2}\), \(\rho(x)=r(x,0)\); \(S(D)\) is the class of bounded measurable functions with norm \(\|f\|_{S(D)}=\sup_{x\in D}|f(x)|\), \(S(\beta,D)\) is the class of functions representable in the form \(f(x)=\rho^{-\beta}(x)f_0(x)\), where \(f_0(x)\in S(D)\). If the norm \(\|f\|_{S(\beta,D)}=\|f_0\|_{S(D)}\) is introduced, then \(S(\beta,D)\) will be isometric to \(S(D)\).
Let there be \(p\) distinct points \(c_1,c_2,\ldots,c_p\) inside and on the boundary of the domain \(D\), and let \(\rho_i(x)=r(x,c_i)\). By \(\Pi(x)\) we denote the distance from the point \(x\) to the set \(c_1,c_2,\ldots,c_p\), i.e.
\[
\Pi(x)=\min\{\rho_1(x),\rho_2(x),\ldots,\rho_p(x)\};
\]
\(S(\beta,\Pi,D)\) is the class of functions of the form \(f(x)=\Pi^{-\beta}(x)f_0(x)\), where \(f_0(x)\in S(D)\), with norm \(\|f\|_{S(\beta,\Pi,D)}=\|f_0\|_{S(D)}\); \(K\) is a cone with vertex at the point \(0\) and infinite in one direction; \(q(K,\alpha,\beta)=\int_K \rho^{-\beta}(y)r^{n-\alpha}(y,I)\,dy\), where \(0<\alpha<n\), \(\alpha<\beta<n\); \(I\) is a point of the unit sphere, \(I\in K\). If \(K=E\), then we shall write \(q(E,\alpha,\beta)=q(\alpha,\beta)\).
2. A new class of integral equations. Let \(D\) be an arbitrary bounded or unbounded domain. We first consider the simplest operator
\[
T\varphi=\rho^{-\alpha}(x)\int_D r^{\alpha-n}(x,t)\varphi(t)\,dt,
\tag{1}
\]
where \(0<\alpha<n\) and \(0\) is an interior point of the domain \(D\).
Theorem 1. Formula (1) defines a linear (not completely continuous) operator in the Banach spaces \(S(\beta,D)\), \(\alpha<\beta<n\), and
\[
\|T\|_{S(\beta,D)}=q(\alpha,\beta).
\]
Generalization. Let there be several singular points \(c_1,c_2,\ldots,c_p\), which may lie both inside and on the boundary of the domain. If the domain is unbounded, then the infinitely distant point is also included among the singular points; it, in turn, may be interior or boundary and is denoted by \(c_{p+1}\).
Then the formula
\[
T_{\Pi}\varphi=\Pi^{-\alpha}(x)\int_D r^{\alpha-n}(x,t)\varphi(t)\,dt
\]
defines a linear operator in \(S(\beta,\Pi,D)\), \(\alpha<\beta<n\), and
\[
q(K_i,\alpha,\beta)\leq \|T_{\Pi}\|_{S(\beta,\Pi,D)}\leq Bq(V_i,\alpha,\beta),\qquad i=1,2,\ldots,p+1,
\]
where \(K_i, V_i\) are the cones of tangency and visibility of the domain \(D\) from the point \(c_i\); the constant \(B\) does not depend on the domain.
If the point \(c_i\) is interior, we put \(K_i=V_i=E\).
Theorem 2. If \(K(x,t)\in S(D\times D)\), then the formula
\[ K\varphi=\Pi^{-\alpha}(x)\int_D r^{\alpha-n}(x,t)K(x,t)\varphi(t)\,dt \]
defines a linear operator in \(S(\beta,\Pi,D)\), \(\alpha<\beta<n\), and
\[ \mu_i q(K_i,\alpha,\beta)\le \|K\|_{S(\beta,\Pi,D)}\le Mq(V_i,\alpha,\beta), \tag{2} \]
where
\[ M=\sup_{x,t\in D}|K(x,t)|,\qquad \mu_i=\lim_{x\to c_i}\lim_{t\to c_i}|K(x,t)|,\qquad i=1,2,\ldots,p+1. \]
Theorem 3. If \(K(x,t)\) is continuous for \(x\ne t\) and \(\overline{\lim}|K(x,t)|=0\) at all singular points, then the operator \(K\varphi\) is completely continuous in \(S(\beta,\Pi,D)\), \(\alpha<\beta<n\), for every \(\beta\).
Theorems 1–3 make it possible to give certain criteria for the solvability of the equation (cf. \((^{8,6})\)).
\[ \varphi(x)+\int_D \frac{K(x,t)}{\Pi^\alpha(x)\,r^{\,n-\alpha}(x,t)}\varphi(t)\,dt=f(x), \tag{3} \]
where \(f(x)\in S(\beta,\Pi,D)\), \(\beta<n\). For sufficiently small \(K(x,t)\) the solution exists and is unique, but this cannot be achieved by making the domain \(D\) small. Under the conditions of Theorem 3 the Fredholm theorems are valid. However, from Theorems 1 and 3 there follows a more general assertion.
Theorem 4. Let \(K(x,t)\) be continuous for \(x\ne t\) and at all singular points. If
\[ \sum_{i=1}^{p+1}|K(c_i,c_i)|\,q(V_i,\alpha,\beta)<1, \tag{4} \]
then the Fredholm theorems are valid for equation (3).
3. The manifold of solutions. Consider the elliptic equation of second order
\[ Lu\equiv \Delta u+\sum_{i=1}^n \frac{a_i(x)}{\rho(x)}u'_{x_i}+\frac{b(x)}{\rho^2(x)}u=0. \tag{5} \]
The point \(x=0\) (as well as \(x=\infty\)) is called its regular singular point if the functions \(a_i(x)\), \(i=1,2,\ldots,n\), \(b(x)\) are bounded at this point, and a weak singularity if, as \(\rho\to 0\), \(a_i(x),b(x)=O(\rho^\varepsilon)\), \(\varepsilon>0\) (as \(\rho\to\infty\), \(O(\rho^{-\varepsilon})\)).
To study the manifold of solutions we put
\[ u(x)=-\int_D \psi(r)\varphi(t)\,dt+h(x), \tag{6} \]
where \(\psi(r)\) is the fundamental solution of Laplace’s equation and \(h(x)\) is an arbitrary function. Substituting (6) into (5), we obtain for \(\varphi(x)\) an integral equation of the form (3), where \(f(x)\) is arbitrary, \(K=K_1+K_2\), and *
\[ K_1(x,t)=-\frac{1}{\omega_n}\, \frac{\sum_{i=1}^n r'_{x_i}a_i(x)}{\rho(x)\,r^{\,n-1}(x,t)}, \qquad K_2(x,t)=\frac{b(x)}{\omega_n(n-2)}\, \frac{1}{\rho^2(x)\,r^{\,n-2}(x,t)}. \tag{7} \]
\[ \text{* Henceforth the case } n=2,\ b(0)\ne 0 \text{ is everywhere excluded.} \]
For a local investigation of a regular singular point, take as \(D\) a sufficiently small neighborhood of it and pass from \(x=0\) to \(x=\infty\). Then Theorem 2 immediately implies:
Theorem 5. If \(\lim_{x\to 0} a_i(x)\), \(i=1,\ldots,n\), and \(\lim_{x\to 0} b(x)\) are sufficiently small, then there always exist solutions continuous at the singular point.
Consider equations with two regular singular points. By translations and inversions we can always move them to the points \(0,\infty\), so that the equation takes the form (5), where \(a_i(x), b(x)\) are defined and bounded in the whole space \(E\). In this case, and also for the case \(p>2\) of singular points, we put \(D=E\) in (6). From Theorems 2 and 4 we obtain a number of results, of which we shall present one that has some connection with Schrödinger’s problem known in quantum mechanics:
Theorem 6. If, of two regular singular points, one is a weak singularity, and at the other \(a_i(x), b(x)\) are continuous and small, then equation (5) has a continuum of solutions continuous in the whole space, including both singular points.
4. The first boundary-value problem. Let \(D\) be a bounded domain whose boundary consists of a finite number of Lyapunov surfaces. If \(G(x,t)\) is the Green’s function of the Laplace operator, then we shall also require that (cf. \((1^2)\))
\[ |G'_{x_i}(x,t)| \le H r^{1-n}, \qquad i=1,\ldots,n. \tag{8} \]
The formulation of the first boundary-value problem is: for the given values \(u(x)=\psi(x)\) on the boundary of the domain, find solutions of the equation \(Lu=\rho^{-2}(x)g(x)\), \(g(x)\in S(D)\), regular everywhere in \(D\) except at the point \(x=0\), and admitting at this point a singularity whose order is restricted by the condition \(\Delta u\in S(\beta,D)\), \(\beta<n\).
As usual, reducing the boundary values to zero and setting
\[ u(x)=-\int_D G(x,t)\varphi(t)\,dt, \]
we obtain for \(\varphi(x)\) the integral equation (3) with kernels of type (7) (see \((9)\)).
Theorem 7. Let \(b(x)=O(\rho^\varepsilon)\), \(\varepsilon>0\), \(n\ge 2\). If the \(a_i(x)\), \(i=1,\ldots,n\), and either \(b(x)\) or the domain \(D\) are sufficiently small, then the boundary-value problem has a unique solution, continuous at the singular point.
If the number \(\beta\) is fixed by the conditions \(2<\beta<n\),
\[ Hq(1,\beta)\sum_{i=1}^{n}\sup_{x\in D}|a_i(x)| + \frac{q(2,\beta)}{\omega_n(n-2)}\sup_{x\in D}|b(x)|<1, \]
then the boundary-value problem has a unique solution \(u(x)\in S(\beta-2,D)\).
Theorem 8. If \(a_i(x)\), \(i=1,2,\ldots,n\), and \(b(x)\) are continuous at the singular point and
\[ Hq(1,\beta)\sum_{i=1}^{n}|a_i(0)| + \frac{q(2,\beta)}{\omega_n(n-2)}|b(0)|<1 \]
(and for \(n=2\), moreover, \(b(x)=O(\rho^\varepsilon)\), \(\varepsilon>0\)), then the Fredholm theorems hold for the first boundary-value problem.
Theorem 9. If \(b(x)\le 0\) in the domain \(D\), then under the conditions of Theorem 8 the boundary-value problem is solvable uniquely and the solution is bounded at the singular point.
The proof of this theorem is based on the generalized maximum principle \((10)\).
Consider the equation
\[ \Delta u+\sum_{1}^{n} A_i(x)u'_{x_i}+B(x)u=F(x) \tag{9} \]
with several regular singular points \(c_1, c_2,\ldots,c_p\) and \(c_{p+1}=\infty\), which may lie both inside and on the boundary of the domain, being a point of smoothness or a conical point of the surface. In other words, let
\(A_i(x)\in S(1,\Pi,D)\), \(i=1,\ldots,n\); \(B(x), F(x)\in S(\alpha,\Pi,D)\), \(\alpha\leq 2\), where
\(\Pi(x)=\min_{1\leq i\leq p}\{r(x,c_i)\}\). Suppose that the Green’s function of the Laplace equation still satisfies conditions (8).
Then theorems of the type 7, 8, 9 hold. Let us note only that if the singular points lie on the boundary of the domain, then in theorem 9 one cannot assert uniqueness and boundedness of the solution. The results also extend to equations of noncanonical form.
The present work is a further development of the author’s papers \((^{6,7})\), in which the generalized Cauchy—Riemann system and the integral equations associated with it were studied. Singularities below the first order and somewhat more general classes of coefficients were considered in \((^{4,5})\). If (5) is multiplied by \(\rho^2(x)\), then one may speak of degeneration of the order of the equation. Degenerations of various types have been the subject of many works; see, for example, \((^{11,12})\). However, all these works do not cover equations (5).
Department of Physics and Mathematics
Academy of Sciences of the Tajik SSR
Received
25 II 1961
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