MATHEMATICS

A. I. ACHIL’DIEV

THE FIRST AND SECOND BOUNDARY-VALUE PROBLEMS FOR ELLIPTIC EQUATIONS DEGENERATING AT A FINITE NUMBER OF INTERIOR POINTS

(Presented by Academician I. N. Vekua, March 21, 1963)

Equations of elliptic type that allow degeneration of order, or, in other terminology \((^{4,5})\), equations with singular coefficients, are the subject of the works \((^{1,2,4,5})\). In the works of L. G. Mikhailov \((^{4,5})\), equations were investigated that degenerate on certain manifolds \(M_k\) of dimension \(k\) \((0 \leq k \leq n-1;\ n\) is the number of independent variables in the equation). He studied the manifold of solutions, the first and second boundary-value problems, and solutions in the whole space. The apparatus of the investigation is a new class of special integral equations \((^5)\). In the case \(k=0\), he considered the equation

\[ r^\alpha \Delta u + r^{\alpha-1} \sum_{i=1}^{n} b_i(x)\,u_{x_i}(x) + c(x)\,u(x) = f(x) \]

for \(\alpha = 2,\ n > 2\) and for \(\alpha < 2,\ n = 2\); here \(r\) is the distance from the point \(x\) to the point of degeneration. Generalized solutions of elliptic equations degenerating at isolated points were considered by V. K. Zakharov \((^2)\), and those degenerating on manifolds \(M_k\) by V. P. Glushko \((^1)\). In the case of a second-order equation, point singularities of order \(\alpha < 2\) were studied. Singularities of higher order are admitted only for a very particular type of equations.

In the present work we study the generalized maximum principle, the first and second internal and external boundary-value problems, and solutions of the equation in the whole space. All these results are obtained for any equation for which the generalized maximum principle holds \((^7)\).

1. Internal boundary-value problems

Let a finite closed domain \(S\) with boundary \(\Gamma\) be given, and let the origin \(O\) be interior to \(S\). Denote by \(S_0\) the domain \(S-\Gamma-O\). Consider in \(S\) the elliptic equation

\[ Lu \equiv r^\mu \sum_{i,j=1}^{n} a_{ij}(x)\,u_{x_i x_j}(x) + r^\nu \sum_{i=1}^{n} b_i(x)\,u_{x_i}(x) + c(x)\,u(x) = f(x). \tag{1} \]

Here \(x=(x_1,x_2,\ldots,x_n)\) denotes a point of \(n\)-dimensional space, \(r\) is the distance from the point \(x\) to the origin, and \(\mu\) and \(\nu\) are constants. Suppose that the coefficients of equation (1) satisfy the condition of uniform ellipticity of the operator \(L\) in any closed domain \(S-\overline{B}(O,\varepsilon)\), where \(\overline{B}(O,\varepsilon)\) is the ball with center at the origin and arbitrarily small radius \(\varepsilon\). Assume the coefficients \(c(x)\leq 0\), \(a_{ij}(x)\), \(b_i(x)\) to be bounded in \(S\). The functions \(a_{ij}(x)\) are continuous at the point \(x=0\), and, without loss of generality, we may suppose that they have the form

\[ a_{ij}(0)= \begin{cases} 1, & \text{if } i=j=1,2,\ldots,m;\quad 0 \leq m \leq n,\\ 0, & \text{for the remaining } i,j=1,2,\ldots,n. \end{cases} \tag{2} \]

Denote by \(C^2(S,\beta)\) the class of functions \(u(x)\) twice continuously differentiable in \(S_0\) and satisfying the condition \(u(x)=o(r^{-\beta})\) as \(r\to 0\), where \(\beta\) is a positive number. We shall call the operator \(L\) \(\beta\)-normal if, in the ball \(\Ш(O,\delta)\) of sufficiently small radius \(\delta\), the inequality

\[ Lr^{-\beta}\leqslant 0. \tag{3} \]

holds.

Let us note some special cases in which condition (3) is fulfilled.

A. Let \(m>2\) and \(c(x)\leqslant 0\). Then inequality (3) will hold provided

\[ \sup_{0\leqslant r\leqslant \delta} \left(r^{\nu-\mu}\sum_{i=1}^{n} b_i(x)x_i\right) < m-2-\beta . \]

B. Let \(m\geqslant 2\), \(2\leqslant \mu\leqslant \nu+1\), and \(\sup c(x)<0\) in \(\Ш(O,\delta)\). Then, whatever the bounded coefficients \(c(x)\leqslant 0\) and \(b_i(x)\) may be, there exists \(\beta>0\) such that inequality (3) is satisfied.

Theorem 1. Let \(u(x)\) be a nonconstant function, continuous in \(S_0+\Gamma\), of class \(C^2(S,\beta)\), satisfying in \(S_0\) the inequality \(Lu\geqslant 0\), where the operator \(L\) is \(\beta\)-normal. Then, if \(c(x)\equiv 0\),

\[ u(x)\leqslant \max_{\Gamma} u(x) \]

in \(S\), and equality is possible only on \(\Gamma\); if \(c(x)\leqslant 0\), then

\[ u(x)\leqslant \max_{\Gamma} (u(x),0) \]

in \(S\), and equality is possible only on \(\Gamma\) and at the point \(x=0\).

From Theorem 1 one easily obtains analogues of Giraud’s theorem and of the uniqueness theorem for solutions of boundary-value problems in the class of functions \(C^2(S,\beta)\) (7). Suppose that on \(\Gamma\) one of the conditions is prescribed:

\[ u(x)\big|_{\Gamma}=\Psi(x); \tag{4} \]

\[ \frac{\partial u}{\partial \bar{\nu}}+b(x)u=\Psi(x), \tag{5} \]

where \(\bar{\nu}\) is the exterior conormal to \(\Gamma\) with respect to \(S\).

Theorem 2. Let the coefficients \(c(x)\leqslant 0\), \(a_{ij}(x)\), \(b_i(x)\), and \(f(x)\) be bounded in the closed domain \(S\) and of class \(A^{(1,\lambda)}\), i.e. \(\Gamma\) is a Lyapunov surface, and let them satisfy the conditions

\[ a_{ij}(x)\in C^{(1,\lambda)},\qquad b_i(x),c(x)\in C^{(0,\lambda)} \quad \text{in } S_0+\Gamma, \]

the function \(f(x)\) is continuous in \(S_0+\Gamma\) and belongs to the class \(C^{(0,\lambda)}\) in \(S_0\). Let the operator \(\Gamma\) be \(\beta\)-normal for some \(\beta>0\), and suppose there exists a sufficiently small ball \(\Ш(O,R)\) such that \(f(x)\equiv 0\) in it. Then the problem (1), (4) for any continuous function \(\Psi(x)\) (the problem (1), (5) for any continuous \(b(x)\geqslant 0\) and \(\Psi(x)\), with at least one of the functions \(c(x)\) and \(b(x)\) not identically zero) has, and moreover has uniquely, a solution \(u(x)\) bounded in \(S\).

Theorem 3. Let the conditions of Theorem 2 be satisfied, except for the equality \(f(x)\equiv 0\), in place of which we require boundedness of \(f(x)/c(x)\) in some ball \(\Ш(O,R)\). Then the assertion of Theorem 2 is valid.

If the function \(f(x)/c(x)\) has the unique limit

\[ (f/c)_0=\lim_{x\to 0} f(x)/c(x) \]

and the inequality \(Lr^{\beta_1}\leqslant 0\) is satisfied in \(\Ш(O,d)\) for some \(\beta_1>0\) and \(d>0\), then the solution \(u(x)\) will have the unique limit

\[ (u)_0=\lim_{x\to 0} u(x), \]

and moreover the equality

\[ (u)_0=\left(\frac{f}{c}\right)_0 . \tag{6} \]

Formula (6) indicates the singular character of equation (1), since in the regular case the value \((u)_0\) depends on the boundary condition.

Remark to Theorems 2 and 3. Theorems 2 and 3 remain valid if, in the condition of \(\beta\)-normality (3), instead of \(r^{-\beta}\) one takes an arbitrary function \(w(x)>0\) such that \(w(x)\to\infty\) as \(x\to 0\) and \(Lw\leqslant 0\) in \(\operatorname{Ш}(0,d)\). In the second condition of Theorem 3, instead of \(r^{\beta_1}\) one may take a function \(v(x)>0\) such that \(v(x)\to 0\) as \(x\to 0\) and \(Lv\leqslant 0\) in \(\operatorname{Ш}(0,d)\).

2. External boundary-value problems. Denote by \(S^{-}\) the domain which is the complement of the bounded closed domain \(S\) to the whole space \(E_n\). Consider in \(S^{-}\) the elliptic equation

\[ L'u \equiv r^{\mu'} \sum_{i,j=1}^{n} a'_{ij}(x) u_{x_i x_j}(x) + r^{\nu'} \sum_{i=1}^{n} b'_i(x) u_{x_i}(x) + c'(x)u(x)=f'(x), \tag{1'} \]

where \(\mu'\), \(\nu'\) are constants. Suppose that the operator \(L'\) is uniformly elliptic in any closed domain \(\operatorname{Ш}(0,A)-S\), where \(A>0\) is sufficiently large. Let the coefficients \(c'(x)\leqslant 0\), \(a'_{ij}(x)\), \(b'_i(x)\), and \(f'(x)\) be bounded in \(S^{-}\), and let the functions \(a'_{ij}(x)\) have, respectively, unique limits which are defined by the equalities

\[ (a'_{ij})_{\infty}=\lim_{x\to\infty} a'_{ij}(x)= \begin{cases} 1 & \text{for } i=j=1,2,\ldots,m';\quad 0\leqslant m'\leqslant n,\\ 0 & \text{for the remaining } i,j=1,2,\ldots,n. \end{cases} \tag{2'} \]

Denote by \(C^2(S^{-},\beta^{-})\) the class of functions \(u(x)\), twice continuously differentiable in \(S^{-}\), satisfying the condition \(u(x)=o(r^{\beta'})\) as \(r\to\infty\), where \(\beta'\) is a positive number. We shall call the operator \(L'\) \(\beta'\)-normal if, outside some ball \(\operatorname{Ш}(0,B)\) of sufficiently large radius \(B\), the inequality

\[ L'r^{\beta'}\leqslant 0 \tag{3'} \]

is satisfied.

For \(\beta'\)-normal operators, theorems analogous to Theorems 1–3 hold. We state two of them.

Theorem \(2'\). Suppose the coefficients \(c'(x)\leqslant 0\), \(a'_{ij}(x)\), \(b'_i(x)\), \(f'(x)\) are bounded in the closed domain \(S^{-}\), \(S\in A^{(1,\lambda)}\), and satisfy the conditions: \(a'_{ij}(x)\in C^{1,\lambda}\); \(b'_i(x)\), \(c'(x)\in C^{(0,\lambda)}\) in \(S^{-}+\Gamma\), \(f'(x)\) is continuous in \(S^{-}+\Gamma\), and \(u\) belongs to the class \(C^{(0,\lambda)}\) in \(S^{-}\). Suppose the operator \(L'\) is \(\beta'\)-normal for some \(\beta'>0\) and that there exists a ball \(\operatorname{Ш}(0,R_1)\) so large that \(f'(x)\equiv 0\) outside it. Then problem \((1')\), (4) for any continuous function \(\Psi(x)\) (problem \((1')\), (5) for any continuous \(b(x)\leqslant 0\) and \(\Psi(x)\), with, among the functions \(c'(x)\) and \(b(x)\), at least one not identically equal to zero) has, and moreover has uniquely, a solution \(u(x)\) bounded in \(S^{-}\).

Theorem \(3'\). Suppose the conditions of Theorem \(2'\) are fulfilled, except for the equality \(f'(x)\equiv 0\), in place of which we require the boundedness of \(f'(x)/c'(x)\) outside some ball \(\operatorname{Ш}(0,R_1)\). Then the assertion of Theorem \(2'\) is valid.

If the function \(f'(x)/c'(x)\) has a unique limit

\[ (f'/c')_{\infty}=\lim_{x\to\infty} f'(x)/c'(x) \]

and the inequality \(L'r^{-\beta_1}=0\) is satisfied outside \(\operatorname{Ш}(0,d)\) for some \(\beta_1>0\) and \(d>0\), then the solution \(u(x)\) will have a unique limit

\[ (u)_{\infty}=\lim_{x\to\infty} u(x), \]

and the equality

\[ (u)_{\infty}=\left(\frac{f'}{c'}\right)_{\infty} \tag{7'} \]

holds.

3. Solution of the equation in the whole space

Consider the elliptic equation

\[ \mathcal{L}u \equiv \sum_{i,j=1}^{n} A_{ij}(x) u_{x_i x_j}(x) + \sum_{i=1}^{n} B_i(x) u_{x_i}(x) + C(x)u(x)=F(x) \tag{8} \]

in the whole space \(E_n\). Denote by \(K(a,b)\) the domain \(a \le r(x) \le b\). Suppose that the operator \(\mathcal{L}\) is uniformly elliptic in every domain \(K(\varepsilon,\varepsilon^{-1})\), where \(\varepsilon\) is an arbitrarily small positive number. Let equation (8) reduce in some ball \(\Ш(O,\delta)\) to equation (1), and outside the ball \(\Ш(O,\delta^{-1})\) to \((1')\). Consequently, we shall consider an equation with two singular points \(O,\infty\) (cf. \((5)\)).

Theorem 4. Let the coefficients of the operator \(\mathcal{L}\) in any domain \(K(\varepsilon,\varepsilon^{-1})\) satisfy the conditions:
\[ A_{ij}(x)\in C^{(1,\lambda)};\qquad B_i(x),\ C(x)\le 0;\qquad F(x)\in C^{(0,\lambda)}. \]

Then the following results hold:

1) If there exists such a small \(R\) that \(F(x)\equiv 0\) in \(\Ш(O,R)\), the conditions of Theorem 2 are satisfied, and all the conditions of Theorem \(3'\) are satisfied outside \(\Ш(O,R^{-1})\), then there exists, and moreover uniquely, a bounded in \(E_n\) solution \(u(x)\) of equation (8), regular for \(0<r<\infty\), for which equality (7) holds.

2) If the first part of the conditions of Theorem 3 is satisfied in \(\Ш(O,R)\) and all the conditions of Theorem \(3'\) outside \(\Ш(O,R^{-1})\), then assertion 1) holds. If the second part of Theorem 3 is satisfied, then the solution \(u(x)\) satisfies equality (6).

We note that in Theorem 4 the points \(O\) and \(\infty\) may be interchanged, and we obtain a new result analogous to Theorem 4.

The proof of Theorems 2–4 is carried out according to the following scheme. By a method analogous to that used by M. V. Keldysh \({}^{(3)}\), the existence of a solution of the first boundary-value problem for the ball \(\Ш(O,R)\) or for the exterior of the ball \(\Ш(O,R^{-1})\) is proved. Uniqueness of the solution follows from the generalized maximum principle. For domains of general form the boundary-value problems are solved by a modified method of alternation \({}^{(8)}\), based on the results of \({}^{(7)}\).

I take this opportunity to express my gratitude to L. G. Mikhailov for supervising this work.

Department of Physics and Mathematics
Academy of Sciences of the Tajik SSR

Received
16 III 1963

CITED LITERATURE

\({}^{1}\) V. P. Glushko, DAN, 129, No. 3 (1959).
\({}^{2}\) V. K. Zakharov, DAN, 124, No. 4 (1959).
\({}^{3}\) M. V. Keldysh, DAN, 77, No. 2 (1951).
\({}^{4}\) L. G. Mikhailov, Izv. AN SSSR, Ser. Matem. 26, No. 2, 293 (1962).
\({}^{5}\) L. G. Mikhailov, A New Class of Singular Integral Equations and Its Application to Differential Equations with Singular Coefficients, Dushanbe, 1963.
\({}^{6}\) K. Miranda, Equations with Partial Derivatives of Elliptic Type, Moscow, 1957.
\({}^{7}\) A. I. Achil’diev, Izv. AN Tajik SSR, Division of Geological-Chemical Sciences, 2(8) (1962).
\({}^{8}\) A. I. Achil’diev, Dokl. AN Tajik SSR, 6, No. 1 (1963).