A. G. TETEREV

CHAPLYGIN’S THEOREM FOR ELLIPTIC EQUATIONS

(Presented by Academician S. L. Sobolev on 23 IV 1960)

In the present article we shall consider some basic properties of Green’s function, Chaplygin’s theorem, the limits of its applicability, and comparison theorems. In what follows, everywhere, unless the contrary is stated, the notation and definitions from (¹) are used.

We shall need the following.

Remark. The results of the author’s work (³) are based on general properties of integral equations and therefore, without substantial changes, carry over to the Ries—Schauder equations in the space \(\mathscr{L}_2\) (²). Let us also recall that an integral operator with a weak singularity is completely continuous in \(\mathscr{L}_2\) (²).

Let in the domain \(T\) a linear differential operator of second order of elliptic type be given:

\[ \mathfrak{M} = \sum_{i,k=1}^{m} a_{ik}\frac{\partial^2}{\partial x_i \partial x_k} + \sum_{i=1}^{m} b_i \frac{\partial}{\partial x_i} + c. \]

We shall assume that the operator \(\mathfrak{M}\) is self-adjoint. Further, with respect to the domain \(T\) and the operator \(\mathfrak{M}\) we shall consider as satisfied one or another set of conditions ensuring the existence of the Green’s function \(F = F(x,y)\) of the Dirichlet problem

\[ \mathfrak{M}u = -f,\qquad f \geqslant 0, \tag{1} \]

\[ u|_{\mathfrak{S}T}=\varphi,\qquad \varphi \geqslant 0. \tag{2} \]

If \(c \leqslant 0\), then regular solutions of equation (1) possess the property of a maximum (minimum) ((¹), Theorems 3, I and 3, II).

Definition. The coefficient \(c(x)\) will be called critical for a given domain \(T\) and, conversely, the domain \(T\) will be called critical with respect to \(c(x)\), if for these \(c\) and \(T\) the uniqueness theorem 21, II (¹) is violated. Critical values \(c(x)\) will be denoted by \(\lambda=\lambda(x)\), and domains by \(T=T(c)\).

We shall call \(\lambda_1\) the least or first critical value, if there exists no \(\lambda\) such that \(\lambda \leqslant \lambda_1,\ \lambda \ne \lambda_1\) in the domain \(T\). \(T_1(c)\) is defined analogously. The totality of first critical values forms a set of functions \(\Lambda_1\), whose elements have the property that all of them intersect one another.

We shall say that \(c<\lambda_1\) if there are no critical values \(\lambda \leqslant c\). Denote the set of all critical values of \(c\) for the given domain \(T\) by \(\Lambda\). \(\lambda_k\) is defined as the least among the critical values

\[ \Lambda-\sum_{i=1}^{k-1}\Lambda_i. \]

Here “least” is understood in the sense of the definition of \(\lambda_1\).

The meaning of the notations and inequalities \(T_k(c),\ T(c),\ \lambda_{k-1}<c<\lambda_k,\ T_{k-1}<T<T_k\) is clear from the preceding definitions.

Theorem 1. If \(c<\lambda_1\), then the Green’s function of the Dirichlet problem is positive in the domain \(T\): \(F\geqslant 0\).

Proof. Consider two cases.

\(1^\circ.\ c\leqslant 0.\) It is known [(1)] that for \(c\leqslant 0\) the Green’s function \(F\) exists. Further,

\[ \begin{aligned} \mathfrak M_x F &= 0 &&\text{for} && x\in T-\mathfrak F T-y;\\ F &= 0 &&\text{for} && x\in \mathfrak F T,\quad y\in T-\mathfrak F T; \tag{3}\\ F(x,y)&\to\infty &&\text{as} && x\to y. \end{aligned} \]

Exclude some neighborhood \(O(y)\) of the point \(y\), where \(F>0\). In the domain \(T-O(y)\), \(F\) is a regular solution of equation (3), which assumes nonnegative values on the boundary. From the minimum property \(((^1),\) Theorems 3, I and 3, II) it follows that \(F\geqslant 0\) in \(T-O(y)\), and hence also in the entire domain \(T\).

\(2^\circ.\ c\) is arbitrary. Choose a function \(\chi\in C^{(0,\alpha)}\) such that \(c-\chi<0\), \(\chi\geqslant 0\). Construct the Green’s function \(F^*\) for the operator \(\mathfrak M-\chi\), which, by \(1^\circ\), will be positive. Then \(F\) is determined from the equation \(((^1),\) pp. 82–83)

\[ F(x,y)=F^*(x,y)+\int_T F^*(x,t)\chi(t)F(t,y)\,dt. \tag{4} \]

By the hypothesis of the theorem \(c<\lambda_1\). It follows that \(1<\mu_1\), where \(\mu_1\) is the first eigenvalue of equation (4).

Indeed, by the definition of \(\lambda_1\), the Green’s function \(F\) exists and is unique for every \(c<\lambda_1\). On the other hand, it is a solution of equation (4). If we assume that \(\mu_1\leqslant 1\), then for the existence and uniqueness of a solution for \(\mu=\mu_1\) it is necessary that the free term \(F^*\) be orthogonal to all eigenfunctions of the kernel \(K=F^*\chi\) corresponding to \(\mu_1\). But \(F^*\geqslant 0\), \(\chi\geqslant 0\), hence \(K\geqslant 0\), and among the eigenfunctions there will be one positive. This leads to a contradiction, since \(F^*\) is also positive.

Fredholm theory is applicable to equation (4).

From Theorem 1 \((^3)\) and the remark it follows that \(F\geqslant 0\) in the domain \(T\). We note that if \(c>\lambda_1\), then for any choice of \(\chi\) we shall have \(1>\mu_1\), and from Theorem 1 \((^3)\) it follows that the Green’s function cannot be positive everywhere in the domain \(T\).

Theorem 2. For \(c<\lambda_1\), Chaplygin’s theorem holds: if \(f\geqslant 0,\ \varphi\geqslant 0\), then the solution of equation (1)—(2) is positive, \(u\geqslant 0\), in the domain \(T\).

Indeed, the solution \(u\) can be represented as the sum

\[ u=u_1+u_2, \]

where \(u_1\) is the solution of the problem

\[ \mathfrak M u_1=-f,\qquad u_1|_{\mathfrak F T}=0, \]

and \(u_2\) is the solution of the problem

\[ \mathfrak M u_2=0,\qquad u_2|_{\mathfrak F T}=\varphi. \]

1) \(u_1\geqslant 0\), since, by Theorem 1, from the condition \(c<\lambda_1\) it follows that \(F\geqslant 0\), and

\[ u_1=\int_T Ff\,dy. \]

2) \(u_2\geqslant 0\): suppose the contrary. Then in some domain \(T_1\subset T\) we have

\[ \mathfrak M u_2=0,\qquad u_2|_{\mathfrak F T_1}=0,\qquad u_2\leqslant 0,\qquad u_2\ne 0. \]

This cannot be, since the domain \(T\), and hence also \(T_1\), is smaller than the critical one.

Corollary. We obtain the obvious generalization of the theorem on maximum (minimum) values: for \(c<\lambda_1\), the regular solution of the Dirichlet problem (1)—(2) has no points of negative relative minimum in the domain \(T-\mathfrak{F}T\).

Let \(F\) and \(F_1\) be the Green’s functions corresponding to \(c\) and \(c_1\), respectively, where \(c_1\le c\) and \(F\ge 0\). Then it is not difficult to see that \(F_1\ge 0\) as well, and the difference \(F-F_1\) satisfies the relation

\[ \mathfrak{M}_x(F-F_1)=-(c-c_1)F_1, \]

\[ F-F_1=0\quad \text{for } x\in\mathfrak{F}T,\quad y\in T-\mathfrak{F}T. \]

Hence

\[ F-F_1=\int_T F(c-c_1)F_1\,dt\ge 0. \]

The requirement \(F\ge 0\) is equivalent to \(c<\lambda_1\).

Thus we have proved the following.

Theorem 3. The Green’s function is a monotonically increasing function of the coefficient \(c\), provided that \(c\), while varying, remains below the critical value \(\lambda_1\).

From this it follows:

Theorem 4 (comparison theorem). If \(c_1\le c<\lambda_1\), then \(u_1\le u\), where \(u,u_1\) are the solutions of problem (1)—(2) corresponding to \(c,c_1\).

Next we shall consider the dependence of the Green’s function on the domain. Let the operator \(\mathfrak{M}\) be defined in the domain \(T\), \(T_1\in T\), and let \(F_1\) be the Green’s function of the Dirichlet problem for the domain \(T_1\). Suppose, moreover, that the domain \(T\) is smaller than critical. Then the functions \(F\) and \(F_1\) exist and, by Theorem 1, are positive. Further, under these conditions there exists the principal fundamental solution \(G\) of the equation \(\mathfrak{M}u=0\) ((\(^{1}\), pp. 69—76), and the Green’s function for any domain \(T\) has the form

\[ F=G+g, \]

where \(g\) is the regular solution of the equation \(\mathfrak{M}g=0\), \(g=-G\) for \(x\in\mathfrak{F}T\) ((\(^{1}\), p. 81).

Therefore the difference \(F-F_1=g_1\) is a regular solution of the equation \(\mathfrak{M}g_1=0\), \(g_1|_{\mathfrak{F}T_1}\ge 0\). By Theorem 2 we have \(g_1\ge 0\), i.e. \(F\ge F_1\) in the domain \(T_1\).

We have proved:

Theorem 5. The Green’s function is a monotonically increasing function of the domain \(T\), provided that \(T\), while varying, remains smaller than critical.

Hence it follows:

Theorem 6 (comparison theorem with respect to the domain). Let \(u,u_1\) be solutions of the Dirichlet problem for the domains \(T,T_1\); \(u|_{\mathfrak{F}T}=0\), \(u_1|_{\mathfrak{F}T_1}=0\), and let the domain \(T\) be smaller than critical. Then \(u\ge u_1\) in the domain \(T_1\).

Theorems 3 and 5 give broad scope for estimating the Green’s function from below and from above, while Theorems 2, 4, and 6 give such estimates for solutions.

We note that the results set forth, with certain changes, carry over also to non-self-adjoint equations.

Kazan State University
named after V. I. Ulyanov-Lenin

Received
7 IV 1960

CITED LITERATURE

\(^{1}\) K. Miranda, Equations with Partial Derivatives of Elliptic Type, IL, 1957.
\(^{2}\) S. G. Mikhlin, Lectures on Linear Integral Equations, 1959.
\(^{3}\) A. G. Teterev, Reports of the Scientific Conference on Theoretical and Applied Problems of Mathematics and Mechanics, Tomsk, 1960.