UDC 517.955.81

MATHEMATICS

B. N. KHIMCHENKO

ON THE BEHAVIOR OF SOLUTIONS OF ELLIPTIC EQUATIONS NEAR THE BOUNDARY OF A DOMAIN OF TYPE \(A^{(1)}\)

(Presented by Academician A. N. Tikhonov on 8 I 1970)

Let, in a closed bounded domain \((T+\partial T)\) of \(n\)-dimensional Euclidean space \(R^n\), a function \(u(M)\) be defined which is a regular solution of the equation

\[ \sum_{i,k=1}^{n} a_{ik}(M)\frac{\partial^2 u}{\partial x_i \partial x_k} + \sum_{i=1}^{n} b_i(M)\frac{\partial u}{\partial x_i} + c(M)u = f(M). \]

Here the functions \(a_{ik}\) are bounded in \(T\), and

\[ \sum_{i,k=1}^{n} a_{ik}\lambda_i\lambda_k \ge \alpha \sum_{i=1}^{n}\lambda_i^2, \qquad \alpha>0. \]

Let further \(l\) be a ray issuing from the considered point \(M_0\) of the boundary \(\partial T\), such that \(\cos(l,n)>0\), where \(n\) is the inner normal to the surface \(\partial T\) at the point \(M_0\).

Denote by \(\varphi(\rho)\) a function satisfying the conditions:

\[ \varphi(\rho)\in C^{(1)}([0,\rho_0])\cap C^{(\infty)}((0,\rho_0]); \tag{1} \]

\[ \varphi(0)=\varphi'(0)=0; \tag{2} \]

\[ \varphi'(\rho)>0,\qquad \varphi''(\rho)>0; \tag{3} \]

\[ \int_{0}^{\rho_0}\frac{\varphi(t)\,dt}{t^2}<\infty. \tag{4} \]

We shall call a \(\varphi\)-paraboloid (see \((^1)\)) the body \(z_0\ge z\ge \varphi(\rho)\). By \(A_\varphi^*\) and \(A_{\varphi *}\) we denote the classes of domains from \(A^{(1)}\) each boundary point of which can be touched by a \(\varphi\)-paraboloid, respectively from outside and from inside.

The construction of barriers for \(u(M)\), analogous to those considered in \((^2)\), leads to the proof of the following theorems. It is assumed here that the maximum principle is valid.

Theorem 1. If \((T+\partial T)\in A_{\varphi *}^{*}\) and at the point \(M_0\in\partial T\) \(u(M)\) attains its minimum value \(u_0\), then for every ray \(l\) there exists a constant \(c_1>0\) such that, for \(M\in l\) in a neighborhood of \(M_0\),

\[ u(M)-u_0\ge c_1 r_{10}. \]

Here (and below) \(r_{10}\) is the distance from \(M\) to \(M_0\).

Denote by \(\Omega_\varphi^*\) the body \(z\ge -\varphi(\rho)\), \(r\le r_0\) \((r=\sqrt{z^2+\rho^2})\), where \(\varphi(\rho)\) satisfies conditions (1)—(3), but

\[ \int_{0}^{\rho_0}\frac{\varphi(t)\,dt}{t^2}=\infty. \tag{5} \]

Theorem 2. If the function \(u(M)\) is defined in \(\Omega_{\varphi}^{*}\) and at the origin of coordinates \(M_{0}\) attains its minimum value \(u_{0}\), then for each ray \(l\) there exists a constant \(c_{2}>0\) such that, for \(M\in l\) in a neighborhood of \(M_{0}\),

\[ u(M)-u_{0}\geq c_{2}r_{10}\int_{r_{10}}^{r_{0}}\frac{\varphi(t)\,dt}{t^{2}}. \]

Let now \(d(M)\) be the distance from the point \(M\) to the boundary \(\partial T\), and suppose the functions \(b_i(M)\), \(c(M)\), and \(f(M)\) everywhere in \(T\) satisfy the conditions

\[ |b_i(M)|<c_3;\qquad |c(M)|d^{\lambda}(M)<c_3;\qquad |f(M)|d^{\lambda}(M)<c_3 \tag{6} \]

\[ (0<\lambda<1) \]

and, moreover, on the surface \(\partial T\) the function \(u(M)\) satisfies the requirement

\[ \left|u(M)-u_0-\sum_{i=1}^{n-1}a_i x_i\right|\leq c_4\varphi(r_{10}), \tag{7} \]

where \(u_0=u(M_0)\), \(a_i=\partial u(M_0)/\partial x_i\), and for the function \(\varphi(\rho)\) conditions (1)—(4) hold.

Theorem 3. If \(u(M)\) is defined in \((T+\partial T)\in A_{\varphi}^{*}\) and \(M_0\) is an arbitrary fixed point of the boundary \(\partial T\), then for each ray \(l\) there exists \(c_5>0\) such that in a neighborhood of \(M_0\)

\[ |u(M)-u_0|\leq c_5 r_{10}. \]

For harmonic functions, in the case where \((T+\partial T)\in A_{\varphi}^{*}\cap A_{\varphi *}\), the assertion of Theorem 3 follows directly from the results given in \({}^{3}\).

Denote by \(\Omega_{\varphi *}\) the \(\varphi\)-paraboloid for which \(\varphi(\rho)\) satisfies conditions (1)—(3) and (5), and suppose that, for the function \(u(M)\) defined in \(\Omega_{\varphi *}\), on the surface \(z=\varphi(\rho)\) one has

\[ \left|u(M)-u_0-\sum_{i=1}^{n-1}a_i x_i\right| \leq c_6\varphi(r_{10}) \exp\left[-c_7\int_{r_{10}}^{r_0}\frac{\varphi(t)\,dt}{t^2}\right], \]

where \(u_0=u(M_0)\), \(M_0\) is the vertex of the \(\varphi\)-paraboloid, and \(c_6\) and \(c_7\) are fixed positive constants.

Theorem 4. For each ray \(l\) there exist \(c_8\) and \(c_9\) such that, in a neighborhood of \(M_0\),

\[ |u(M)-u_0| \leq c_8 r_{10} \exp\left[-c_9\int_{r_{10}}^{r_0}\frac{\varphi(t)\,dt}{t^2}\right]. \]

Moscow State University
named after M. V. Lomonosov

Received
2 XII 1969

REFERENCES

\({}^{1}\) B. N. Khimchenko, DAN, 192, No. 1 (1970).
\({}^{2}\) B. N. Khimchenko, Differential Equations, 5, No. 10 (1969).
\({}^{3}\) Kjell-Ove Widman, Math. Scand., 21, 17 (1967).