ON ESTIMATES IN A CLOSED REGION OF THE EIGENFUNCTIONS OF AN ELLIPTIC OPERATOR WITH DISCONTINUOUS COEFFICIENTS

T. V. LOSSIEVSKAYA

INTRODUCTION

Let a domain \(G\), bounded by a surface \(\Gamma\), be given in \(N\)-dimensional space. Inside \(G\) let there be given an \((N-1)\)-dimensional closed surface \(\Gamma_1\), which divides \(G\) into two parts: \(G_1\), bounded by \(\Gamma_1\), and \(G_2\), bounded by \(\Gamma_1\) and \(\Gamma\). Consider the eigenvalue problem:

\[ c_i \Delta u+\lambda u=0 \quad \text{in } G_i \qquad (i=1,2), \tag{1} \]

\[ [u]\big|_{\Gamma_1}=u\big|_{\Gamma_1+0}-u\big|_{\Gamma_1-0}=0, \tag{2} \]

\[ \left[\frac{\partial u}{\partial \nu}\right]\bigg|_{\Gamma_1} = c_2\frac{\partial u}{\partial \nu}\bigg|_{\Gamma_1+0} - c_1\frac{\partial u}{\partial \nu}\bigg|_{\Gamma_1-0} =0, \tag{3} \]

\[ u\big|_{\Gamma}=0, \tag{4} \]

where \(\nu\) is the direction of the normal to \(\Gamma_1\), exterior with respect to \(G_1\), and the symbols \(\Gamma_1+0\) and \(\Gamma_1-0\) mean that the limits are taken, respectively, from the exterior and interior sides of the surface \(\Gamma_1\) with respect to \(G_1\).

In the present paper, for the orthonormal eigenfunctions \(u_n(x)\) of problem (1)—(4), the following estimates in the closed domain \(G\) are proved:

\[ |u_n(x)|=O\!\left(\lambda_n^{\frac14}\right) \quad \text{for } N=2, \tag{5} \]

\[ |u_n(x)|=O\!\left(\lambda_n^{\frac{N-1}{4}}\sqrt{\ln\lambda_n}\right)^{*)} \quad \text{for } N\ge 3. \tag{6} \]

It is assumed that \(\Gamma,\Gamma_1\in A^{(2)}\). If, however, \(\Gamma,\Gamma_1\in A^{(k,\gamma)}\), \((0<\gamma<1)\), then the eigenfunctions of problem (1)—(4) in each of the closed domains \(G_1+\Gamma_1\) and \(G_2+\Gamma_1+\Gamma\) belong to the class \(C^{(k,\gamma)**)}\) (see [1]), and then the estimates

\[ u_l=O\!\left(\lambda^{\frac14+\frac{l}{2}}\right), \qquad u_{l,\gamma}=O\!\left(\lambda^{\frac14+\frac{l}{2}+\frac{\gamma}{2}}\right) \tag{7} \]

hold for \(l\le k,\ N=2\);

\[ u_l=O\!\left(\lambda^{\frac{N-1}{4}+\frac{l}{2}}\sqrt{\ln\lambda}\right), \qquad u_{l,\gamma}=O\!\left(\lambda^{\frac{N-1}{4}+\frac{l}{2}+\frac{\gamma}{2}}\sqrt{\ln\lambda}\right), \tag{8} \]

\[ \text{*) Everywhere below, for brevity, we omit the number } n \text{ of the eigenfunctions.} \]
\[ \text{We note that all constants entering into } O \text{ do not depend on } n. \]

\[ \text{**) The definition of the classes } C^{(k)},\ C^{(k,\gamma)} \text{ and } A^{(k)},\ A^{(k,\gamma)} \text{ can be found in [2].} \]

for \(l \leq k,\ N \geq 3\). Here \(u_l\) denotes the sum of the maxima of the moduli of all \(l\)-th derivatives of an eigenfunction, and \(u_{l,\gamma}\) the sum of the Hölder coefficients of all \(l\)-th derivatives.

Since the eigenfunctions of an \(N\)-dimensional ball that possess radial symmetry take at the center of the ball a value not less than \(\operatorname{const}\lambda^{\frac{N-1}{4}}\), estimate (5) is sharp. The same is also true for estimates (7).

Estimates (5) and (7) are the first sharp estimates of eigenfunctions in a closed region for a problem with discontinuous coefficients. Until now, sharp estimates in a closed region had been obtained only for eigenfunctions of the Laplace operator (see [3]).

§ 1. INTEGRAL ESTIMATES OF EIGENFUNCTIONS AND THEIR NORMAL DERIVATIVES

Lemma. Let \(u(x)\) be orthonormal eigenfunctions of problem (1)—(4). If \(\Gamma, \Gamma_1 \in A^{(2)}\), then the following estimates hold:

\[ \int_{\Gamma}\left(\frac{\partial u}{\partial \nu}\right)^2 ds = O(\lambda), \tag{9} \]

\[ \int_{\Gamma_1}\left(\frac{\partial u}{\partial \nu}\right)^2\bigg|_{\Gamma_1+0} ds = O(\lambda), \tag{10} \]

\[ \int_{\Gamma_1}\left(\frac{\partial u}{\partial \nu}\right)^2\bigg|_{\Gamma_1-0} ds = O(\lambda), \tag{11} \]

\[ \int_{\Gamma_1} u^2\,ds = O(1). \tag{12} \]

Proof. Write Green’s first formula over the domain \(G_2\) for the functions \(u\) and

\[ v=\sum_{j=1}^{N}\frac{\partial u}{\partial x_j}q_j, \]

where

\[ q_j\big|_{\Gamma_1}=\cos(\nu,x_j),\qquad q_j\big|_{\Gamma}=0 \]

(thus, for the function \(v\) we have

\[ v\big|_{\Gamma_1}=\frac{\partial u}{\partial \nu}\bigg|_{\Gamma_1+0},\qquad v\big|_{\Gamma}=0 \]

)\(^*\):

\[ -\int_{\Gamma_1}\left(\frac{\partial u}{\partial \nu}\right)^2\bigg|_{\Gamma_1+0} ds = \int_{G_2}\sum_{i=1}^{N}\frac{\partial u}{\partial x_i} \frac{\partial}{\partial x_i}\left[\sum_{j=1}^{N}\frac{\partial u}{\partial x_j}q_j\right]dx + \int_{G_2}\sum_{j=1}^{N}\frac{\partial u}{\partial x_j}q_j\Delta u\,dx. \tag{13} \]

Differentiating the expression in square brackets and substituting \(\Delta u = -\dfrac{\lambda}{c_2}u\) into the right-hand side of (13), we obtain

\[ -\int_{\Gamma_1}\left(\frac{\partial u}{\partial \nu}\right)^2\bigg|_{\Gamma_1+0} ds = \int_{G_2}\sum_{i,j=1}^{N}\frac{\partial q_j}{\partial x_i} \frac{\partial u}{\partial x_i} \frac{\partial u}{\partial x_j}\,dx+ \]

\[ {}^*\) This device was first applied by D. M. Eidus [4] in obtaining estimate [9] for eigenfunctions of a problem with continuous coefficients. <!-- source-page: 003 --> \[ + \int_{G_2} \sum_{i,j=1}^{N} q_j \frac{\partial u}{\partial x_i}\, \frac{\partial^2 u}{\partial x_i \partial x_j}\, dx - \frac{\lambda}{c_2} \int_{G_2} \sum_{j=1}^{N} q_j \frac{\partial u}{\partial x_j}\, u\, dx = I_1 + I_2 + I_3 . \]

Consider \(I_2\). Taking into account that
\[ \frac{\partial u}{\partial x_i}\, \frac{\partial^2 u}{\partial x_i \partial x_j} = \frac{1}{2}\frac{\partial}{\partial x_j} \left(\frac{\partial u}{\partial x_i}\right)^2, \]
we integrate \(I_2\) by parts:
\[ I_2 = -\frac{1}{2}\int_{\Gamma_1} \sum_{i=1}^{N}\left(\frac{\partial u}{\partial x_i}\right)^2 \bigg|_{\Gamma_1+0} ds -\frac{1}{2}\int_{G_2}\sum_{i,j=1}^{N} \frac{\partial q_j}{\partial x_j} \left(\frac{\partial u}{\partial x_i}\right)^2 dx . \]

We may write
\[ \sum_{i=1}^{N}\left(\frac{\partial u}{\partial x_i}\right)^2 \bigg|_{\Gamma_1+0} = \left(\frac{\partial u}{\partial \nu}\right)^2 \bigg|_{\Gamma_1+0} + \left(\frac{\partial u}{\partial t}\right)^2 \bigg|_{\Gamma_1+0}, \]
where
\[ \frac{\partial u}{\partial t}\bigg|_{\Gamma_1+0} \]
is the right-hand limit of the tangential derivative of the function \(u\) on \(\Gamma_1\). Hence
\[ I_2 = -\frac{1}{2}\int_{\Gamma_1} \left(\frac{\partial u}{\partial \nu}\right)^2 \bigg|_{\Gamma_1+0} ds -\frac{1}{2}\int_{\Gamma_1} \left(\frac{\partial u}{\partial t}\right)^2 \bigg|_{\Gamma_1+0} ds \]
\[ -\frac{1}{2}\int_{G_2}\sum_{i,j=1}^{N} \frac{\partial q_j}{\partial x_j} \left(\frac{\partial u}{\partial x_i}\right)^2 dx . \]

Similarly,
\[ I_3 = -\frac{\lambda}{2c_2}\int_{\Gamma_1} u^2\bigg|_{\Gamma_1+0} ds + \frac{\lambda}{2c_2}\int_{G_2} \sum_{j=1}^{N}\frac{\partial q_j}{\partial x_j} u^2\, dx, \]
and then
\[ -\int_{\Gamma_1} \left(\frac{\partial u}{\partial \nu}\right)^2 \bigg|_{\Gamma_1+0} ds -\frac{\lambda}{c_2}\int_{\Gamma_1} u^2\, ds = \]
\[ = -\int_{\Gamma_1} \left(\frac{\partial u}{\partial t}\right)^2 \bigg|_{\Gamma_1+0} ds -\int_{G_2}\sum_{i,j=1}^{N} \frac{\partial q_j}{\partial x_j} \left(\frac{\partial u}{\partial x_i}\right)^2 dx \]
\[ +2\int_{G_2}\sum_{i,j=1}^{N} \frac{\partial q_j}{\partial x_i} \frac{\partial u}{\partial x_i} \frac{\partial u}{\partial x_j}\, dx + \frac{\lambda}{c_2}\int_{G_2}\sum_{j=1}^{N} \frac{\partial q_j}{\partial x_j}u^2\, dx . \tag{14} \]

In exactly the same way, applying Green’s first formula in the domain \(G_1\) to the functions \(u\) and
\[ v=\sum_{j=1}^{N}\frac{\partial u}{\partial x_j}q_j \]
(where \(q_j\) is the same function as before, only continued into \(G_1\)), we obtain
\[ \int_{\Gamma_1} \left(\frac{\partial u}{\partial \nu}\right)^2 \bigg|_{\Gamma_1-0} ds + \frac{\lambda}{c_1}\int_{\Gamma_1} u^2\, ds = \int_{\Gamma_1} \left(\frac{\partial u}{\partial t}\right)^2 \bigg|_{\Gamma_1-0} ds - \]

\[ -\int_{G_1}\sum_{i,j=1}^{N}\frac{\partial q_j}{\partial x_j} \left(\frac{\partial u}{\partial x_i}\right)^2 dx +2\int_{G_1}\sum_{i,j=1}^{N}\frac{\partial q_j}{\partial x_i} \frac{\partial u}{\partial x_i}\frac{\partial u}{\partial x_j}\,dx +\frac{\lambda}{c_1}\int_{G_1}\sum_{j=1}^{N}\frac{\partial q_j}{\partial x_j}u^2\,dx . \tag{15} \]

Taking into account (3) and the fact that
\[ \left(\frac{\partial u}{\partial t}\right)_{\Gamma_1-0} = \left(\frac{\partial u}{\partial t}\right)_{\Gamma_1+0} \]
(which follows from (2)), in accordance with (14) and (15), we have

\[ \begin{aligned} \left(1-\frac{c_1^2}{c_2^2}\right) \int_{\Gamma_1}\left(\frac{\partial u}{\partial \nu}\right)^2_{\Gamma_1-0}ds +\frac{\lambda}{c_1}\left(1-\frac{c_1}{c_2}\right) \int_{\Gamma_1}u^2\,ds &= -\sum_{k=1}^{2}\int_{G_k}\sum_{i,j=1}^{N} \frac{\partial q_j}{\partial x_j} \left(\frac{\partial u}{\partial x_i}\right)^2 dx \\ &\quad +2\sum_{k=1}^{2}\int_{G_k}\sum_{i,j=1}^{N} \frac{\partial q_j}{\partial x_i} \frac{\partial u}{\partial x_i}\frac{\partial u}{\partial x_j}\,dx \\ &\quad +\lambda\sum_{k=1}^{2}\frac{1}{c_k} \int_{G_k}\sum_{j=1}^{N}\frac{\partial q_j}{\partial x_j}u^2\,dx . \end{aligned} \tag{16} \]

It remains to estimate the right-hand side of (16). Since \(q_j\in C^{(1)}\) in \((G+\Gamma)\), it follows that
\(\left|\dfrac{\partial q_j}{\partial x_i}\right|\leq M_1\) *); hence

\[ \left| \sum_{k=1}^{2}\int_{G_k}\sum_{i,j=1}^{N} \frac{\partial q_j}{\partial x_j} \left(\frac{\partial u}{\partial x_i}\right)^2 dx \right| \leq M_1\int_G\sum_{i=1}^{N}\left(\frac{\partial u}{\partial x_i}\right)^2 dx, \tag{17} \]

\[ \begin{aligned} \left| \sum_{k=1}^{2}\int_{G_k}\sum_{i,j=1}^{N} \frac{\partial q_j}{\partial x_i} \frac{\partial u}{\partial x_i} \frac{\partial u}{\partial x_j}\,dx \right| &\leq M_1\int_G\sum_{i,j=1}^{N} \left|\frac{\partial u}{\partial x_i}\frac{\partial u}{\partial x_j}\right|dx \\ &\leq M_1\int_G\sum_{i=1}^{N} \left(\frac{\partial u}{\partial x_i}\right)^2 dx . \end{aligned} \tag{18} \]

The last inequality follows from the Cauchy—Bunyakovsky inequality,

\[ \left| \sum_{k=1}^{2}\frac{1}{c_k}\int_{G_k}\sum_{j=1}^{N} \frac{\partial q_j}{\partial x_j}u^2\,dx \right| \leq M_2\int_G u^2\,dx = M_2 . \tag{19} \]

We shall prove that

\[ \int_{G_k}\sum_{i=1}^{N}\left(\frac{\partial u}{\partial x_i}\right)^2 dx = O(\lambda). \tag{20} \]

*) Here and below \(M_k\) \((k=1,2,\ldots)\) denote constants independent of \(\lambda\).

Let us write the first Green formula for the function \(u\) in the domain \(G_2\):

\[ -\int_{\Gamma_1} u\left.\frac{\partial u}{\partial \nu}\right|_{\Gamma_1+0} ds = \int_{G_2} \sum_{i=1}^{N}\left(\frac{\partial u}{\partial x_i}\right)^2 dx + \int_{G_2} u\Delta u\,dx = \int_{G_2} \sum_{i=1}^{N}\left(\frac{\partial u}{\partial x_i}\right)^2 dx - \frac{\lambda}{c_2}\int_{G_2} u^2\,dx . \tag{21} \]

We do the same for the domain \(G_1\):

\[ \int_{\Gamma_1} u\left.\frac{\partial u}{\partial \nu}\right|_{\Gamma_1-0} ds = \int_{G_1} \sum_{i=1}^{N}\left(\frac{\partial u}{\partial x_i}\right)^2 dx - \frac{\lambda}{c_1}\int_{G_1} u^2\,dx . \tag{22} \]

From (21) and (22) it follows that

\[ \int_{\Gamma_1} u \left[ c_2\left.\frac{\partial u}{\partial \nu}\right|_{\Gamma_1+0} - c_1\left.\frac{\partial u}{\partial \nu}\right|_{\Gamma_1-0} \right] ds = \sum_{k=1}^{2} c_k \int_{G_k} \sum_{i=1}^{N}\left(\frac{\partial u}{\partial x_i}\right)^2 dx - \lambda \int_G u^2\,dx . \]

Using (3) and the fact that \(\int_G u^2\,dx=1\), we obtain

\[ \sum_{k=1}^{2} c_k \int_{G_k} \sum_{i=1}^{N}\left(\frac{\partial u}{\partial x_i}\right)^2 dx = \lambda . \]

Estimate (20) is proved. From (17)—(20) it is seen that the right-hand side of (16) is estimated as \(O(\lambda)\), whence estimates (11) and (12) follow, and from (3) and (11)—estimate (10). To prove formula (9) in \(G_2\), it is necessary to specify functions \(q_j\) that satisfy the following conditions on \(\Gamma_1\) and \(\Gamma\): \(q_j|_{\Gamma}=\cos(\nu,x_j)\), \(q_j|_{\Gamma_1}=0\). Then the proof of estimate (9) completely coincides with the proof of the same estimate for the problem with continuous coefficients (the Laplacian case), which was carried out by D. M. Eidus [4].

§ 2. ESTIMATE OF EIGENFUNCTIONS FOR THE \(N\)-DIMENSIONAL CASE \((N \ge 3)\)

In order to obtain an estimate of the eigenfunctions in the closed domain \(G\), it is necessary to estimate them in \(G_1\), \(G_2\), and on \(\Gamma_1\). For this we shall need estimates of

\[ \left.\frac{\partial u}{\partial \nu}\right|_{\Gamma_1\pm 0} \quad\text{and}\quad \left.\frac{\partial u}{\partial \nu}\right|_{\Gamma}. \]

Since

\[ \left.\frac{\partial u}{\partial \nu}\right|_{\Gamma_1+0} = \frac{c_1}{c_2} \left.\frac{\partial u}{\partial \nu}\right|_{\Gamma_1-0}, \]

it is clear that it is sufficient to estimate only

\[ \left.\frac{\partial u}{\partial \nu}\right|_{\Gamma_1-0}. \]

\(1^\circ\). Thus, let \(x\in G_1\). Then, taking into account that the principal fundamental solution of the equation

\[ \Delta u+\frac{\lambda}{c_1}u=0 \]

is

\[ -\frac{1}{4}\left(\frac{1}{2\pi}\right)^{\frac{N-2}{2}} \left(\frac{\lambda}{c_1}\right)^{\frac{N}{4}-\frac{1}{2}} \times \frac{ N_{\frac{N-2}{2}}\left(r\sqrt{\frac{\lambda}{c_1}}\right) }{ r^{\frac{N-2}{2}} } \,^{*)} \]

(\(N_{\frac{N-2}{2}}(t)\) is the Neumann function), we write \(u(x)\) in the form

\({}^{*)}\) In what follows, everywhere, for the sake of brevity, we shall denote the quantity

\[ \frac{1}{4}\left(\frac{1}{2\pi}\right)^{\frac{N-2}{2}} \]

by \(\gamma_N\).

\[ \begin{aligned} u(x)={}&-\chi_N\left(\frac{\lambda}{c_1}\right)^{\frac N4-\frac12} \int_{\Gamma_1} \frac{ N_{\frac{N-2}{2}}\left(r\sqrt{\frac{\lambda}{c_1}}\right) }{ r^{\frac{N-2}{2}} } \\ &\times \left.\frac{\partial u}{\partial \nu}(y)\right|_{\Gamma_1-0}\,ds_y +\chi_N\left(\frac{\lambda}{c_1}\right)^{\frac N4-\frac12} \\ &\times \int_{\Gamma_1} \frac{\partial}{\partial \nu_y} \left[ \frac{ N_{\frac{N-2}{2}}\left(r\sqrt{\frac{\lambda}{c_1}}\right) }{ r^{\frac{N-2}{2}} } \right]u(y)\,ds_y \\ ={}&-\chi_N\left(\frac{\lambda}{c_1}\right)^{\frac N4-\frac12} \int_{\Gamma_1} \frac{ N_{\frac{N-2}{2}}\left(r\sqrt{\frac{\lambda}{c_1}}\right) }{ r^{\frac{N-2}{2}} } \\ &\times \left.\frac{\partial u}{\partial \nu}(y)\right|_{\Gamma_1-0}\,ds_y -\chi_N\left(\frac{\lambda}{c_1}\right)^{\frac N4} \int_{\Gamma_1} \frac{ N_{\frac N2}\left(r\sqrt{\frac{\lambda}{c_1}}\right) }{ r^{\frac{N-2}{2}} } \frac{\partial r}{\partial \nu_y}\,u(y)\,ds_y . \tag{23} \end{aligned} \]

Here \(r=\sqrt{\sum_{i=1}^{N}(x_i-y_i)^2}\). Using the properties of derivatives of simple- and double-layer potentials, from (23) we obtain

\[ \begin{aligned} \left.\frac{\partial u}{\partial \nu}\right|_{\Gamma_1-0} ={}& 2\chi_N\left(\frac{\lambda}{c_1}\right)^{\frac N4} \int_{\Gamma_1} \frac{ N_{\frac N2}\left(r\sqrt{\frac{\lambda}{c_1}}\right) }{ r^{\frac{N-2}{2}} } \frac{\partial r}{\partial \nu_x} \left.\frac{\partial u}{\partial \nu}\right|_{\Gamma_1-0} \,ds \\ &- 2\chi_N\left(\frac{\lambda}{c_1}\right)^{\frac N4} \frac{\partial}{\partial \nu_x} \int_{\Gamma_1} \frac{ N_{\frac N2}\left(r\sqrt{\frac{\lambda}{c_1}}\right) }{ r^{\frac{N-2}{2}} } \frac{\partial r}{\partial \nu_y}\,u\,ds =I_4+I_5 . \end{aligned} \]

Thus, in order to estimate \(\left.\dfrac{\partial u}{\partial \nu}\right|_{\Gamma_1-0}\), we must estimate \(I_4\) and \(I_5\).

1) Estimate of \(I_4\). Introduce a coordinate system as follows: place the origin at the point under consideration \(x\in\Gamma_1\), and direct the \(x_N\)-axis along the normal \(\nu\) to \(\Gamma_1\) at the point \(x\). Replace integration over the part of \(\Gamma_1\) lying inside the Lyapunov sphere with center at the point \(x\) by integration over the plane tangent to \(\Gamma_1\) at the origin. Denote \(\rho^2=\sum_{i=1}^{N-1}y_i^2\). Now split \(\Gamma_1\) into three parts: \(\Gamma_1^{(1)}\)—the set of points of \(\Gamma_1\) for which \(\rho<\dfrac{a}{\sqrt{\lambda}}\), where \(a\)—

some constant, which will be chosen later; \(\Gamma_1^{(2)}\) is the set of points of \(\Gamma_1\) for which
\[ \frac{\alpha}{\sqrt{\lambda}}\leqslant \rho \leqslant R, \]
where \(R\) is the radius of the Lyapunov sphere, and
\[ \Gamma_1^{(3)}=\Gamma_1-\Gamma_1^{(1)}-\Gamma_1^{(2)}. \]

We take into account that
\[ N_{\frac N2}\left(r\sqrt{\frac{\lambda}{c_1}}\right) = O\left(r^{-\frac N2}\lambda^{-\frac N4}\right) \quad \text{on } \Gamma_1^{(1)}, \tag{24} \]
\[ N_{\frac N2}\left(r\sqrt{\frac{\lambda}{c_1}}\right) = O\left(r^{-\frac12}\lambda^{-\frac14}\right) \quad \text{on } \Gamma_1^{(2)}+\Gamma_1^{(3)} \tag{25} \]
and
\[ \frac1{r^\beta}\leqslant \frac1{R^\beta} \quad \text{for any } \beta>0 \text{ on } \Gamma_1^{(3)}. \tag{26} \]

Moreover, on \(\Gamma_1^{(1)}+\Gamma_1^{(2)}\),
\[ \frac{\partial r}{\partial \nu_x}=O(\rho), \tag{27} \]
which follows from the fact that
\[ \left|\frac{\partial r}{\partial \nu_x}\right| = \left|\frac{\partial r}{\partial x_N}\right| = \frac{|x_N-y_N|}{r} = \frac{|y_N|}{r}, \qquad \Gamma \in A^{(2)} \quad \text{and} \quad \rho \leqslant r. \]

Everywhere below, when considering the values of the function \(u\) and its derivatives on \(\Gamma_1\), we shall use the coordinate system introduced above. In doing so we shall divide the surface \(\Gamma_1\) into the parts \(\Gamma_1^{(1)}\), \(\Gamma_1^{(2)}\), and \(\Gamma_1^{(3)}\), as we have just done.

We represent the integral \(I_4\) in the form
\[ I_4=\int_{\Gamma_1} = \int_{\Gamma_1^{(1)}}+ \int_{\Gamma_1^{(2)}}+ \int_{\Gamma_1^{(3)}} = I_4^{(1)}+I_4^{(2)}+I_4^{(3)}. \tag{28} \]

Then, using (24), (27), and the fact that
\[ \left.\frac{\partial u}{\partial \nu}\right|_{\Gamma_1-0} = O\left(\lambda^{\frac N4+\frac12}\right) \quad \text{(see [3])}, \]
we obtain
\[ |I_4^{(1)}| \leqslant M_3\lambda^{\frac N4+\frac12} \int_0^{\frac{\alpha}{\sqrt{\lambda}}} \frac{\rho^{N-1}}{r^{N-1}}\,d\rho \leqslant M_3\lambda^{\frac N4}. \tag{29} \]

In estimating \(I_4^{(2)}\) we proceed differently. First we substitute (25) and (27) into this integral, then apply Bunyakovsky’s inequality, and, finally, use estimate (11), obtained by us in the preceding paragraph:
\[ |I_4^{(2)}| \leqslant M_4\lambda^{\frac{N-1}{4}} \int_{\Gamma_1^{(2)}} \frac{\rho}{r^{\frac{N-1}{2}}} \left| \left.\frac{\partial u}{\partial \nu}\right|_{\Gamma_1-0} \right|\,ds \leqslant M_4\lambda^{\frac{N-1}{4}} \sqrt{ \int_{\Gamma_1^{(2)}} \frac{\rho^2\,ds}{r^{N-1}} } \]
\[ {}\times \sqrt{ \int_{\Gamma_1^{(2)}} \left(\left.\frac{\partial u}{\partial \nu}\right|_{\Gamma_1-0}\right)^2 ds } \leqslant M_5\lambda^{\frac{N+1}{4}} \sqrt{ \int_{\frac{\alpha}{\sqrt{\lambda}}}^{R} \frac{\rho^N\,d\rho}{r^{N-1}} } \leqslant M_6\lambda^{\frac{N+1}{4}}. \tag{30} \]

Finally, when considering \(I_4^{(3)}\) it is necessary to take into account (25), (26), apply Bunyakovsky’s inequality and estimate (11). Then

\[ \left| I_4^{(3)} \right| \leq M_7 \lambda^{\frac{N+1}{4}} . \tag{31} \]

Combining (29)—(31), we have

\[ I_4 = O\left(\lambda^{\frac{N+1}{4}}\right). \tag{32} \]

2) Estimate of \(I_5\). Similarly to (28), we write

\[ I_5=\int_{\Gamma_1} =\int_{\Gamma_1^{(1)}}+\int_{\Gamma_1^{(2)}}+\int_{\Gamma_1^{(3)}} =I_5^{(1)}+I_5^{(2)}+I_5^{(3)} . \]

We have:

\[ \begin{aligned} I_5^{(2)}+I_5^{(3)} &= -2\chi_N\left(\frac{\lambda}{c_1}\right)^{\frac{N}{4}} \times \\ &\quad \times \int_{\Gamma_1^{(2)}+\Gamma_1^{(3)}} \frac{\partial}{\partial \nu_x} \left[ \frac{ N_{\frac{N}{2}}\!\left(r\sqrt{\frac{\lambda}{c_1}}\right) }{ r^{\frac{N-2}{2}} } \frac{\partial r}{\partial \nu_y} \right] u\,ds \\ &= -2\chi_N\left(\frac{\lambda}{c_1}\right)^{\frac{N}{4}} \times \\ &\quad \times \int_{\Gamma_1^{(2)}+\Gamma_1^{(3)}} \frac{ N_{\frac{N}{2}}\!\left(r\sqrt{\frac{\lambda}{c_1}}\right) }{ r^{\frac{N-2}{2}} } \frac{\partial^2 r}{\partial \nu_x \partial \nu_y} u\,ds \\ &\quad -2\chi_N\left(\frac{\lambda}{c_1}\right)^{\frac{N}{4}} \int_{\Gamma_1^{(2)}+\Gamma_1^{(3)}} \frac{ N_{\frac{N}{2}}\!\left(r\sqrt{\frac{\lambda}{c_1}}\right) }{ r^{\frac{N}{2}} } \frac{\partial r}{\partial \nu_x} \frac{\partial r}{\partial \nu_y} u\,ds + \\ &\quad +2\chi_N\left(\frac{\lambda}{c_1}\right)^{\frac{N}{4}+\frac{1}{2}} \int_{\Gamma_1^{(2)}+\Gamma_1^{(3)}} \frac{ N_{\frac{N+2}{2}}\!\left(r\sqrt{\frac{\lambda}{c_1}}\right) }{ r^{\frac{N-2}{2}} } \frac{\partial r}{\partial \nu_x} \frac{\partial r}{\partial \nu_y} u\,ds . \end{aligned} \]

On \(\Gamma_1^{(1)}+\Gamma_1^{(2)}\) the estimate

\[ \frac{\partial r}{\partial \nu_y}=O(\rho). \tag{33} \]

is valid. Indeed,

\[ \frac{\partial r}{\partial \nu_y} = \sum_{i=1}^{N}\frac{\partial r}{\partial y_i}\cos(\nu,y_i) = \sum_{i=1}^{N-1}\frac{\partial r}{\partial y_i}\cos(\nu,y_i) + \frac{\partial r}{\partial \nu_x}\cos(\nu,y_N). \]

Since

\[ \left|\frac{\partial r}{\partial y_i}\right|\leqslant 1,\qquad \cos(\nu,y_i)=O(\rho)\qquad (i=1,\ldots,N-1) \]

(in view of the fact that \(\Gamma\in A^{(2)}\)) and

\[ \frac{\partial r}{\partial \nu_x}=O(\rho),\qquad |\cos(\nu,y_N)|\leqslant 1, \]

it follows that (33) holds. Moreover,

\[ \frac{\partial^2 r}{\partial \nu_x\partial \nu_y} = O\left(\frac1r\right). \tag{34} \]

This is seen from the fact that

\[ \frac{\partial^2 r}{\partial \nu_x\partial \nu_y} = \sum_{i=1}^N \frac{\partial^2 r}{\partial x_N\partial y_i}\cos(\nu,y_i) = -\frac1r \sum_{i=1}^N \left[ \delta_{iN}+ \left(\frac{\partial r}{\partial x_N}\right) \left(\frac{\partial r}{\partial y_i}\right) \right]\cos(\nu,y_i). \]

Taking these remarks into account, one can, just as we obtained the estimates for \(I_4^{(2)}\) and \(I_4^{(3)}\), obtain

\[ I_5^{(2)}+I_5^{(3)}=O\left(\lambda^{\frac{N+1}{4}}\right). \tag{35} \]

The only difference is that instead of estimate (11), estimate (12) is used here.

Finally, let us estimate \(I_5^{(1)}\). For this purpose we represent it in the form

\[ I_5^{(1)} = 2\int_{\Gamma_1^{(1)}} \frac{\partial}{\partial \nu_x} \left\{ \left[ -\chi_N\left(\frac{\lambda}{c_1}\right)^{\frac N4} \frac{N_{\frac N2}\left(r\sqrt{\frac{\lambda}{c_1}}\right)} {r^{\frac{N-2}{2}}} - \frac{1}{\omega_N r^{N-1}} \right] \frac{\partial r}{\partial \nu_y} \right\}u\,ds + \]

\[ + 2\frac{\partial}{\partial \nu_x} \int_{\Gamma_1^{(1)}} \frac{1}{\omega_N r^{N-1}} \frac{\partial r}{\partial \nu_y} u\,ds, \tag{36} \]

where

\[ \omega_N=\frac{2\pi^{\frac N2}}{\Gamma\left(\frac N2\right)} \]

is the area of the unit sphere in \(N\)-dimensional space.

Let us rewrite the first integral on the right-hand side of (36) in the following form:

\[ 2\int_{\Gamma_1^{(1)}} \frac{\partial}{\partial \nu_x} \left\{ \left[ -\chi_N\left(\frac{\lambda}{c_1}\right)^{\frac N4} \frac{N_{\frac N2}\left(r\sqrt{\frac{\lambda}{c_1}}\right)} {r^{\frac{N-2}{2}}} - \frac{1}{\omega_N r^{N-1}} \right] \frac{\partial r}{\partial \nu_y} \right\}u\,ds = \]

\[ = 2\int_{\Gamma_1^{(1)}} \left[ -\chi_N\left(\frac{\lambda}{c_1}\right)^{\frac N4} \frac{N_{\frac N2}\left(r\sqrt{\frac{\lambda}{c_1}}\right)} {r^{\frac{N-2}{2}}} - \frac{1}{\omega_N r^{N-1}} \right] \frac{\partial^2 r}{\partial \nu_x\partial \nu_y} u\,ds + \]

\[ + 2\int_{\Gamma_1^{(1)}} \left[ \chi_N\left(\frac{\lambda}{c_1}\right)^{\frac N4+\frac12} \frac{N_{\frac{N+2}{2}}\left(r\sqrt{\frac{\lambda}{c_1}}\right)} {r^{\frac{N-2}{2}}} \right. \]

\[ -\chi_N\left(\frac{\lambda}{c_1}\right)^{\frac N4} \left[ \frac{ N_{\frac N2}\left(r\sqrt{\frac{\lambda}{c_1}}\right) }{r^{\frac N2}} + \frac{N-1}{\omega_N r^N} \right] \frac{\partial r}{\partial \nu_x}\, \frac{\partial r}{\partial \nu_y}\,u\,ds = \]

\[ = \int_{\Gamma_1^{(1)}} \Phi_1(r,\lambda) \frac{\partial^2 r}{\partial \nu_x\partial \nu_y}\,u\,ds + \int_{\Gamma_1^{(1)}} \Phi_2(r,\lambda) \frac{\partial r}{\partial \nu_x}\, \frac{\partial r}{\partial \nu_y}\,u\,ds. \tag{37} \]

The following estimates hold:

\[ \Phi_1(r,\lambda) = O\left( \lambda r^{3-N}\ln\frac r2\sqrt{\frac{\lambda}{c_1}} \right), \tag{38} \]

\[ \Phi_2(r,\lambda) = O\left( \lambda r^{2-N}\ln\frac r2\sqrt{\frac{\lambda}{c_1}} \right). \tag{39} \]

They can be obtained by using the expansion of the Neumann function in a series. From (38) and (39) it follows that the first integral on the right-hand side of (37) does not exceed \(M_8\alpha|\ln\alpha|\lambda^{-\frac12}\max|u|_{\Gamma_1}\), and the second does not exceed \(M_9\lambda^{\frac N4-\frac12}\). Thus,

\[ I_5^{(1)} = 2\frac{\partial}{\partial \nu_x} \int_{\Gamma_1^{(1)}} \frac{1}{\omega_N r^{N-1}} \frac{\partial r}{\partial \nu_y}\,u\,ds + O\left( \alpha|\ln\alpha|\lambda^{-\frac12}\max|u|_{\Gamma_1} + \lambda^{\frac N4-\frac12} \right), \tag{40} \]

and, combining (32) and (40), we have

\[ \left. \frac{\partial u}{\partial \nu} \right|_{\Gamma_1-0} = 2\frac{\partial}{\partial \nu_x} \int_{\Gamma_1^{(1)}} \frac{1}{\omega_N r^{N-1}} \frac{\partial r}{\partial \nu_y}\,u\,ds + \]

\[ + O\left( \lambda^{\frac{N+1}{4}} + \alpha|\ln\alpha|\lambda^{-\frac12}\max|u|_{\Gamma_1} \right). \tag{41} \]

Similarly, if one represents \(\left.\dfrac{\partial u}{\partial \nu}\right|_{\Gamma_1+0}\) in the form

\[ \left. \frac{\partial u}{\partial \nu} \right|_{\Gamma_1+0} = -2\chi_N\left(\frac{\lambda}{c_2}\right)^{\frac N4-\frac12} \int_{\Gamma} \frac{\partial}{\partial \nu_x} \left[ \frac{ N_{\frac{N-2}{2}}\left(r\sqrt{\frac{\lambda}{c_2}}\right) }{r^{\frac{N-2}{2}}} \right] \frac{\partial u}{\partial \nu}\,ds + \]

\[ + 2\chi_N\left(\frac{\lambda}{c_2}\right)^{\frac N4-\frac12} \int_{\Gamma_1} \frac{\partial}{\partial \nu_x} \left[ \frac{ N_{\frac{N-2}{2}}\left(r\sqrt{\frac{\lambda}{c_2}}\right) }{r^{\frac{N-2}{2}}} \right] \left. \frac{\partial u}{\partial \nu} \right|_{\Gamma_1+0} \,ds - \]

\[ - 2\chi_N\left(\frac{\lambda}{c_2}\right)^{\frac N4-\frac12} \frac{\partial}{\partial \nu_x} \int_{\Gamma_1} \frac{\partial}{\partial \nu_y} \left[ \frac{ N_{\frac{N-2}{2}}\left(r\sqrt{\frac{\lambda}{c_2}}\right) }{r^{\frac{N-2}{2}}} \right] u\,ds, \tag{42} \]

then one can obtain

\[ -\left.\frac{\partial u}{\partial \nu}\right|_{\Gamma_1+0} = -2\frac{\partial}{\partial \nu_x} \int_{\Gamma_1^{(1)}} \frac{1}{\omega_N r^{N-1}} \frac{\partial r}{\partial \nu_y}\,u\,ds + O\!\left(\lambda^{\frac{N+1}{4}}+\alpha|\ln\alpha|\lambda^{\frac12}\max |u|_{\Gamma_1}\right). \tag{43} \]

In considering the first integral on the right-hand side of (42), it should be taken into account that the distance between \(\Gamma_1\) and \(\Gamma\) is nonzero. Therefore, for sufficiently large \(\lambda\) it is estimated in the same way as the integrals over \(\Gamma_1^{(3)}\).

Now from (3), (41), and (43) we have

\[ \left.\frac{\partial u}{\partial \nu}\right|_{\Gamma_1-0} = O\!\left(\lambda^{\frac{N+1}{4}}+\alpha|\ln\alpha|\lambda^{\frac12}\max |u|_{\Gamma_1}\right). \tag{44} \]

From these arguments, with the obvious simplification caused by the fact that \(u(x)|_{\Gamma}=0\), we obtain the estimate

\[ \left.\frac{\partial u}{\partial \nu}\right|_{\Gamma} = O\!\left(\lambda^{\frac{N+1}{4}}\right). \tag{45} \]

\(2^\circ\). To estimate \(u|_{\Gamma_1}\), it is necessary to use the representation of the function \(u(x)\) in the form

\[ u(x)|_{\Gamma_1} = -2\chi_N\left(\frac{\lambda}{c_1}\right)^{\frac{N}{4}-\frac12} \int_{\Gamma_1} \frac{ N_{\frac{N-2}{2}}\!\left(r\sqrt{\frac{\lambda}{c_1}}\right) }{ r^{\frac{N-2}{2}} } \left.\frac{\partial u}{\partial \nu}\right|_{\Gamma_1-0} \,ds - \]

\[ -2\chi_N\left(\frac{\lambda}{c_1}\right)^{\frac{N}{4}} \int_{\Gamma_1} \frac{ N_{\frac{N}{2}}\!\left(r\sqrt{\frac{\lambda}{c_1}}\right) }{ r^{\frac{N-2}{2}} } \frac{\partial r}{\partial \nu_y}\,u\,ds. \]

Then, using the same method as in obtaining the estimate for \(I_4\) in the preceding item, we have

\[ \max |u|_{\Gamma_1} \le M_{10}\alpha\lambda^{-\frac12} \left|\left.\frac{\partial u}{\partial \nu}\right|_{\Gamma_1-0}\right| + M_{11}\lambda^{\frac{N-1}{4}}\sqrt{\ln\lambda}, \]

and, substituting (44) here, we obtain

\[ \max |u|_{\Gamma_1} \le M_{12}\left( \lambda^{\frac{N-1}{4}}\sqrt{\ln\lambda} + \alpha^2|\ln\alpha|\max |u|_{\Gamma_1} \right). \]

Then, choosing \(\alpha\) so that \(\alpha^2|\ln\alpha|<\frac12\), we see that

\[ u|_{\Gamma_1} = O\!\left(\lambda^{\frac{N-1}{4}}\sqrt{\ln\lambda}\right). \tag{46} \]

Combining (44) and (46):

\[ \left.\frac{\partial u}{\partial \nu}\right|_{\Gamma_1-0} = O\!\left(\lambda^{\frac{N+1}{4}}\sqrt{\ln\lambda}\right). \tag{47} \]

\(3^\circ\). Finally, let us estimate \(u(x)\) when \(x \in G_1\) or \(G_2\). Let first \(x \in G_1\). Then, using the representation of \(u(x)\) in the form (23), one can write:

\[ \begin{aligned} u(x)={}&-\chi_N\left(\frac{\lambda}{c_1}\right)^{\frac N4-\frac12} \int_{\Gamma_1} \frac{N_{\frac{N-2}{2}}\left(r\sqrt{\frac{\lambda}{c_1}}\right)} {r^{\frac{N-2}{2}}} \left.\frac{\partial u}{\partial \nu}\right|_{\Gamma_1-0}\,ds+ \\ &+\int_{\Gamma_1}\left[ -\chi_N\left(\frac{\lambda}{c_1}\right)^{\frac N4} \frac{N_{\frac N2}\left(r\sqrt{\frac{\lambda}{c_1}}\right)} {r^{\frac{N-2}{2}}} -\frac{1}{\omega_N r^{N-1}} \right]\frac{\partial r}{\partial \nu_y}\,u\,ds+ \\ &+\int_{\Gamma_1}\frac{1}{\omega_N r^{N-1}}\frac{\partial r}{\partial \nu_y}\,u\,ds \\ ={}&-\chi_N\left(\frac{\lambda}{c_1}\right)^{\frac N4-\frac12} \int_{\Gamma_1} \frac{N_{\frac{N-2}{2}}\left(r\sqrt{\frac{\lambda}{c_1}}\right)} {r^{\frac{N-2}{2}}} \left.\frac{\partial u}{\partial \nu}\right|_{\Gamma_1-0}\,ds+ \\ &+\int_{\Gamma_1}\Phi_1(r,\lambda)\frac{\partial r}{\partial \nu_y}\,u\,ds +\int_{\Gamma_1}\frac{1}{\omega_N r^{N-1}}\frac{\partial r}{\partial \nu_y}\,u\,ds . \end{aligned} \tag{48} \]

The last integral on the right-hand side of (48) is a double-layer potential with density \(u(x)\), and therefore is estimated in terms of the maximum of the modulus of the density ([5]), i.e., it is
\(O(|u|_{\Gamma_1})\), or
\(O\!\left(\lambda^{\frac{N-1}{4}}\sqrt{\ln\lambda}\right)\).
As for the first two integrals on the right-hand side of (48), to estimate them it is necessary to introduce a coordinate system in the following way: from the point under consideration \(x\in G_1\) we drop to \(\Gamma_1\) the shortest normal \(\nu\); the origin of coordinates is placed at the point of intersection of \(\nu\) with \(\Gamma_1\); the \(x_N\)-axis is directed along \(\nu\). As above, we divide the surface \(\Gamma_1\) into three parts:
\(\Gamma_1^{(1)}\) is the set of points of \(\Gamma_1\) for which
\(r<1/\sqrt{\lambda}\);
\(\Gamma_1^{(2)}\) is the set of points of \(\Gamma_1\) for which
\(1/\sqrt{\lambda}\le r\le R\),
where \(R\) is the radius of the Lyapunov sphere, and
\(\Gamma_1^{(3)}=\Gamma_1-\Gamma_1^{(1)}-\Gamma_1^{(2)}\).
In contrast to the case when \(x\in\Gamma\), \(\Gamma_1^{(1)}\) or
\(\Gamma_1^{(1)}+\Gamma_1^{(2)}\) may be empty sets. Using these remarks, the integrals under consideration can be estimated in the same way as we did with \(I_4\). Thus, if \(x\in G_1\), then
\(u=O\!\left(\lambda^{\frac{N-1}{4}}\sqrt{\ln\lambda}\right)\).
If \(x\in G_2\), then the same estimate is valid for \(u(x)\). In this case we start from the equality

\[ \begin{aligned} u(x)={}&-\chi_N\left(\frac{\lambda}{c_2}\right)^{\frac N2-\frac14} \int_{\Gamma} \frac{N_{\frac{N-2}{2}}\left(r\sqrt{\frac{\lambda}{c_2}}\right)} {r^{\frac{N-2}{2}}} \frac{\partial u}{\partial \nu}\,ds +\chi_N\left(\frac{\lambda}{c_2}\right)^{\frac N2-\frac14}\times \\ &\times \int_{\Gamma_1} \frac{N_{\frac{N-2}{2}}\left(r\sqrt{\frac{\lambda}{c_2}}\right)} {r^{\frac{N-2}{2}}} \left.\frac{\partial u}{\partial \nu}\right|_{\Gamma_1+0}\,ds- \\ &-\chi_N\left(\frac{\lambda}{c_2}\right)^{\frac N2-\frac14} \int_{\Gamma_1} \frac{\partial}{\partial \nu_y} \left[ \frac{N_{\frac{N-2}{2}}\left(r\sqrt{\frac{\lambda}{c_2}}\right)} {r^{\frac{N-2}{2}}} \right]u\,ds . \end{aligned} \]

Thus, estimate (6) is proved.

§ 3. ESTIMATE OF EIGENFUNCTIONS FOR THE TWO-DIMENSIONAL CASE

1°. In proving (5) we shall follow the method proposed in [3] for proving the same estimate in the case of continuous coefficients.

Let \(x \in G_1\). Describe a circle of radius \(a\) with center at \(x\). Write for \(u(x)\) the mean-value formula ([6]):

\[ \int_{\Gamma K(x,a)} u\,ds = 2\pi a u(x) J_0\left(a\sqrt{\frac{\lambda}{c_1}}\right), \tag{49} \]

where \(\Gamma K(x,a)\) is the circle of radius \(a\) with center at \(x\). If the point \(x\) is farther from \(\Gamma_1\) than \(2a_0 \sqrt{\dfrac{c_1}{\lambda}}\), where \(a_0\) is the first zero of the Bessel function, then one may choose \(a=a_0 \sqrt{\dfrac{c_1}{\lambda}}\). Then (49) becomes

\[ \int_{\Gamma K\left(x,a_0\sqrt{\frac{c_1}{\lambda}}\right)} u\,ds = 0. \]

Hence it is clear that on the circle \(\Gamma K\left(x,a_0\sqrt{\dfrac{c_1}{\lambda}}\right)\) there exists at least one point \(x_1\) at which \(u(x)=0\). From Lagrange’s theorem it follows that

\[ |u(x)|=|u(x)-u(x_1)| = \left|\frac{\partial u(\bar{x})}{\partial l}\right| a_0\sqrt{\frac{c_1}{\lambda}}, \tag{50} \]

where \(\bar{x}\) is a point lying on the radius \(xx_1\), and \(l\) is the direction of the radius \(xx_1\). From (50) we have

\[ |u(x)| \le |\operatorname{grad} u(\bar{x})|\, a_0\sqrt{\frac{c_1}{\lambda}}, \tag{51} \]

and the distance of the point \(\bar{x}\) from \(\Gamma_1\) is greater than \(a_0 \sqrt{\dfrac{c_1}{\lambda}}\). From (51) it is clear that, in order to obtain the desired estimate for
\(r_{x\Gamma_1}>2a_0\sqrt{\dfrac{c_1}{\lambda}}\), it is necessary for us to have

\[ |\operatorname{grad} u(\bar{x})| = O\left(\lambda^{\frac{N+1}{4}}\right) \quad\text{for}\quad r_{\bar{x}\Gamma_1}>a_0\sqrt{\frac{c_1}{\lambda}}. \tag{52} \]

If we show that the derivatives of \(u\) in two mutually perpendicular directions are
\(O\left(\lambda^{\frac{N+1}{4}}\right)\) for
\(r_{\bar{x}\Gamma_1}>a_0\sqrt{\dfrac{c_1}{\lambda}}\), then in this way we shall obtain (52).

Let one of the directions coincide with the direction of the shortest normal to \(\Gamma_1\) passing through the point \(\bar{x}\). Denote the derivative in this direction by \(\dfrac{\partial u}{\partial \nu}\), and the derivative in the direction perpendicular to it by \(\dfrac{\partial u}{\partial t}\).

Next we must obtain an estimate of \(u(x)\) for \(r_{x\Gamma_1}\le 2a_0\sqrt{\dfrac{c_1}{\lambda}}\). To this end, through the point \(x\) we draw the shortest normal to \(\Gamma_1\). Consider a point \(x_2\) lying on this normal and such that
\[ 2a_0\sqrt{\frac{c_1}{\lambda}}<r_{x\Gamma_1}<3a_0\sqrt{\frac{c_1}{\lambda}}, \]
i.e. \(x_2\) lies in the region where estimate (5) has already been proved. Now, applying Lagrange’s theorem, we obtain
\[ |u(x)|\le |u(x_2)|+\left|\frac{\partial u}{\partial \nu}(\bar x)\right|\,3a_0\sqrt{\frac{c_1}{\lambda}}, \]
where \(\bar x\) is a point on the segment \(xx_2\). Hence it is clear that we need the estimate
\[ \frac{\partial u}{\partial \nu}(\bar x)=O\left(\lambda^{\frac{N+1}{4}}\right) \]
for \(r_{\bar x\Gamma_1}<2a_0\sqrt{\dfrac{c_1}{\lambda}}\). Summarizing what has been said, we arrive at the conclusion that, in order to obtain estimate (5) in the closed region \(G_1+\Gamma_1\), it is necessary for us to have
\[ \frac{\partial u}{\partial \nu}(x)=O\left(\lambda^{\frac{N+1}{4}}\right), \tag{53} \]
in the closed region \(G_1+\Gamma_1\) (\(\nu\) is the direction of the shortest normal to \(\Gamma_1\) passing through \(x\)) and
\[ \frac{\partial u}{\partial t}(x)=O\left(\lambda^{\frac{N+1}{4}}\right) \tag{54} \]
for
\[ r_{x\Gamma_1}>a_0\sqrt{\frac{c_1}{\lambda}} \]
(\(t\) is a direction perpendicular to \(\nu\)).

In order to estimate the function \(u(x)\) in the closed region \(G_2+\Gamma_1+\Gamma\), it is necessary to carry out analogous arguments. The presence of the boundary \(\Gamma\) essentially does not change the matter. Therefore we shall consider only the case \(x\in G_1+\Gamma_1\).

\(2^\circ\). To estimate
\[ \frac{\partial u}{\partial \nu}(x) \]
in the closed region \(G_1+\Gamma_1\), it is necessary to estimate
\[ \frac{\partial u}{\partial \nu}(x) \]
on \(\Gamma_1\) and
\[ \frac{\partial u}{\partial \nu}(x) \]
inside \(G_1\). The obtaining of an estimate for
\[ \left.\frac{\partial u}{\partial \nu}(x)\right|_{\Gamma_1-0} \]
in the two-dimensional case almost coincides with the obtaining of an estimate for
\[ \left.\frac{\partial u}{\partial \nu}(x)\right|_{\Gamma_1-0} \]
in the \(N\)-dimensional case \((N\ge 3)\). The difference consists only in the estimate of the integral
\[ \int_{\Gamma_1^{(1)}} \Phi_1(r,\lambda)\,\frac{\partial^2 r}{\partial \nu_x\,\partial \nu_y}\,u\,ds \]
(see (37)). We now estimate it as follows. We substitute (34) and (38) into this integral, then apply Bunyakovsky’s inequality and estimate (12); hence it follows that
\[ \left| \int_{\Gamma_1^{(1)}} \Phi_1(r,\lambda)\,\frac{\partial^2 r}{\partial \nu_x\,\partial \nu_y}\,u\,ds \right| \le M_{13}\lambda^{\frac{3}{4}}. \]
Thus, the estimate
\[ \left.\frac{\partial u}{\partial \nu}\right|_{\Gamma_1-0} = O\left(\lambda^{\frac{3}{4}}\right). \tag{55} \]

We now proceed to estimate \(\dfrac{\partial u}{\partial \nu}(x)\) inside \(G_1\). We have

\[ \frac{\partial u}{\partial \nu}(x) = -\frac{1}{4} \int_{\Gamma_1} \frac{\partial}{\partial \nu_x} \left[ N_0\left(r\sqrt{\frac{\lambda}{c_1}}\right) \right] \left. \frac{\partial u}{\partial \nu} \right|_{\Gamma_1-0} \,ds + \]

\[ + \frac{1}{4} \int_{\Gamma_1} \frac{\partial^2}{\partial \nu_x \partial \nu_y} \left[ N_0\left(r\sqrt{\frac{\lambda}{c_1}}\right) \right] u\,ds = I_6+I_7 . \tag{56} \]

The proof of the estimate for \(I_7\) coincides word for word with the proof of the estimate of the same integral in the case \(x \in \Gamma_1\). As for \(I_6\), we rewrite it in the form

\[ I_6 = -\frac{1}{4} \sqrt{\frac{\lambda}{c_1}} \int_{\Gamma_1} N_1\left(r\sqrt{\frac{\lambda}{c_1}}\right) \frac{\partial r}{\partial \nu_x} \left. \frac{\partial u}{\partial \nu} \right|_{\Gamma_1-0} \,ds = \]

\[ = \frac{1}{4} \int_{\Gamma_1} \left[ \sqrt{\frac{\lambda}{c_1}} N_1\left(r\sqrt{\frac{\lambda}{c_1}}\right) + \frac{2}{\pi r} \right] \frac{\partial r}{\partial \nu_x} \left. \frac{\partial u}{\partial \nu} \right|_{\Gamma_1-0} \,ds - \]

\[ - \frac{1}{2\pi} \int_{\Gamma_1} \frac{\partial r}{\partial \nu_x} \frac{1}{r} \left. \frac{\partial u}{\partial \nu} \right|_{\Gamma_1-0} \,ds = I_6' + I_6'' . \tag{57} \]

\(I_6''\) can be estimated at once, since it is the derivative of a simple-layer potential and is estimated in terms of the maximum of the modulus of the density [5], i.e. it is \(O\left(\lambda^{3/4}\right)\).

Let us consider \(I_6'\). Suppose first that \(r_{x\Gamma_1}<\dfrac{1}{\sqrt{\lambda}}\). Here one should pay attention to the integral over \(\Gamma_1^{(2)}\), since the integrals over \(\Gamma_1^{(1)}\) and \(\Gamma_1^{(3)}\) are easily estimated. We use the fact that

\[ \left| \frac{\partial r}{\partial \nu_x} \right| \le \frac{r_{x\Gamma_1}+|y_2|}{r} \le \frac{r_{x\Gamma_1}+M_{14}\rho^2}{\rho} = \frac{r_{x\Gamma_1}}{\rho}+M_{14}\rho . \tag{58} \]

Then

\[ \left| \lambda^{1/2} \int_{\Gamma_1^{(2)}} N_1\left(r\sqrt{\frac{\lambda}{c_1}}\right) \frac{\partial r}{\partial \nu_x} \left. \frac{\partial u}{\partial \nu} \right|_{\Gamma_1-0} \,ds \right| \le \]

\[ \le M_{15}\lambda^{1/4} \sqrt{ \int_{\Gamma_1^{(2)}} \left( \left. \frac{\partial u}{\partial \nu} \right|_{\Gamma_1-0} \right)^2 ds } \left[ r_{x\Gamma_1} \sqrt{ \int_{\Gamma_1^{(2)}} \frac{ds}{\rho^3} } + \sqrt{ \int_{\Gamma_1^{(2)}} \rho\,ds } \right] \le \]

\[ \le M_{16}\lambda^{3/4} \left[ \frac{1}{\sqrt{\lambda}} \sqrt{ \int_{1/\sqrt{\lambda}}^{R} \frac{d\rho}{\rho^3} } +1 \right] \le M_{17}\lambda^{3/4}. \tag{59} \]

If \(r_{x\Gamma_1}>\dfrac{1}{\sqrt{\lambda}}\), then \(\Gamma_1^{(1)}\) disappears, while the integrals over \(\Gamma_1^{(3)}\) have already been estima-

remained. We divide the remaining domain of integration into the following two parts: \(\overline{\Gamma}_1^{(1)}\) is the set of points \(y \in \Gamma_1\) for which \(0 \leq \rho \leq r_{x\Gamma_1}\), and \(\overline{\Gamma}_1^{(2)}\) is the set of points \(y \in \Gamma_1\) for which \(r_{x\Gamma_1} < \rho \leq R\) (\(R\) is the radius of the Lyapunov sphere).

Then, since \(r_{xy} \geq r_{x\Gamma_1}\), we have
\[ \left| \lambda^{\frac12} \int\limits_{\Gamma_1^{(1)}} N_1\left(r\sqrt{\frac{\lambda}{c_1}}\right) \frac{\partial r}{\partial \nu_x} \frac{\partial u}{\partial \nu}\bigg|_{\Gamma_1-0} \,ds \right| \leq \]
\[ \leq M_{18}\lambda^{\frac14} r_{x\Gamma_1}^{-\frac12} \int\limits_{\Gamma_1^{(1)}} \left| \frac{\partial u}{\partial \nu}\bigg|_{\Gamma_1-0} \right|\,ds \leq \]
\[ \leq M_{19}\lambda^{\frac14} r_{x\Gamma_1}^{-\frac12} \sqrt{\int\limits_0^{r_{x\Gamma_1}} d\rho}\, \sqrt{ \int\limits_{\Gamma_1^{(1)}} \left( \frac{\partial u}{\partial \nu} \right)^2\bigg|_{\Gamma_1-0} \,ds } \leq M_{20}\lambda^{\frac34}. \tag{60} \]

From (59), analogously to (60), it follows that
\[ \left| \lambda^{\frac12} \int\limits_{\Gamma_1^{(2)}} N_1\left(r\sqrt{\frac{\lambda}{c_1}}\right) \frac{\partial r}{\partial \nu_x} \frac{\partial u}{\partial \nu}\bigg|_{\Gamma_1-0} \,ds \right| \leq \]
\[ \leq M_{21}\lambda^{\frac14} \left[ r_{x\Gamma_1} \sqrt{\int\limits_{\Gamma_1^{(2)}} \frac{ds}{\rho^3}} + \sqrt{\int\limits_{\Gamma_1^{(2)}} \rho\,ds} \right] \times \]
\[ \times \sqrt{ \int\limits_{\Gamma_1^{(2)}} \left( \frac{\partial u}{\partial \nu} \right)^2\bigg|_{\Gamma_1-0} \,ds } \leq M_{22}\lambda^{\frac34}(1+r_{x\Gamma_1}) \leq M_{23}\lambda^{\frac34}. \tag{61} \]

Thus, we have verified that estimate (53) is valid.

\(3^\circ.\) We now proceed to the estimate of \(\dfrac{\partial u}{\partial t}(x)\) for
\[ r_{x\Gamma_1}>a_0\sqrt{\frac{c_1}{\lambda}}. \]
Since
\[ \frac{\partial r}{\partial x_1}=\cos\theta=1-2\sin^2\frac{\theta}{2}, \]
where \(\theta\) is the angle between \(r_{xy}\) and the \(x_1\)-axis, in order to obtain the required estimate we must estimate the quantities:
\[ \lambda^{\frac12} \int\limits_{\Gamma_1} \left| N_1\left(r\sqrt{\frac{\lambda}{c_1}}\right) \frac{\partial u}{\partial \nu}\bigg|_{\Gamma_1-0} \right| \sin^2\frac{\theta}{2}\,ds + \]
\[ + \lambda \int\limits_{\Gamma_1} \left| N_2\left(r\sqrt{\frac{\lambda}{c_1}}\right) \frac{\partial r}{\partial \nu_y} \right| \sin^2\frac{\theta}{2}\,|u|\,ds + \]
\[ + \lambda^{\frac12} \int\limits_{\Gamma_1} \left| N_1\left(r\sqrt{\frac{\lambda}{c_1}}\right) \frac{\partial r}{\partial \nu_x} \frac{\partial r}{\partial \nu_y} \right| \frac{|u|}{r}\,ds + \]
\[ + \lambda^{\frac12} \int\limits_{\Gamma_1} \left| N_1\left(r\sqrt{\frac{\lambda}{c_1}}\right) \frac{\partial^2 r}{\partial \nu_x\partial \nu_y}\,u \right|\,ds, \tag{62} \]

\[ \left| \sqrt{\frac{\lambda}{c_1}}\int_{\Gamma_1} N_1\left(r\sqrt{\frac{\lambda}{c_1}}\right) \left.\frac{\partial u}{\partial \nu}\right|_{\Gamma_1-0}\,ds + \frac{\lambda}{c_1}\int_{\Gamma_1} N_2\left(r\sqrt{\frac{\lambda}{c_1}}\right) \frac{\partial r}{\partial \nu_y}\,u\,ds \right|. \tag{63} \]

Estimates for (62) can be obtained in the same way as the estimate for \(\dfrac{\partial u}{\partial \nu}\) in \(G_1+\Gamma_1\).

From the fact that
\[ N_1\left(r\sqrt{\frac{\lambda}{c_1}}\right) = J_0\left(r\sqrt{\frac{\lambda}{c_1}}\right) + O\left(r^{-\frac32}\lambda^{-\frac34}\right), \]
\[ N_2\left(r\sqrt{\frac{\lambda}{c_1}}\right) = J_1\left(r\sqrt{\frac{\lambda}{c_1}}\right) + O\left(r^{-\frac32}\lambda^{-\frac34}\right) \]
for \(r\sqrt{\dfrac{\lambda}{c_1}}\ge 1\), it follows that
\[ \sqrt{\frac{\lambda}{c_1}}\int_{\Gamma_1} N_1\left(r\sqrt{\frac{\lambda}{c_1}}\right) \left.\frac{\partial u}{\partial \nu}\right|_{\Gamma_1-0}\,ds + \frac{\lambda}{c_1}\int_{\Gamma_1} N_2\left(r\sqrt{\frac{\lambda}{c_1}}\right) \frac{\partial r}{\partial \nu_y}\,u\,ds = \]
\[ = \sqrt{\frac{\lambda}{c_1}}\int_{\Gamma_1} J_0\left(r\sqrt{\frac{\lambda}{c_1}}\right) \left.\frac{\partial u}{\partial \nu}\right|_{\Gamma_1-0}\,ds + \]
\[ + \frac{\lambda}{c_1}\int_{\Gamma_1} J_1\left(r\sqrt{\frac{\lambda}{c_1}}\right) \frac{\partial r}{\partial \nu_y}\,u\,ds + \]
\[ + O\left(\lambda^{-\frac14}\right)\int_{\Gamma_1} \frac{1}{r\sqrt r}\, \left.\left|\frac{\partial u}{\partial \nu}\right|\right|_{\Gamma_1-0}\,ds + O\left(\lambda^{\frac14}\right)\int_{\Gamma_1} \frac{|u|}{r\sqrt r}\,ds . \tag{64} \]

For the last two integrals in (64) the estimate \(O\left(\lambda^{\frac34}\right)\) is valid. To estimate the first two integrals on the right-hand side of (64), we use Green’s second formula, written for \(J_0\left(r\sqrt{\dfrac{\lambda}{c_1}}\right)\) and \(u(x)\) in \(G_1\):
\[ 0= \int_{\Gamma_1} J_0\left(r\sqrt{\frac{\lambda}{c_1}}\right) \left.\frac{\partial u}{\partial \nu}\right|_{\Gamma_1-0}\,ds - \int_{\Gamma_1} \frac{\partial}{\partial \nu_y} \left[ J_0\left(r\sqrt{\frac{\lambda}{c_1}}\right) \right]u\,ds = \]
\[ = \int_{\Gamma_1} J_0\left(r\sqrt{\frac{\lambda}{c_1}}\right) \left.\frac{\partial u}{\partial \nu}\right|_{\Gamma_1-0}\,ds + \sqrt{\frac{\lambda}{c_1}}\int_{\Gamma_1} J_1\left(r\sqrt{\frac{\lambda}{c_1}}\right) \frac{\partial r}{\partial \nu_y}\,u\,ds . \]

Thus (54) is proved.

4°. Carrying out arguments analogous to those used in [1] for proving estimates of the derivatives of eigenfunctions, we obtain (7) and (8) from (5) and (6).

Remark. The results obtained are also valid for the case when in the domain \(G\) there are \(n\) nonintersecting closed surfaces \(\Gamma_i\) of discontinuity of the coefficients, on each of which the conditions

\[ [u]_{\Gamma_i}=0,\qquad \left[\frac{\partial u}{\partial \nu}\right]_{\Gamma_i}=0 \]

are satisfied.

In conclusion, I wish to express my gratitude to I. A. Shishmarev for his attention to the work and for valuable suggestions.

References

1. Shishmarev I. A. DAN SSSR 137, No. 1, 45—47, 1961.
2. Miranda C. Partial Differential Equations of Elliptic Type. IL, 1957.
3. Il’in V. A., Shishmarev I. A. Proceedings of the Joint Soviet-American Symposium on Partial Differential Equations. Novosibirsk, 1963, pp. 3—5.
4. Eidus D. M. DAN SSSR, 107, No. 6, 796—798, 1956.
5. Günter N. M. Potential Theory and Its Applications to the Basic Problems of Mathematical Physics. Moscow, Gostekhizdat, 1953.
6. Courant R. Partial Differential Equations. Moscow, “Mir,” 1964.

Received by the editors
April 22, 1966

Moscow