MATHEMATICS

I. A. SHISHMAREV

UNIFORM ESTIMATES OF DERIVATIVES OF SOLUTIONS OF THE DIRICHLET PROBLEM AND OF THE EIGENFUNCTION PROBLEM FOR THE OPERATOR

\[ Lu=\operatorname{div}(p(x)\operatorname{grad}u)+q(x)\cdot u \]

WITH DISCONTINUOUS COEFFICIENTS

(Presented by Academician I. G. Petrovskii on 14 X 1960)

In the present note uniform estimates are established for derivatives of arbitrary order and their Hölder coefficients for solutions of the Dirichlet problem and of the eigenfunction problem for the operator
\(Lu=\operatorname{div}(p(x)\operatorname{grad}u)+q(x)\cdot u\) with discontinuous coefficients.

1. Let there be given an open \(N\)-dimensional domain \(g\) with boundary manifold \(\Gamma_2\), and inside it an \((N-1)\)-dimensional geometrically closed surface \(\Gamma_1\), dividing the domain \(g\) into subdomains \(g_1\) and \(g_2\).

Consider in the closed domain \((g+\Gamma_2)\) the following Dirichlet problem:
\[ \begin{gathered} L_1u=f_1 \quad \text{in the domain } g_1,\\ L_2u=f_2 \quad \text{in the domain } g_2,\\ [u]_{\Gamma_1}=\varphi,\qquad \left[\frac{\partial u}{\partial \nu}\right]_{\Gamma_1}=\psi,\qquad u|_{\Gamma_2}=\chi . \end{gathered} \tag{1} \]

Here
\[ L_lu=\operatorname{div}(p_l(x)\operatorname{grad}u)+q_l(x)\cdot u \tag{2} \]
is an elliptic operator given in the domain \(g_l\) \((l=1,2)\), \(p_l>0\), \(q_l\leqslant 0\) in \(g_l\);
\[ [u]_{\Gamma_1}\equiv u|_{x\to\Gamma_1-0}-u|_{x\to\Gamma_1+0}; \qquad \left[\frac{\partial u}{\partial \nu}\right]_{\Gamma_1} \equiv \frac{\partial u}{\partial \nu_1}\bigg|_{x\to\Gamma_1-0} + \frac{\partial u}{\partial \nu_2}\bigg|_{x\to\Gamma_1+0}, \]
where \(\dfrac{\partial}{\partial \nu_l}\) denotes differentiation in the direction of the conormal, equal to
\[ p_l\frac{\partial}{\partial n_l} \]
(\(n_l\) is the outward normal for the domain \(g_l\)), and the signs \(\Gamma_1-0\) and \(\Gamma_1+0\) mean that the limiting values are taken respectively from the inner and from the outer side of the surface \(\Gamma_1\) with respect to the domain \(g_1\).

Definition. We shall call a function \(u(x)\) a classical solution of problem (1) if it satisfies the following requirements: 1) \(u(x)\in C^{(0)}\) in each of the closed domains \((g_1+\Gamma_1)\) and \((g_2+\Gamma_1+\Gamma_2)\); 2) \(u(x)\in C^{(1)}\) in the domains \((g_1+\Gamma_1)\) and \((g_2+\Gamma_1)\); 3) \(u(x)\in C^{(2)}\) in the open domains \(g_1\) and \(g_2\); 4) \(u(x)\) satisfies all conditions of problem (1) in the usual classical sense.

Theorem 1. Let the boundary manifolds \(\Gamma_1\) and \(\Gamma_2\) belong to the class \(A^{(n,\mu)}\), and let the coefficients of the operators \(L_1\) and \(L_2\) and the functions \(f_1, f_2, \varphi, \psi\) and \(\chi\) satisfy the requirements:
\(p_l\in C^{(n,\mu)}\), \(q_l\in C^{(n-2,\mu)}\), \(f_l\in C^{(n-2,\mu)}\) in the closed

* The classes \(A^{(n,\mu)}\) and \(C^{(n,\mu)}\) are defined in (1), p. 10.

smooth domain \((g_1+\Gamma_1)\); \(p_2\in C^{(n,\mu)}\), \(q_2\in C^{(n-2,\mu)}\), \(f_2\in C^{(n-2,\mu)}\) in the closed domain \((g_2+\Gamma_1+\Gamma_2)\); \(\varphi\in C^{(n,\mu)}\) and \(\psi\in C^{(n-1,\mu)}\) on the surface \(\Gamma_1\); \(\chi\in C^{(n,\mu)}\) on the surface \(\Gamma_2\) \((n\geqslant 2)\).

Then there exists a unique classical solution of problem (1)—a function \(u(x)\); \(u(x)\) belongs to the class \(C^{(n,\mu)}\) in each of the closed domains \((g_1+\Gamma_1)\) and \((g_2+\Gamma_1+\Gamma_2)\), and in these domains the estimate* holds

\[ u^{(n,\mu)}= O\left\{ F^{(0)}+\varphi^{(0)}+\psi^{(0)}+\chi^{(0)} +\sum_{i=1}^{n}\psi^{(i-1,\mu)} +\sum_{i=0}^{n}\varphi^{(i,\mu)} +\right. \]
\[ \left. +\sum_{i=0}^{n}\chi^{(i,\mu)} +\sum_{i=2}^{n}F^{(i-2,\mu)} \right\}. \tag{3} \]

The constant entering the estimate \(O\) depends on the coefficients of the operators \(L_1\) and \(L_2\) and on the form of the domains \(g_1\) and \(g_2\).

For functions belonging to the class \(C^{(n,\mu)}\), the following estimate is known (see (1), p. 137)

\[ u^{(n)}=O\left\{\left[u^{(n,\mu)}\right]^{\frac{n}{n+\mu}} \left[u^{(0)}\right]^{\frac{\mu}{n+\mu}}+u^{(0)}\right\}. \tag{4} \]

Relying on formulas (3) and (4) and on the estimate of \(u^{(0)}\) obtained in (2), we easily estimate \(u^{(n)}\).

Remark 1. The estimates \(u^{(n,\mu)}\) and \(u^{(n)}\) are uniform with respect to operators \(L_l\) with uniformly bounded values of the quantities \(1/p_l\), \(p_l^{(n,\mu)}\), \(q_l^{(n-2,\mu)}\), \(p_l^{(0)}\), \(q_l^{(0)}\) \((l=1,2)\).

Remark 2. Theorem 1 is readily extended to the case when the operator \(L\) has the form

\[ L_lu=\operatorname{div}(p_l(x)\operatorname{grad}u)+ \sum_{i=1}^{N} b_{li}(x)\frac{\partial u}{\partial x_i} +q_l(x)u. \]

2. The proof of Theorem 1 is carried out as follows. Using the existence theorems proved in (3), we represent the solution of problem (1) in the form of the sum of three functions \(u=v_1+v_2+w\). The functions \(v_1\) and \(v_2\) are solutions of two ordinary Dirichlet problems, and therefore the known results of Schauder and Caccioppoli (see (1)) may be used for their estimates, while \(w(x)\) is the solution of the following problem with discontinuous coefficients:

\[ \begin{aligned} \tilde Lw&=0 && \text{in the domain } g_1,\\ \Delta w&=0 && \text{in the domain } g_2,\\ [w]\big|_{\Gamma_1}&=0,\qquad \left[\frac{\partial w}{\partial \nu}\right]\bigg|_{\Gamma_1}=\theta,\qquad w\big|_{\Gamma_2}=0. \end{aligned} \tag{5} \]

Here \(\Delta\) is the Laplace operator,

\[ \tilde L=\sum_{i=1}^{N}\frac{\partial}{\partial x_i} \left(\frac{p_1}{p_2}\frac{\partial}{\partial x_i}\right) +\frac{q_1}{p_2} \]

(the coefficient \(p_2(x)\) is extended to the whole closed domain \((g+\Gamma_2)\) with preservation of membership in the class \(C^{(n,\mu)}\), which is possible by known results of Gever; see (1), p. 52).

Let \(F(x,y)\) denote the Green’s function of the Dirichlet problem for the Laplace operator in the closed domain \((g+\Gamma_2)\). Define the function \(\tilde F(x,y)\) by means of

\[ \text{* If } z(x)\text{ is some function defined in a domain }T\text{ and belonging there to the class }C^{(n,\mu)},\text{ then by }z^{(k)}\text{ we shall denote the sum of the maxima of the absolute values of all derivatives of order }k\ (k\leq n),\text{ and by }z^{(k,\mu)}\text{ the sum of the Hölder coefficients (taken for exponent }\mu\text{) of these derivatives.} \]

by means of the equality

\[ \widetilde F(x,y)=F(x,y)+\int_{g_1} F(x,t)\,\frac{p_2(t)}{p_1(t)}\,\widetilde L_t F(t,y)\,dt. \tag{6} \]

We shall seek the solution of problem (5) in the form of the sum of a volume potential and a single-layer potential

\[ w(x)=\int_{g_1} F(x,y)\,\mu(y)\,dy+\int_{\Gamma_1}\widetilde F(x,s)\,\nu(s)\,ds \tag{7} \]

with unknown functions \(\mu\) and \(\nu\). Substituting (7) into (5), we arrive at the following system of integral equations:

\[ \begin{aligned} \mu(x)&-\int_{g_1}K_{11}(x,y)\mu(y)\,dy-\int_{\Gamma_1}K_{12}(x,s)\nu(s)\,ds=0,\\ \nu(s)&-\int_{g_1}K_{21}(s,y)\mu(y)\,dy-\int_{\Gamma_1}K_{22}(s,s_1)\nu(s_1)\,ds_1=\theta(s). \end{aligned} \tag{8} \]

Formulas (7) and (8) make it possible to estimate the function \(w(x)\) by means of the known theorems of potential theory \((^4)\).

1. Let us consider the eigenfunction problem for the operator
   \(Lu=\operatorname{div}(p(x)\operatorname{grad}u)+q(x)\cdot u\) with discontinuous coefficients

\[ \begin{gathered} L_1u+\lambda u=0 \quad \text{in the domain } g_1,\\ L_2u+\lambda u=0 \quad \text{in the domain } g_2,\\ [u]_{\Gamma_1}=0,\qquad \left[\frac{\partial u}{\partial \nu}\right]_{\Gamma_1}=0,\qquad u|_{\Gamma_2}=0. \end{gathered} \tag{9} \]

Relying on Theorem 1 and on the estimate of the eigenfunctions of problem (9) established in \((^3)\), we arrive at the following theorem.

Theorem 2. If the surfaces \(\Gamma_1\) and \(\Gamma_2\) belong to the class \(A^{(n,\mu)}\), and the coefficients of the operators \(L_1\) and \(L_2\) satisfy the conditions: \(p_1\in C^{(n,\mu)}\), \(q_1\in C^{(n-2,\mu)}\) in the closed domain \((g_1+\Gamma)\); \(p_2\in C^{(n,\mu)}\), \(q_2\in C^{(n-2,\mu)}\) in the closed domain \((g_2+\Gamma_1+\Gamma_2)\), then the eigenfunctions of problem (9) belong to the class \(C^{(n,\mu)}\) in each of the closed domains \((g_1+\Gamma_1)\) and \((g_2+\Gamma_1+\Gamma_2)\), and for them the following uniform estimates hold in the closed domain \((g+\Gamma_2)\):

\[ u_l^{(k)}=O\!\left(\lambda_l^{\frac{N}{4}+\frac{k}{2}}\right),\qquad u_l^{(k,\mu)}=O\!\left(\lambda_l^{\frac{N}{4}+\frac{k}{2}+\frac{\mu}{2}}\right) \tag{10} \]

(\(l\) is the number of the eigenfunction).

Remark 3. Theorems 1 and 2 are also valid in the case when, inside the surface \(\Gamma_2\), there lie \(m\) surfaces of discontinuity of the coefficients.

I take this opportunity to express my gratitude to V. A. Il’in for his attention to the work and to the participants of the seminar directed by A. N. Tikhonov for useful discussion of the results.

Moscow State University
named after M. V. Lomonosov

Received
13 X 1960

References

1. K. Miranda, Equations with Partial Derivatives of Elliptic Type, IL, 1957.
2. I. A. Shishmarev, DAN, 131, No. 2 (1960).
3. V. A. Il’in, I. A. Shishmarev, DAN, 135, No. 4 (1960).
4. I. N. Günter, Potential Theory, 1953.