Reports of the Academy of Sciences of the USSR

1. Volume 133, No. 6

MATHEMATICS

Z. V. REITBLAT

A MEAN-VALUE THEOREM FOR LINEAR ELLIPTIC EQUATIONS WITH COEFFICIENTS OF THE LIPSCHITZ CLASS

(Presented by Academician S. L. Sobolev on 15 IV 1960)

In the present paper a mean-value theorem is proved for solutions of the equation

\[ \mathfrak{M}u=\sum_{i,k}^{m} a_{ik}u_{ik}+\sum_i b_i u_i+cu=f \tag{1} \]

in an \(m\)-dimensional open domain \(\Omega=\Omega(x)\) with boundary \(S\), under the assumption that\(^*\) \(f\in L_p;\ p>m/2;\ a_{ik}, b_i, c\in C^{(0,\lambda)},\ 0<\lambda\leq 1;\ c<0;\ \sum_{i,k}^{m} a_{ik}t_i t_k>\gamma\sum_i t_i^2,\ \gamma>0.\)

Namely, it is proved that the solution of equation (1), for arbitrary \(y\in\Omega,\ |y-S|>\delta\), satisfies the equality

\[ u(y)=\frac{1}{\varkappa(\delta)}X(y;\delta)+\frac{1}{\varkappa(\delta)} \int_{\Omega} F^{(\delta)}(y,z)u(z)\,d\Omega \tag{2} \]

(the definition of the functions \(\varkappa(\delta), X(y;\delta), F^{(\delta)}(y,z)\) will be given below), and conversely, every function \(u\in L_p\) for which equality (2) is valid is a solution of equation (1).

This was proved by other authors under the assumption that \(a_{ik}\in C^{(2,\lambda)},\lambda\geq 0\), and the concept of the adjoint operator was used (see the bibliography in (1)). In the present paper, the proof uses the maximum principle with the Green’s function for equation (1) for a sphere of small radius.

Let \(y\in\Omega\), and let \(\delta\) be so small that the sphere \(\Omega_\delta(y)\{|x-y|<\delta\}\) lies entirely in \(\Omega\) together with its boundary \(S_\delta(y)\). Let \(F(x,z;y,\delta)\), \(x,z\in\Omega_\delta(y)\), be the Green’s function for solving the Dirichlet problem for equation (1) in the domain \(\Omega_\delta(y)\), so that the solution of this problem in the domain \(\Omega_\delta(y)\) is given by the formula \((u\in C(S_\delta), f\in L_p(\Omega_\delta))\)

\[ u(x)=-\int_{\Omega_\delta(y)} F(x,z;y,\delta)f(z)\,d\Omega_z -\int_{S_\delta(y)} \mathfrak{D}_z F(x,z;y,\delta)u(z)\,dS_z . \tag{3} \]

Put \(\delta=\rho,\ x=y\) in (3), multiply both sides of the resulting equality by \(\varphi_\delta(\rho)\rho^{m-1}\), where
\[ \varphi_\delta(\rho)=\exp\left(-\frac{1}{\delta^2-\rho^2}\right) \]
for \(\rho<\delta\) and \(\varphi_\delta(\rho)=0\) for \(\rho\geq\delta\), and integrate with respect to \(\rho\) from 0 to \(\delta\). We obtain

\[ \varkappa(\delta)u(y)=X(y;\delta)+ \int_{\Omega_\delta(y)} F^{(\delta)}(y,z)u(z)\,d\Omega_z, \tag{2_1} \]

\(^*\) The notation of the monograph (1) is used.

where

\[ \varkappa(\delta)=\int_{0}^{\delta}\varphi_{\delta}(\rho)\rho^{m-1}\,d\rho; \tag{2_2} \]

\[ X(y;\delta)=-\int_{0}^{\delta}\varphi_{\delta}(\rho)\rho^{m-1} \int_{\Omega_{\rho}(y)} F(y,z;y,\rho)f(z)\,d\Omega_z\,d\rho; \tag{2_3} \]

\[ F^{(\delta)}(y,z)=-\varphi_{\delta}(\rho)\rho^{m-1}\mathfrak D_zF(y,z;y,\rho). \tag{2_4} \]

Here in formula \((2_4)\) \(\rho=\rho(z)=|z-y|\), and \(\mathfrak D_z\) means that the operator \(\mathfrak D\) is applied to \(F\) as a function of \(z\) on the surface of the sphere \(|z-y|=\rho\). The integral over \(\Omega_{\delta}(y)\) in formula \((2_1)\) may be regarded as extended over \(\Omega\), since \(F^{(\delta)}\equiv 0\) for \(z\notin \Omega_{\delta}(y)\).

Lemma 1. If \(u\in L_p(\Omega)\), \(\psi(y)=\int_{\Omega}F^{(\delta)}(y,z)u(z)\,d\Omega_z\), then \(\psi(y)\in C(\widetilde\Omega_{\delta})\),

where \(\widetilde\Omega_{\delta}\subset\Omega,\ |\widetilde\Omega_{\delta}-S|>\delta\).

Lemma 2. If \(f\in L_p(\Omega)\), then \(X(y;\delta)\in C(\widetilde\Omega_{\delta})\).

The proof of Lemmas 1 and 2 is based on the properties of integral equations for the construction of the Green function \(({}^{1-3})\).

Lemma 3. If \(u\in C(\Omega)\) and, for all \(y\) and all sufficiently small \(\delta\) such that \(\Omega_{\delta}(y)\subset\Omega\), the equality

\[ u(y)=\frac{1}{\varkappa(\delta)} \int_{\Omega_{\delta}(y)}F^{(\delta)}(y,z)u(z)\,d\Omega_z, \tag{4} \]

holds, then \(u(x)\) has in \(\Omega\) neither a positive maximum nor a negative minimum.

Proof. 1. We have \(-\mathfrak D_zF(y,z;y,\rho)>0\), for otherwise \(u(x)\)—the solution of the Dirichlet problem for equation (1) with \(f=0\), equal to zero on \(S_{\rho}(y)\) everywhere except for a small neighborhood of a point \(z\in S_{\rho}(y)\), in which \(u>0\)—would be negative at the center of the sphere \(\Omega_{\rho}(y)\), which is impossible.

1. Next, if \(v(x)\) is the solution of the Dirichlet problem for equation (1) with \(f=0\), \(v|_{S_{\rho}(y)}=1\), then \(0<v(y)<1\), and therefore, according to (3),

\[ 0<-\int_{S_{\rho}(y)}\mathfrak D_zF(y,z;y,\rho)\,dS_z<1. \]

Multiplying this inequality by \(\varphi_{\rho}(\rho)\rho^{m-1}\) and integrating with respect to \(\rho\) from \(0\) to \(\delta\), we obtain

\[ 0<\frac{1}{\varkappa(\delta)} \int_{\Omega_{\delta}(y)}F^{(\delta)}(y,z)\,d\Omega_z<1. \tag{5} \]

1. Let \(u(x)\) satisfy the conditions of the lemma. Suppose that \(u(x)\) attains a maximum at \(x=y\), \(u(y)>0\). Then, for sufficiently small \(\delta\), one has \(u(z)<u(y)\), \(z\in\Omega_{\delta}(y)\), \(z\ne y\), and, by virtue of inequality (5),

\[ \frac{1}{\varkappa(\delta)} \int_{\Omega_{\delta}(y)}F^{(\delta)}(y,z)u(z)\,d\Omega_z < \frac{1}{\varkappa(\delta)} \int_{\Omega_{\delta}(y)}F^{(\delta)}(y,z)u(y)\,d\Omega_z <u(y), \]

which contradicts identity (4). The absence of negative minima of the function \(u(x)\) in \(\Omega\) is proved analogously.

Lemmas 1–3 make it possible to prove the following theorem.

Theorem (mean-value theorem). If \(u(x)\) is a solution of equation (1), then for all \(y\in\Omega\) and all \(\delta<|y-S|\) equality (2) is satisfied. Convers-

but if \(u\in L_p\) and for all \(y\in\Omega\), for all \(\delta<\delta_1/2\), where \(\delta_1<|y-S|\) is arbitrary, equality (2) holds, then \(u\) is a solution of equation (1).

Proof. The first part of the theorem is obvious; let us prove the second. Thus, let \(u(x)\) satisfy the conditions of the theorem. Let \(\widetilde{\Omega}_{\delta_1}\subset\Omega\) be such a subdomain of \(\Omega\) that \(|\widetilde{\Omega}_{\delta_1}-S|>\delta_1\). Then, by Lemmas 1 and 2, we obtain \(u\in C(\widetilde{\Omega}_{\delta_1})\). We may assume that \(\widetilde{S}_{\delta_1}\), the boundary of \(\widetilde{\Omega}_{\delta_1}\), belongs to the class \(A^{(2,\lambda)}\). Construct \(\bar u\), a solution of equation (1), such that \((\bar u-u)|_{\widetilde{S}_{\delta_1}}=0\). Then \(\bar u\) satisfies equality (2), and the difference \(\bar u-u\) satisfies equality (4). But then, by Lemma 3, from the fact that \((\bar u-u)|_{\widetilde{S}_{\delta_1}}=0\) it follows that \(\bar u-u=0\) in \(\Omega\). Since \(\delta_1\) is arbitrary, \(u(x)\) is a solution of equation (1) in the whole domain \(\Omega\), as was required to prove.

Corollary 1. The totality of solutions of equation (1) for \(f=0\) forms a subspace in \(L_p\).

Corollary 2. If a sequence \(u_n\in L_p\cap D_{\mathfrak M}\) is such that \(\|u_n-u\|_{L_p}\to0\), \(\|\mathfrak M u_n-f\|_{L_p}\xrightarrow[n\to\infty]{}0\), then \(u\) is a solution of equation (1).

The validity of the corollaries follows from the fact that, in equality (2), written for \(u_n\), passage to the limit as \(n\to\infty\) under the integral sign is permissible.

In conclusion I express my gratitude to Yu. G. Reshetnyak for advice in writing the present article.

REFERENCES

1. K. Miranda, Equations with Partial Derivatives of Elliptic Type, IL, 1957.
2. G. Giraud, C. R., 202, 380 (1936).
3. G. Giraud, Bull. Sci. Math., 61, 172 (1937).