MATHEMATICS

A. I. KOSHELEV

ON THE DIFFERENTIABILITY OF SOLUTIONS OF ELLIPTIC DIFFERENTIAL EQUATIONS

(Presented by Academician S. L. Sobolev on 22 IX 1956)

Let (\Omega) be an open bounded domain of variation of the variable (x(x_1,x_2,\ldots,x_n)), and let (\Gamma) be the boundary of this domain. We shall require that the function defining (\Gamma) in local coordinates be twice continuously differentiable.

Consider the elliptic differential equation

[
L(u)=\sum_{i,k=1}^{n}\frac{\partial}{\partial x_i}
\left(a_{ik}(x)\frac{\partial u}{\partial x_k}\right)=f(x),
\tag{1}
]

which is to be integrated inside (\Omega) under the homogeneous boundary condition

[
u\big|_{\Gamma}=0.
\tag{2}
]

We assume that the coefficients (a_{ik}(x)) are twice continuously differentiable in the closed domain (\Omega+\Gamma), and that the quadratic form composed of the coefficients of equation (1) satisfies the inequality

[
\sum_{i,k=1}^{n} a_{ik}\xi_i\xi_k \ge \mu \sum_{i=1}^{n}\xi_i^2,
]

where (\mu) is a positive constant.

It is known ((^1)) that if (f) satisfies a Lipschitz condition with exponent (\alpha) ((0<\alpha<1)) in (\Omega+\Gamma), then the second derivatives of the solution of problem (1)—(2) also possess this property. In the work ((^2)) of O. A. Ladyzhenskaya it was proved that if (f\in L_2(\Omega)), then the second generalized (in the sense of S. L. Sobolev) derivatives of the solution belong to (L_2(\Omega)).* We are interested in the analogous question for the case (f\in L_p(\Omega)) ((p>n/2)).

By a generalized solution of problem (1)—(2) we shall mean a function (u\in W_p^{(2)}(\Omega)) which: 1) almost everywhere in (\Omega) satisfies equation (1), and 2) almost everywhere on the boundary (\Gamma) vanishes.

Lemma 1. The linear elliptic equation with constant coefficients

[
\sum_{i,k=1}^{n}\bar a_{ik}\frac{\partial^2 u}{\partial x_i\partial x_k}
+\sum_{j=1}^{n}\bar b_j\frac{\partial u}{\partial x_j}
=f,\quad f\in L_p(\Omega)\quad (p>1)
\tag{3}
]

has a generalized solution which vanishes on the ellipsoid

[
\sum_{i,k=1}^{n}\bar a_{ik}x_ix_k=r^2 \quad (r\text{ constant}).
]

* Following S. L. Sobolev ((^3)), spaces of functions having generalized derivatives of order (l) in (\Omega), summable to the power (p), will be denoted below by (W_p^{(l)}). The notation of the norms corresponds to the notation in ((^3)).

Proof. The equation under consideration, by means of a suitable change of variables and of the function, can be reduced to the canonical form

[
\Delta u-a^{2}u=f .
]

The existence of a solution of the latter equation which vanishes on the sphere of radius (r) can be proved in exactly the same way as in ((^{4})), by means of the transformation indicated by S. G. Mikhlin ((^{5})). Moreover, for a solution of equation (3) satisfying the indicated boundary condition, the estimate

[
|u|{W \leq C|f|}^{(2)}(\Omega){L}(\Omega)
]

will hold, where (C) is a constant independent of (u) and (f).

Lemma 2. If there exists a continuous solution (u(x)) of problem (1)—(2) and (f \in L_{p}(\Omega)) for (p>n/2), then the estimate

[
|u|{C} \leq A|f|,}
]

holds, where (C) is a constant independent of (u), (f).

Proof. As is known ((^{6})), the solution (u(x)) can be written in the form

[
u(x)=\int_{\Omega}\Gamma(x,\xi)f(\xi)\,d\Omega,
]

where the integration is performed with respect to the variable point (\xi). The function (\Gamma(x,\xi)) has a singularity at (x=\xi) of order (\rho^{2-n}), where (\rho) is the distance between (x) and (\xi). By S. L. Sobolev’s theorem on potentials ((^{3})), the function

[
\varphi(x)=\int_{\Omega} f(\xi)\frac{d\Omega}{\rho^{\lambda}}
]

will be continuous for (\lambdan/2).

Lemma 3. If (u(x)) is a solution of problem (1)—(2) twice continuously differentiable inside (\Omega), and (f) is a continuous function, then for (p>n/2) the inequality

[
\int_{\Omega'}\left|\frac{\partial^{2}u}{\partial x_{i}\partial x_{k}}\right|^{p}\,d\Omega
\leq
B\int_{\Omega}|f|^{p}\,d\Omega
\qquad (i,k=1,2,\ldots,n),
]

holds, where (B) is a constant independent of (u,f), and (\Omega') is any interior subdomain of the domain (\Omega).

Proof. It suffices to prove the lemma for a rectangular parallelepiped entirely contained inside (\Omega'). Take a point (P_{0}) lying inside (\Omega'), and construct a cube with edge (\delta) and center (P_{0}). Choose (\delta) so that the lateral surface of the cube is at a distance greater than (d/2) from (\Gamma). Consider the ellipsoid

[
\sum_{i,k=1}^{n} a_{ik}(P_{0})(x_{i}-x_{i}^{0})(x_{k}-x_{k}^{0})=r^{2}
\qquad
(P_{0}=P_{0}(x_{1}^{0},\ldots,x_{n}^{0})).
]

Since the eigenvalues of the matrix (|a_{ik}|) are strictly positive, the number (r) can be chosen so that the ellipsoid (T_{0}), lying outside (Q_{0}) and inside (\Omega'), has small diameter when (\delta) is small.

By the uniqueness of the solution of problem (1)—(2), the function (u(x)) will satisfy equation (1) inside the ellipsoid, and on the boundary (T_{0}) the condition (u|{T=u). The function (u) can be decomposed into two summands}

[
u=v+w,
\tag{4}
]

where

[
L(v)=0,\quad v|{T=u;\qquad L(w)=f,\quad w|}{T=0.}
\tag{5}
]

The function (v(x)), as a solution of the homogeneous equation (1), can be represented in the form of the surface integral (6)

[
v(x)=\int_{T_0} R(x,\xi)u(\xi)\,d\xi,
]

where (\xi) lies on (T_0), and (R) is continuous for (x\ne \xi). Therefore

[
\int_{Q_0}\left|\frac{\partial^2 v}{\partial x_i\partial x_k}\right|^p d\Omega
\le C\max_{x\in\Omega}|u(x)|.
\tag{6}
]

Equation (1) for the function (w(x)) can be rewritten as follows:

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

Consider inside (T_0) the approximate equation with constant coefficients

[
\sum_{i,k=1}^{n} a_{ik}(P_0)\frac{\partial^2 \widetilde w}{\partial x_i\partial x_k}
+
\sum_{i,k=1}^{n}\frac{\partial a_{ik}(P_0)}{\partial x_i}\frac{\partial \widetilde w}{\partial x_k}
=f(x)
]

and the same boundary condition for (\widetilde w). The function (\widetilde w(x)), which vanishes on (T_0), satisfies the conditions of Lemma 1. Therefore

[
\int_{\widetilde K_0}\left|\frac{\partial^2\widetilde w}{\partial x_i\partial x_k}\right|^p d\Omega
\le C\int_{\widetilde K_0}|f|^p d\Omega,
]

where (K_0) is the interior of the ellipsoid (T_0). Since the diameter of (K_0) is small together with (\delta), and the coefficients are continuously differentiable, it follows that

[
\int_{K_0}\left|\frac{\partial^2 w}{\partial x_i\partial x_k}\right|^p d\Omega
\le C\int_{K_0}|f|^p d\Omega.
]

Since (Q_0\subset K_0), strengthening the inequality, we obtain

[
\int_{Q_0}\left|\frac{\partial^2 w}{\partial x_i\partial x_k}\right|^p d\Omega
\le C\int_{\Omega}|f|^p d\Omega.
\tag{7}
]

Inequalities (6) and (7) give

[
\int_{Q_0}\left|\frac{\partial^2 u}{\partial x_i\partial x_k}\right|^p d\Omega
\le
C\left(\int_{\Omega}|f|^p d\Omega+\max_{x\in\overline{\Omega}}|u(x)|\right).
]

Hence, and from Lemma 2, for (p>n/2) we obtain the inequality

[
\int_{Q_0}\left|\frac{\partial^2 u}{\partial x_i\partial x_k}\right|^p d\Omega
\le C\int_{\Omega}|f|^p d\Omega,
]

which was to be proved.

We shall say that a domain (G'\subset G) rests on (\Gamma) if the intersection of the boundary of (G') with (\Gamma) is an ((n-1))-dimensional manifold of nonzero measure.

Lemma 4. Suppose that the conditions of Lemma 3 are satisfied. Then

[
\int_{G'}\left|\frac{\partial^2 u}{\partial x_i\partial x_k}\right|^p d\Omega
\le B\int_{\Omega}|f|^p d\Omega,
]

where (G') rests on (\Gamma).

Proof. Without loss of generality, one may assume that the common part of the boundary (G') and (\Gamma) is a piece of the plane (x_n=0). Take inside (\overline G'\cap\Gamma) a square of sufficiently small diameter (\delta), and construct the cube (Q_\delta) resting on the indicated square. Denote all faces of this cube by (Q'\delta). Inside (Q\delta), take a domain (G_1) resting on (x_n=0). Again represent the function (u) in the form of a sum of functions (u=v+w) satisfying equations (4) and (5) and the boundary conditions

[
v\big|{Q'\delta}=u,\qquad w\big|{Q'\delta}=0.
]

Using the smallness of (\delta), it is easy to show(^*) that

[
\int_{Q_\delta}\left|\frac{\partial^2 w}{\partial x_i\,\partial x_k}\right|^p d\Omega
\leq
C\int_{Q_\delta}|f|^p d\Omega
]

and, a fortiori,

[
\int_{G_1}\left|\frac{\partial^2 w}{\partial x_i\,\partial x_k}\right|^p d\Omega
\leq
C\int_{\Omega}|f|^p d\Omega .
]

Schauder ((^1)) showed that

[
\max_{x\in \overline{G}1}\left|\frac{\partial^2 v}{\partial x_i\,\partial x_k}\right|
\leq
C\max|u|.}
]

From this it is easy to obtain that

[
\int_{G_1}\left|\frac{\partial^2 v}{\partial x_i\,\partial x_k}\right|^p d\Omega
\leq
C\int_{\Omega}|f|^p d\Omega .
]

Then

[
\int_{G_1}\left|\frac{\partial^2 u}{\partial x_i\,\partial x_k}\right|^p d\Omega
\leq
C\int_{\Omega}|f|^p d\Omega,
]

as was required to be proved.

From Lemmas 3 and 4 it follows that, for a twice differentiable solution of problem (1)—(2), the estimate

[
\int_{\Omega}\left|\frac{\partial^2 u}{\partial x_i\,\partial x_k}\right|^p d\Omega
\leq
C\int_{\Omega}|f|^p d\Omega .
]

is valid.

From this inequality, by means of the usual arguments, one obtains the theorem:

Theorem. If (a_{ik}(x)) are continuously differentiable in (\Omega+\Gamma) and (f\in L_p(\Omega)) ((p>n/2)), then there exists a generalized solution of problem (1)—(2) satisfying the inequality

[
|u|{W_p^{(2)}(\Omega)}\leq C|f|,
]

where (C) is a constant independent of (u), (f).

Leningrad Textile Institute
named after S. M. Kirov

Received
22 IX 1956

REFERENCES

(^1) J. Schauder, Math. Zs., 38 (2) (1934).
(^2) O. A. Ladyzhenskaya, DAN, 79, No. 5 (1951).
(^3) S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, 1950.
(^4) A. I. Koshelev, Matem. sborn., 32 (74), 3 (1953).
(^5) S. G. Mikhlin, DAN, 78, No. 3 (1951).
(^6) V. I. Smirnov, A Course of Higher Mathematics, 4, 1951.

(^*) In work ((^4)) we showed that if (L(u)=\Delta u), then Lemma 1 will also be valid when (\Omega) is a square. It is not difficult to extend this result to the case of an arbitrary number of variables and then to argue in the same way as in the proof of Lemmas 1 and 3.