Reports of the Academy of Sciences of the USSR

1. Volume 116, No. 4

MATHEMATICS

A. I. KOSHELEV

ON THE BOUNDEDNESS IN \(L_p\) OF DERIVATIVES OF SOLUTIONS OF ELLIPTIC EQUATIONS AND SYSTEMS

(Presented by Academician V. I. Smirnov on 19 IV 1957)

Consider in \(n\)-dimensional space \(x(x_1,\ldots,x_n)\) a domain \(\Omega\), whose boundary \(S\) is a closed simply connected surface. Suppose that \(S\), in a neighborhood of each of its points, can be represented in local coordinates by means of a sufficiently many times differentiable function. Let an elliptic system of equations be given in \(\Omega\) (see, for example, \((^1)\))

\[ Lu=\sum_{k_1,\ldots,k_{2m}=1}^{n} a^{(k_1,\ldots,k_{2m})}(x) \frac{\partial^{2m}u}{\partial x_{k_1}\cdots \partial x_{k_{2m}}} +Tu=f(x), \tag{1} \]

where \(u(x)=[u_1(x),\ldots,u_N(x)]\); \(f(x)=[f_1(x),\ldots,f_N(x)]\); \(a^{(k)}(x)\) are square matrices of order \(N^2\); \(Tu\) is a linear differential operator of order lower than \(2m\).

By a generalized solution of system (1) we shall mean such a function\(^*\) \(u\in W_p^{(2m)}(\Omega)\) \((^2)\), which almost everywhere in \(\Omega\) satisfies system (1). We shall consider a generalized solution satisfying the boundary conditions

\[ u\big|_S=\frac{\partial u}{\partial \nu}\bigg|_S=\cdots= \frac{\partial^{m-1}u}{\partial \nu^{m-1}}\bigg|_S=0, \tag{2} \]

where \(\nu\) is the direction of the exterior normal to \(S\).

We formulate the main result of the present note.

Theorem. If the coefficients of system (1) are continuous, \(f\in L_p(\Omega)\) \((p>1)\), and there exists a generalized solution of problem (1)—(2), then this solution satisfies the inequality

\[ \|u\|_{W_p^{(2m)}(\Omega)} \le C_1\|f\|_{L_p(\Omega)}+C_2\|u\|_{L_p(\Omega)}, \tag{3} \]

where \(C_1, C_2\) do not depend on \(u, f\).

Inequality (3) for any interior subdomain \(\Omega^*\) of the domain \(\Omega\) was obtained in our note \((^3)\). For the case \(m=1,\ N=1\), inequality (3) was obtained earlier in our notes \((^{4,5})\). The proof of the inequality

\(^*\) A vector function \(u\) belongs to \(W_p^{(2m)}(\Omega)\) if each of its \(N\) components belongs to \(W_p^{(2m)}(\Omega)\). The norm of a vector function is defined by the rule

\[ \|u\|_{W_p^{(2m)}}= \left(\sum_{i=1}^{N}\|u_i\|_{W_p^{(2m)}}^{p}\right)^{1/p}. \]

(3) in the interior subdomain \(\Omega^*\) is carried out in the same way as in (3). To obtain inequality (3) in the boundary strip, the following lemma is first proved.

Lemma. Let the domain \(\Omega\) be the cube \(D: 0 \leq x_i \leq 1\) \((i=1,\ldots,n)\), and let the coefficients of system (1) be constant. Suppose that in \(D\) there exists a generalized solution of system (1) with constant coefficients, satisfying on the boundary \(x_n=0\) the conditions (2). Then, if \(f\in L_p(\Omega)\) \((p>1)\), the estimate holds

\[ \|u\|_{W_p^{(2m)}(D_r)} \leq A\|f\|_{L_p(D)}+\frac{B}{r^\gamma}\|u\|_{L_p(D)}, \tag{4} \]

where \(D_r\) is any subdomain of the cube \(D\) whose distance from all faces of \(D\), except \(x_n=0\), is not less than \(r>0\); \(A,B\) are positive constants depending only on \(m,n,p\).

For the proof of the lemma, the Green’s function of system (1) with constant coefficients is constructed for the half-space \(x_n>0\). In constructing the regular part of the Green’s function, a method analogous to that of O. V. Guseva is used, by means of which inequality (3) was proved for \(p=2\) \((^6)\). After certain transformations, the solution \(u(x)\) and its derivatives up to order \(2m\) inclusive are represented in the form of Fourier integrals. Under the sign of these integrals stands the Fourier transform of the function \(f\) with multipliers satisfying the conditions formulated in a note of S. G. Mikhlin \((^7)\). The considerations listed lead to inequality (4). To prove estimate (3) in the boundary strip, a part of the boundary \(S\) is straightened by means of a change of coordinates. Without loss of generality one may assume that the equation of this part of the boundary will be \(x_n=0\). A system of cubes \(D_\rho\) is constructed, one of whose faces coincides with the plane \(x_n=0\) and whose edges have length \(\rho>0\). Inside each such cube \(D_\rho\), system (1) is replaced by a system with constant coefficients obtained by substituting into the coefficients the coordinates of the center of the corresponding cube. Since the constant \(A\) in the corresponding estimate (4) for the cube \(D_\rho\) will not depend on \(\rho\), it is possible to apply a method analogous to Schauder’s method (see, for example, \((^4)\)), which after simple transformations gives estimate (3). We note that if the estimate

\[ \|u\|_{L_p(\Omega)} \leq C_3\|f\|_{L_p(\Omega)}, \tag{5} \]

is known from the very beginning, then estimate (3) will have the form

\[ \|u\|_{W_p^{(2m)}(\Omega)} \leq C\|f\|_{L_p(\Omega)}. \tag{6} \]

In some cases estimate (5), and consequently also (6), is easy to obtain if one assumes that system (1) is strongly elliptic \((^8)\), more precisely, that on functions \(u\in W_p^{(2m)}(\Omega)\) satisfying the equalities (2), the inequality

\[ |(Lu,u)|>\mu^2\|u\|_{W_2^{(m)}(\Omega)}^2, \tag{7} \]

holds, where

\[ (u,v)=\sum_{i=1}^{N}\int_{\Omega} u_i v_i\,d\Omega \]

and \(\mu\) is some real constant different from zero. Thus, for example, if inequality (7) is fulfilled, then estimate (6) is obtained for \(p>1\), if \(m\leq 2n\), and for \(p>\dfrac{2n}{n+2m}\), if \(m>2n\).

As is known (see, for example, (⁹)), the a priori estimate (6) makes it possible to prove the existence theorem for a generalized solution by the method of continuation with respect to a parameter.

Let us now consider a quasilinear system of the form

\[ Mu=\sum a_{\lambda}^{(k_1,\ldots,k_{2m})} \left(x;u,Du,\ldots,D^{2m-1}u\right) \frac{\partial^{2m}u}{\partial x_{k_1}\cdots \partial x_{k_{2m}}} = \]

\[ = f_\lambda\left(x;u,\ldots,D^{2m-1}u\right), \tag{8} \]

where
\[ a_{\lambda}^{(k)}\left(x;u,\ldots,D^{2m-1}u\right) = a^{(k)}(x)+\lambda b^{(k)}\left(x;u,Du,\ldots,D^{2m-1}u\right), \]
\[ f_\lambda\left(x;u,Lu,\ldots,D^{2m-1}u\right) = f(x)+\lambda\varphi\left(x,u,Lu,\ldots,D^{2m-1}u\right); \]
by \(L^k u\) are denoted the derivatives of the function of order \(k\), and by \(\lambda\) a real parameter.

If the coefficients \(a^{(k)}\), \(b^{(k)}\), and \(\varphi\) of system (8) are continuous in all variables and \(f\in L_p(\Omega)\) for \(p>n\), then \(Mu\) is an operator mapping \(W_p^{(2m)}(\Omega)\) into \(L_p(\Omega)\). This follows from the fact that, according to the embedding theorems (²), all derivatives of the function \(u\) up to order \(2m-1\), inclusive, will be continuous in \(\Omega\).

If, for system (8), when \(\lambda=0\), condition (7) is satisfied with regard to the boundary conditions (2), then it is not difficult, with the aid of the contraction mapping principle, to prove the following theorem.

Theorem. If the coefficients \(a^{(k)}\), \(b^{(k)}\), \(\varphi\) are continuous in all variables, the coefficients \(b^{(k)}\), \(\varphi\) are differentiable in all variables except \(x\), and \(f\in L_p(\Omega)\) for \(p>n\), then system (8) with boundary conditions (2) has a generalized solution for \(|\lambda|<\varepsilon\), where \(\varepsilon\) is some positive number.

Leningrad Textile Institute
named after S. M. Kirov

Received
11 IV 1957

REFERENCES

¹ I. G. Petrovskii, Uspekhi Mat. Nauk, 1, no. 3–4 (13–14) (1946).
² S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, L., 1950.
³ A. I. Koshelev, DAN, 110, no. 3 (1956).
⁴ A. I. Koshelev, Mat. Sb., 38 (80), no. 3 (1956).
⁵ A. I. Koshelev, DAN, 105, no. 1 (1955).
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⁷ S. G. Mikhlin, DAN, 109, no. 4 (1956).
⁸ M. I. Vishik, Mat. Sb., 29 (71), no. 3 (1951).
⁹ O. A. Ladyzhenskaya, Vestn. LGU, no. 11 (1955).