Reports of the Academy of Sciences of the USSR

1. Volume 146, No. 4

MATHEMATICS

N. N. URAL'TSEVA

GENERAL QUASILINEAR EQUATIONS OF SECOND ORDER AND SOME CLASSES OF SYSTEMS OF EQUATIONS OF ELLIPTIC TYPE

(Presented by Academician V. I. Smirnov on 24 IV 1962)

1. We consider the quasilinear elliptic equation

\[ a_{ij}(x,u,u_{x_k})u_{x_i x_j}+a(x,u,u_{x_k})=0 \tag{1} \]

in a bounded domain \(\Omega\) of variation of the independent variables \(x_1, x_2, \ldots, x_n\), \(n\geqslant 2\).

For equations having a divergent principal part

\[ \frac{\partial}{\partial x_i}\bigl(a_i(x,u,u_{x_k})\bigr)+a(x,u,u_{x_k})=0, \tag{2} \]

in the papers \((^{1,\,2a,\,б,\,в})\) all estimates necessary for proving the existence of classical (twice continuously differentiable) solutions of the first boundary-value problem were obtained: successively there are estimated \(\max_{\Omega}|u|\), \(|u|_{C_{0,\alpha}(\Omega)}\), \(\max_{\Omega}|u_{x_k}|\), \(|u_{x_k}|_{C_{0,\alpha}(\Omega)}\).

In the general case equation (1) cannot be represented in the form (2). For solutions \(u(x_1,x_2)\) of equations (1) with two independent variables there are known the theorems of S. N. Bernstein \((^3)\) and L. Nirenberg \((^4)\), which make it possible to estimate \(|u_{x_k}|_{C_{0,\alpha}(\Omega)}\) through \(\max_{\Omega}|u_{x_k}|\), \(k=1,\ldots,n\).

In the present paper this result is generalized to the case of equations (1) with arbitrary \(n\geqslant 2\).

2. Theorem 1. Let \(u(x)\) belong to \(W^2_2(\Omega)\), satisfy equation (1) almost everywhere in \(\Omega\), and let \(\operatorname{vrai}\max_{\Omega}|u|\leqslant M\), \(\operatorname{vrai}\max_{\Omega}|u_{x_k}|\leqslant M_1\), \(k=1,\ldots,n\). Suppose, moreover, that for \(x\in\Omega\), \(|u|\leqslant M\), \(p_k\leqslant M_1\), the inequalities

\[ a_{ij}\xi_i\xi_j\geqslant \nu\sum \xi_i^2, \]

\[ |a_{ij}|,\quad \left|\frac{\partial a_{ij}}{\partial p_k}\right|,\quad \left|\frac{\partial a_{ij}}{\partial u}\right|,\quad \left|\frac{\partial a_{ij}}{\partial x_k}\right|,\quad |a|\leqslant \mu \tag{3} \]

hold for \(a_{ij}(x,u,p_k)\) and \(a(x,u,p_k)\), with positive constants \(\nu\) and \(\mu\). Then \(u\in C_{1,\alpha}(\Omega')\), where \(\alpha>0\), \(|u_{x_k}|_{C_{0,\alpha}(\Omega')}\), \(\Omega'\Subset\Omega\), is determined only by the quantities \(\nu,\mu,n,M_1\) and by the distance of \(\Omega'\) to the boundary \(S\) of the domain \(\Omega\).

If \(S\) belongs to \(C_2\) (see \((^{2a})\)) and \(u|_S\in C_{1,\beta}(S)\), then \(|u_{x_k}|_{C_{0,\alpha}(\Omega)}\) is estimated in terms of \(\nu,\mu,n,M_1\), the norm of \(S\) in \(C_2\), and \(|u|_{C_{1,\beta}(S)}\), \(\beta\geqslant\alpha\). To prove this theorem, the method developed in \((^{5,1,2a,г})\) is sharpened.

We explain the proof of the first part of the theorem.

Consider the \(2n\) functions

\[ w_\pm^r=(\operatorname{grad}u)^2+5n(1\pm M_1^{-1}u_{x_r}),\qquad r=1,\ldots,n. \tag{4} \]

Each of these functions satisfies the integral identity

\[ \int_\Omega \left( a_{ij}u_{x_kx_j}u_{x_kx_i}\eta +\frac12 a_{ij}w^r_{x_j}\eta_{x_i} +\frac12 b^l_{ij}u_{x_lx_j}w^r_{x_i}\eta + c^r_{ij}u_{x_ix_j}\eta +d^r_i\eta_{x_i} +f^r\eta \right)\,dx=0. \tag{5} \]

Here \(b^l_{ij}, c^r_{ij}, d^r_i, f^r\) are expressed in terms of \(u_{x_k}, a\), and the derivatives of \(a_{ij}\) with respect to the arguments, while \(\eta(x)\) is an arbitrary bounded function from \(\overset{0}{W}{}^1_2(\Omega)\).

We shall denote by \(A^{r\pm}_{\lambda,\rho}\) the set of points \(x\) of the ball \(K(\rho)\subset\Omega\) for which \(w^r_\pm(x)>\lambda\), and by \(B^{r\pm}_{\lambda,\rho}\) the set of points in \(K(\rho)\) where \(w^r_\pm(x)<\lambda\).

From (5) it is established that \(w^r_\pm\) obey the inequalities

\[ \int_{A^{r\pm}_{\lambda,\rho}}(\operatorname{grad} w^r_\pm)^2 \xi^2\,dx \le \gamma \int_{A^{r\pm}_{\lambda,\rho}} (w^r_\pm-\lambda)^2(\operatorname{grad}\xi)^2\,dx +\gamma\,\operatorname{mes} A^{r\pm}_{\lambda,\rho}. \tag{6} \]

for an arbitrary bounded nonnegative \(\xi(x)\) from \(\overset{0}{W}{}^1_2(\Omega)\) and for \(\lambda \ge \max_{K(\rho)} w^r_\pm-\delta\), where \(\delta>0\) and \(\gamma\) are constants depending only on \(\nu\) and \(\mu\) from (3).

These inequalities proved sufficient for estimating \(|u_{x_k}|_{C_{0,\alpha}(\Omega')}\), \(k=1,\ldots,n\), for some \(\alpha>0\).

Theorem 1, together with the previously obtained estimates \((^{6,2a})\) of the maxima of the moduli \(u_{x_k}\), makes it possible to prove the following theorems.

Theorem 2. Let \(u(x)\) be a solution of equation (1) from \(C_3(\Omega)\), and let \(a_{ij}(x,v,p_k)\) and \(a(x,v,p_k)\), for \(x\in\Omega\), \(|v|\le \max_\Omega |u(x)|=M\), and \(|p_k|<\infty\), satisfy the natural compatibility conditions on the orders of growth with respect to \(p=\left(\sum_{k=1}^n p_k^2\right)^{1/2}\) as \(p\to\infty\), namely:

\[ a_{ij}(x,v,p_k)\xi_i\xi_j\ge \nu_1\sum \xi_i^2,\qquad \nu_1>0, \]

\[ |a|+\left|\frac{\partial a}{\partial v}\right| +\left|\frac{\partial a}{\partial x_k}\right| +(1+p)\left|\frac{\partial a}{\partial p_k}\right| +(1+p^2)\left( |a_{ij}|+\left|\frac{\partial a_{ij}}{\partial v}\right| +\left|\frac{\partial a_{ij}}{\partial x_k}\right| \right) + (1+p)^3\left|\frac{\partial a_{ij}}{\partial p_k}\right| \le \mu_1(1+p)^2. \tag{7} \]

Then \(|u|_{C_{1,\alpha}(\Omega')}\) and \(\alpha>0\) are determined by the quantities \(\nu_1, \mu_1, n, |u|_{C_{0,\beta}(\Omega)}\), \(\beta>0\), and by the distance from \(\Omega'\) to \(S\).

If \(S\in C_2\) and \(u|_S\in C_{1,\alpha}(S)\), then \(|u|_{C_{1,\alpha}(\Omega)}\) and \(\alpha\) depend only on \(\nu_1,\mu_1,n, |u|_{C_{0,\beta}(\Omega)}\), the norm of \(S\) in \(C_2\), and \(|u|_{C_{1,\alpha}(S)}\).

Remark. We have somewhat modified the method of obtaining the estimates \(\max_\Omega |u_{x_k}|\) given in \((^{6,2a})\). This makes it possible to lower the smoothness requirements on \(u\), so that in Theorems 2 and 3 \(u\) may be taken from \(C_1(\Omega)\cap \overset{*}{W}{}^2_2(\Omega)\).

Theorem 3. Suppose that, in addition to all the requirements of Theorem 2, the conditions of “weakened occurrence of \(v\) in \(a_{ij}\) and \(a\)” are fulfilled:

\[ 1)\quad \left|\frac{\partial a_{ij}(x,v,p_k)}{\partial v}\right|\le \varepsilon; \]

\[ 2)\quad \left|\frac{\partial a(x,v,p_k)}{\partial v}\right| \le \varepsilon p^2+P(p) \quad\text{or}\quad \frac{\partial a(x,v,p_k)}{\partial v}\le 0, \]
where \(\varepsilon\) is a small con-

constant, and \(p^{2}P(p)\to 0\) as \(p\to\infty\). Then in all the assertions of the preceding theorem \(|u|_{C_{0,\beta}(\Omega)}\) may be replaced by \(M=\max\limits_{\Omega}|u|\).

Let us formulate one of the existence theorems following from the estimates obtained and from Schauder estimates for solutions of linear equations.

Theorem 4. Let \(S\in C_{2,\alpha}\), \(\varphi(s)\in C_{2,\alpha}(S)\), and let the functions \(a_{ij}(x,v,p_k)\) and \(a(x,v,p_k)\), for \(x\in\Omega\) and arbitrary \(v,p_k\), have derivatives with respect to their arguments that are bounded for any finite \(v,p_k\), and also satisfy inequalities (7) and conditions 1), 2) of Theorem 3. If, for solutions \(u(x,\tau)\) of the boundary-value problems
\[ L_\tau u\equiv \tau Lu+(1-\tau)\Delta u=0,\qquad u|_S=\tau\varphi(s),\qquad \tau\in[0,1], \]
one can give an estimate \(\max\limits_{\Omega}|u(x,\tau)|\) uniformly with respect to \(\tau\in[0,1]\), then there exists a solution \(u(x)\in C_{2,\alpha}(\Omega)\) of the boundary-value problem \(Lu=0,\quad u|_S=\varphi(s)\).

1. We considered the first boundary-value problem for elliptic systems of quasilinear equations of the form
   \[ Mu\equiv a_{ij}(x,\mathbf u)\,\mathbf u_{x_i x_j}+b_i(x,\mathbf u,\mathbf u_{x_k})\,\mathbf u_{x_i}+\mathbf a(x,\mathbf u,\mathbf u_{x_k})=0, \tag{8} \]
   where \(\mathbf u=(u^1,\ldots,u^N)\),
   \[ a_{ij}(x,\mathbf u)\xi_i\xi_j\geq \nu(|\mathbf u|)\sum \xi_i^2, \]
   \[ |b_i(x,\mathbf u,\mathbf p_k)|\leq \mu(|\mathbf u|)(1+p),\qquad p=\left[\sum_{l=1}^{N}\sum_{k=1}^{n}(p_k^l)^2\right]^{1/2}, \tag{9} \]
   \[ |\mathbf a(x,\mathbf u,\mathbf p_k)|\leq \mu(|\mathbf u|)\varepsilon p^2+P(p),\qquad \varepsilon\ll 1,\qquad p^{-2}P(p)\to 0,\quad p\to\infty, \]
   \(\nu(t)\) and \(\mu(t)\) are positive continuous functions, \(t\in[0,\infty)\). Among the works in the literature on systems of quasilinear elliptic equations with \(n>2\), we point to the work of M. I. Višik \((^7)\) on generalized solutions of systems of a more general form, but such that, in a certain sense, the methods of studying linear equations are applicable to them.

In addition to conditions (9), we shall make the following assumptions on the functions \(a_{ij}(x,\mathbf u)\), \(b_i(x,\mathbf u,\mathbf p_k)\), \(a^l(x,\mathbf u,\mathbf p_k)\): for \(x\in\Omega\) and finite \(|\mathbf u|, |\mathbf p_k|\), the functions \(a_{ij}\) have bounded derivatives, while \(b_i\) and \(a^l\) satisfy a Hölder condition with respect to their arguments. Let \(S\in C_{2,\alpha}\), \(\vec\varphi(s)\in C_{2,\alpha}(S)\).

Theorem 5. If the conditions just stated are satisfied and if one can give an estimate \(\max\limits_{\Omega}|\mathbf u(x,\tau)|\) of the solutions \(\mathbf u(x,\tau)\) of the problem
\[ M_\tau\mathbf u\equiv M\mathbf u+ \]
\[ +(1-\tau)\Delta\mathbf u,\qquad \mathbf u|_S=\tau\vec\varphi(s), \]
uniformly with respect to \(\tau\in[0,1]\), then there exists a solution \(\mathbf u\in C_{2,\alpha}(\Omega)\) of the boundary-value problem \(M\mathbf u=0,\ \mathbf u|_S=\vec\varphi(s)\).

1. For systems of linear equations with unbounded coefficients
   \[ \frac{\partial}{\partial x_i}\bigl(a_{ij}(x)u_{x_j}+A_i(x)u+\mathbf f_i(x)\bigr)+B_i(x)u_{x_i}+B(x)u+\mathbf a(x)=0, \tag{10} \]
   where \(A_i\), \(B_i\), \(B\) are matrices with elements \(a_i^{lm}\), \(b_i^{lm}\), \(b^{lm}\), respectively, \(\mathbf u=(u^1,\ldots,u^N)\), \(\mathbf f_i=(f_i^1,\ldots,f_i^N)\), \(\mathbf a=(a^1,\ldots,a^N)\), we investigated the question of smoothness of generalized solutions. Along this path we obtained all the same results as were established for the case of a single equation in works \((^8,^9,^{2r})\). Namely, the following holds.

Theorem 6. Suppose the following conditions are satisfied:
\[ a_{ij}\xi_i\xi_j\geq \nu\sum \xi_i^2,\qquad x\in\Omega,\qquad \nu>0, \tag{11} \]
\[ \|a_i^{lm}, b_i^{lm}, f_i^l\|_{L_q(\Omega)}+\|b^{lm}, a^l\|_{L_{\frac q2}(\Omega)} +\max_{\Omega}|a_{ij}|\leq \mu,\qquad q>n. \]

Then every solution \(\mathbf{u}(x)\) from \(W_2^1\Omega\) of system (10) satisfies a Hölder condition in \(\Omega'\subset\Omega\), with exponent \(\alpha>0\), and \(|\mathbf{u}|_{C_{0,\alpha}(\Omega')}\) depends only on \(\nu,\ \mu\) and \(q\) from (11), on \(n\), on \(|\mathbf{u}|_{L_2(\Omega)}\), and on the distance from \(\Omega'\) to \(S\). If \(\mathbf{u}|_S\in C_{0,\alpha}(S)\), then, under small assumptions on \(S\) (condition (A) from \((2^a)\)), \(|\mathbf{u}|_{C_{0,\alpha}(\Omega)}\) is bounded.

Let us note that the conditions (11) cannot be weakened, as is evident from the examples constructed in \((2^r)\).

Leningrad State University
named after A. A. Zhdanov

Received
19 IV 1962

REFERENCES

\(^{1}\) N. N. Ural’tseva, DAN, 130, No. 6, 1206 (1960).
\(^{2}\) O. A. Ladyzhenskaya, N. N. Ural’tseva, a) UMN, 16, No. 1, 19 (1961); b) DAN, 138, No. 1, 29 (1961); c) O. A. Ladyzhenskaya, N. N. Ural’tseva, Comm. pure and appl. math., 14, No. 3, 481 (1961); d) Izv. AN SSSR, ser. matem., 26, 5 (1962).
\(^{3}\) S. N. Bernstein, UMN, 8, 32 (1941).
\(^{4}\) L. Nirenberg, Comm. pure and appl. math., 6, 103 (1953).
\(^{5}\) E. De Giorgi, Mem. Acc. Sci. di Torino, Ser. 3, 3, 25 (1957).
\(^{6}\) O. A. Ladyzhenskaya, DAN, 107, No. 5, 636 (1956); Tr. Mosk. matem. obshch., 7, 149 (1958).
\(^{7}\) M. I. Vishik, a) DAN, 134, No. 4, 749 (1960); b) DAN, 137, No. 3, 502 (1961); c) DAN, 138, No. 3, 518 (1961).
\(^{8}\) C. B. Morrey, Math. Zs., 72, No. 2, 146 (1959).
\(^{9}\) G. Stampacchia, Univ. di Genova, febr. 1960.