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MATHEMATICS
O. A. LADYZHENSKAYA and N. N. URAL'TSEVA
ON THE REGULARITY OF GENERALIZED SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATIONS
(Presented by Academician V. I. Smirnov on 15 IV 1961)
In the papers \((^1)\) we proved the classical solvability “in the large” of the Dirichlet problem for elliptic equations of the form
\[ \sum_{i=1}^{n}\frac{\partial}{\partial x_i} \bigl(a_i(x,u,u_{x_k})\bigr)+a(x,u,u_{x_k})=0,\qquad n\geqslant 2, \tag{1} \]
and investigated the question of the smoothness of generalized solutions of linear and quasilinear elliptic equations of the form (1); in particular, we proved that any bounded generalized solution of the variational problem of finding functions for which the functional
\[ I(u)=\int_{\Omega} F(x,u,u_{x_k})\,dx,\qquad u\bigm|_{S}=\varphi(s), \]
takes a stationary value belongs to the class \(C_{k,\alpha}(\Omega)\), if \(F\in C_{k,\alpha}\), \(k\geqslant 3\), and if certain “natural requirements” with respect to \(F\) are satisfied (see \((^{1,a,6})\)). Here we strengthen these results, reducing the smoothness conditions on \(F\) and \(a_i\), in a certain respect, to minimal ones. In addition, the method of proving the smoothness of generalized solutions is simpler than that given by us in \((^{1б})\).
The main theorems of the present work are the following:
Theorem 1. Let \(a_i(x,u,p_k)\) and \(a(x,u,p_k)\), as functions of their arguments, belong to the classes \(C_{1,\alpha}\), \(\alpha>0\), and \(C_{1,0}\), respectively, for \(x\in\Omega\), \(|u|\leqslant M^*\) and all finite \(p=\left(\sum_{k=1}^{n}p_k^2\right)^{1/2}\). Suppose, moreover, that for the same values of \(x,u,p_k\) the conditions
\[ |a_i|\,p+|a|\leqslant \mu(1+p)^m, \qquad a_i p_i\geqslant \nu_1 p^m-\mu_1,\qquad m>1,\ \nu_1>0, \tag{2} \]
\[ \nu_2(1+p)^{m-2}\sum_{i=1}^{n}\xi_i^2 \leqslant \frac{\partial a_i}{\partial p_j}\xi_i\xi_j \leqslant \mu(1+p)^{m-2}\sum_{i=1}^{n}\xi_i^2, \qquad \nu_2>0, \]
\[ \left|\frac{\partial a_i}{\partial p_j}\right|p^2 +\left|\frac{\partial a_i}{\partial u}\right|p +\left|\frac{\partial a}{\partial p_i}\right|p +\left|\frac{\partial a}{\partial u}\right| +\left|\frac{\partial a_i}{\partial x_j}\right|p +\left|\frac{\partial a}{\partial x_i}\right| \leqslant \mu(1+p)^m \tag{3} \]
are fulfilled. Then every generalized solution \(u(x)\) of equation (1), i.e. a function from \(W_m^1(\Omega)\), satisfying, for an arbitrary bounded \(\eta(x)\) from \(\overset{\circ}{W}{}^{1}_{m}(\Omega)\)
* Here and in \((^{1,3})\), \(u\) may be regarded as varying in any interval \([M_1,M_2]\), and not only in an interval of the form \([-M,M]\).
the identity
\[ \int_{\Omega}\left[a_i(x,u,u_{x_k})\eta_{x_i}-a(x,u,u_{x_k})\eta\right]\,dx=0, \tag{4} \]
with \(\operatorname{vrai\,max}_{\Omega}|u|\le M\), belongs to the class \(C_{2,\alpha}(\Omega')\), where \(\Omega'\) is any interior subdomain of the domain \(\Omega\). If, in addition, \(a_i\in C_{l,\alpha}\), and \(a\in C_{l-1,\alpha}\), \(l>1\), then \(u\in C_{l+1,\alpha}(\Omega')\).
If \(S\) and \(u|_S\) belong to \(C_{l+1,\alpha}\), \(l\ge 1\), then \(u(x)\) belongs to \(C_{l+1,\alpha}(\overline{\Omega})\).
A particular, but important, case of Theorem 1 is:
Theorem 2. Let \(F(x,u,p_k)\) be defined for \(x\in\Omega\), \(|u|\le M\), and \(|p_k|<\infty\), and satisfy the following “natural” conditions:
1) On any compact set it belongs to the class \(C_{l,\alpha}\), \(l\ge 2\), \(\alpha>0\).
2) \(F\) has, with respect to
\[ p=\left(\sum_{k=1}^{n}p_k^2\right)^{1/2}, \]
growth of order \(m>1\), more precisely,
\[ \nu_1p^m\le F(x,u,p_k)\le \mu_1'(1+p)^m,\qquad \nu_1>0, \]
and differentiating \(F\) and its derivatives with respect to \(p_k\) lowers the orders of their growth in \(p\) by at least one, while differentiating with respect to \(u\) and \(x_k\) does not increase them.
3) The Euler equation for it is uniformly elliptic:
\[ \nu_2(1+p)^{m-2}\sum_{i=1}^{n}\xi_i^2 \le F_{p_i p_j}(x,u,p_k)\xi_i\xi_j \le \mu_2(1+p)^{m-2}\sum_{i=1}^{n}\xi_i^2, \qquad \nu_2>0. \]
4) For sufficiently large \(p\),
\[ F_{p_i}(x,u,p_k)p_i>\nu_3p^m,\qquad p\gg 1,\quad \nu_3>0. \]
Then any function \(u(x)\) from \(W_m^1(\Omega)\) with \(\operatorname{vrai\,max}_{\Omega}|u|\le M\), satisfying the identity
\[ \delta I(u)=\int_{\Omega}\left[F_{u_{x_i}}(x,u,u_{x_k})\eta_{x_i}+F_u(x,u,u_{x_k})\eta\right]\,dx=0 \]
for arbitrary bounded \(\eta(x)\) from \(\overset{0}{W}{}^{\,1}_m(\Omega)\), belongs to \(C_{l,\alpha}(\Omega')\), \(\Omega'\Subset\Omega\). If \(u|_S\) and the boundary \(S\) belong to \(C_{l,\alpha}\), then \(u(x)\in C_{l,\alpha}(\overline{\Omega})\).
Let us outline the general course of the proof of Theorems 1 and 2.
First we prove (see \((1^{\mathrm{a},\mathrm{b}})\)) that \(u(x)\) belongs to the class \(C_{0,\beta}\) with some \(\beta>0\). Then we establish (see \((1^{\mathrm{c}})\)) that \(u(x)\) has generalized second-order derivatives and satisfies equation (1) almost everywhere in \(\Omega\), and that for \(u(x)\) the estimate
\[ \int_{\Omega'}\left[|\nabla u|^{m+2}+(1+|\nabla u|)^{m-2}\sum_{i,j=1}^{n}u_{x_i x_j}^{2}\right]\,dx\le \mathrm{const} \tag{5} \]
holds (if the boundary data are smooth, then (5) is true for \(\Omega'=\Omega\)). It is now not difficult to verify, putting in (4) \(\eta=\xi_{x_k}\), that \(u(x)\) satisfies the integral identity
\[ \int_{\Omega}\left(\frac{da_i}{dx_k}\xi_{x_i}+a\xi_{x_k}\right)\,dx=0,\qquad k=1,\ldots,n, \tag{6} \]
with any smooth finite function \(\xi(x)\). Let \(b(x)=\min\{|\nabla u(x)|^2,N^2\}\), and let \(\zeta(x)\) be a smooth nonnegative function equal to zero outside the ball \(K(\rho)\Subset\Omega\) of radius \(\rho\). In view of conditions (2), (3) and estimate (5), in identity (6)
one may set \(\xi(x)=b^r u_{x_k}\zeta^2,\ r\geqslant 1\). Moreover, integration by parts easily verifies the inequality
\[
\int_{K(\rho)}
(1+|\nabla u|)^m |\nabla u|^2 b^r \zeta^2\,dx
\leqslant C\operatorname{osc}\{u,K(\rho)\}
\int_{K(\rho)}
\left[(1+|\nabla u|)^{m-2}\zeta^2
\sum_{i,j=1}^n u_{x_i x_j}^2
\right.
\]
\[
\left.
{}+(1+|\nabla u|)^{m+2}\zeta^2
+(1+|\nabla u|)^m |\nabla\zeta|^2
\right] b^r\,dx .
\tag{7}
\]
Using (6) with the indicated \(\xi\), (7), and the fact that \(u\) belongs to the class \(C_{0,\beta}(\Omega)\), we establish the boundedness of the integrals
\[ \int_{\Omega'} \left[ |\nabla u|^{m+2r+2} + |\nabla u|^{m+2r-2} \sum_{i,j=1}^n u_{x_i x_j}^2 \right]dx \leqslant C(r) \tag{8} \]
successively for any \(r=1,2,\ldots\).
Let now \(\zeta(x)\) be a smooth nonnegative finite function in \(\Omega\), and let
\(w(x)=(|\nabla u|^2\zeta^2)^{(m+2)/4}\). We shall denote by \(A_\lambda\) the set of points of the domain \(\Omega\) for which \(w(x)>\lambda\). Then, putting in (6)
\[ \xi(x)= \begin{cases} \bigl(|\nabla u|^2\zeta^2-\lambda^{4/(m+2)}\bigr)/\zeta^m\, u_{x_k}, & |\nabla u|^2\zeta^2\geqslant \lambda^{4/(m+2)},\\ 0, & |\nabla u|^2\zeta^2\leqslant \lambda^{4/(m+2)}, \end{cases} \quad k=1,\ldots,n, \]
and using condition (3), we obtain, for any \(\lambda\geqslant 0\), the inequality
\[ \int_{A_\lambda} |\nabla w|^2\,dx \leqslant C\int_{A_\lambda} (1+|\nabla u|)^{m+4}(1+|\nabla\zeta|^2)\,dx . \tag{9} \]
Taking into account the boundedness of the integrals (8), one can estimate the right-hand side of (9) by Hölder’s inequality, so that
\[ \int_{A_\lambda} |\nabla w|^2\,dx \leqslant C_1\,\operatorname{mes}^{\,1-\varepsilon} A_\lambda, \qquad \varepsilon<\frac{2}{n}. \]
Hence (see \((1^{\mathrm{b}},2)\)) the boundedness of \(w(x)\) follows, and therefore also that of \(|\nabla u|\) in any \(\Omega'\Subset\Omega\). Further, applying Morrey’s theorem, which we proved for any \(n\geqslant 2\) \((3,1^{\mathrm{b}})\), we are convinced of the validity of the first part of Theorem 1.
To prove the assertion of the theorem for a closed domain \(\overline{\Omega}\), we first estimate
\(M_1=\operatorname{vrai\,max}_{S}|\nabla u|\), using the method of auxiliary functions of S. N. Bernstein and the fact that inside \(\Omega\), \(u_{x_i}'(x)\) is a smooth function. After this, in obtaining the estimates (8) and (9), instead of \(|\nabla u|\) one must take
\[ z(x)= \begin{cases} |\nabla u|-M_1, & |\nabla u|\geqslant M_1,\\ 0, & |\nabla u|\leqslant M_1, \end{cases} \]
and regard \(\zeta(x)\) as not equal to zero on some part of the boundary \(S\).
Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR
Received
12 IV 1961
REFERENCES CITED
- O. A. Ladyzhenskaya, N. N. Ural’tseva, a) DAN, 135, No. 6 (1960); b) UMN, 16, issue 1 (97) (1961); c) DAN, 138, No. 1 (1961).
- G. Stampacchia, Ann. Scuola Norm. Super. Pisa, Ser. 3, 12, Fasc. 3 (1958).
- N. N. Ural’tseva, DAN, 130, No. 6 (1960).