O. A. Ladyzhenskaya and N. N. Ural’tseva

On the Differential Properties of Bounded Generalized Solutions of Multidimensional Quasilinear Elliptic Equations and Variational Problems

(Presented by Academician V. I. Smirnov on 24 XII 1960)

The equation

\[ \sum_{i=1}^{n}\frac{\partial}{\partial x_i}\bigl(a_i(x,u,u_x)\bigr)+a(x,u,u_x)=0 \tag{1} \]

is investigated.

We shall assume that \(m>1\); \(a_i\) and \(a\) are measurable functions of their arguments satisfying the conditions

\[ |a_i(x,u,p_i)|\,p+|a(x,u,p_i)|\leq \mu(|u|)(1+p)^m, \]

\[ a_i(x,u,p_i)p_i\geq \nu(|u|)p^m-\mu(|u|), \tag{2} \]

where \(p=\left[\sum_j p_j^2\right]^{1/2}\). In a number of cases we shall assume that \(a_i\) and \(a\), in addition to condition (2), satisfy condition (3), namely, that they have first differentials with respect to all their arguments and

\[ \nu(|u|)(1+p)^{m-2}\sum_i \xi_i^2 \leq \frac{\partial a_i(x,u,p_j)}{\partial p_j}\,\xi_i\xi_j \leq \mu(|u|)(1+p)^{m-2}\sum_i \xi_i^2, \]

\[ \left|\frac{\partial a_i}{\partial p_j}\right|p^2 + \left|\frac{\partial a_i}{\partial u}\right|p + \left|\frac{\partial a}{\partial p_i}\right|p + \left|\frac{\partial a}{\partial u}\right| \leq \mu(|u|)(1+p)^m . \tag{3} \]

Here and below \(\nu(t)\) and \(\mu(t)\) are, respectively, monotonically nonincreasing and nondecreasing positive functions for \(t\geq 0\).

We shall call a function \(u(x)\) from \(W_m^1(\Omega)\)* a generalized solution of equation (1) if it satisfies the identity

\[ I(u,\eta)\equiv \int_{\Omega}\bigl[a_i(x,u,u_x)\eta_{x_i}-a(x,u,u_x)\eta\bigr]\,dx=0 \tag{4} \]

for every bounded \(\eta\) from \(\overset{\circ}{W}{}_{m}^{1}(\Omega)\).

We have proved \((^{1,2})\) that every bounded generalized solution \(u(x)\) of equation (1) belongs to \(C_{0,\alpha}(\Omega)\), with some \(\alpha>0\) depending only on \(\mu(\max_{\Omega}|u|)\) and \(\nu(\max_{\Omega}|u|)\) occurring in (2).

Lemma 1. For a bounded generalized solution \(u(x)\) of equation (1), the inequalities

\[ \int_{K(\rho)} |\nabla u|^m\,dx \leq c\rho^{\,n-m+\alpha}, \tag{5} \]

* For the definition of \(W_m^1(\Omega)\) and \(\overset{\circ}{W}{}_{m}^{1}(\Omega)\), see, for example, \((^3)\).

\[ \int_{K(\rho)} |x-y|^{-n+m-\alpha/2} |\nabla u|^m dx \leq c\rho^{\alpha/2}, \tag{6} \]

where \(K(\rho)\) is any ball of radius \(\rho\) belonging to \(\Omega\), and the constant \(c\) depends only on \(\mu\bigl(\max_\Omega |u|\bigr)\) and \(\nu\bigl(\max_\Omega |u|\bigr)\), appearing in (2).

For the proof one must take in (4) \(\eta(x)=[u(x)-u(x_0)]\zeta^m(x)\), where \(x_0\) is the center of the ball \(K(\rho)\), and \(\zeta(x)\) is a smooth, nonnegative cut-off function equal to zero outside \(K(\rho)\), and use (2) and \(|u(x)-u(x_0)|\leq c\rho^\alpha\). This leads to (5). Inequality (6), as is known, follows from (5) (see, for example, \((^4)\)).

Lemma 2. For any bounded generalized solution \(u(x)\) of equation (1), for \(m\geq 2\), the inequality

\[ \int_{K(\rho)} (1+|\nabla u|)^m \xi^2 dx \leq c\rho^\alpha \int_{K(\rho)} (1+|\nabla u|)^{m-2} |\nabla \xi|^2 dx \tag{7} \]

holds for any bounded \(\xi\) from \(\overset{0}{W}{}_m^1(K(\rho))\), with a constant \(c\) depending only on \(\mu\bigl(\max_\Omega |u|\bigr)\) and \(\nu\bigl(\max_\Omega |u|\bigr)\), appearing in (2).

For the proof one must take in (4), as \(\eta\), the function \([u(x)-u(x_0)]\xi^2(x)\) and use, along with (2), the well-known inequality

\[ \int_{K(\rho)} \xi^2 dx \leq c\rho^2 \int_{K(\rho)} |\nabla \xi|^2 dx, \]

valid for any \(\xi\in \overset{0}{W}{}_2^1(\Omega)\).

Lemma \(2'\). If \(b(x)>0\) and, for any \(\rho>0\) and \(y\in\Omega\), the integrals

\[ \int_{K(\rho)} |x-y|^{-n+m-\alpha/2} b^m(x) dx \leq c_1\rho^{\alpha/2},\quad \alpha>0,\quad 1<m\leq 2, \]

then

\[ \int_{K(\rho)} b^m \xi^2 dx \leq c\rho^{2\alpha/m} \int_{K(\rho)} b^{m-2} |\nabla \xi|^2 dx \tag{8} \]

for any bounded \(\xi\) from \(\overset{0}{W}{}_m^1(K(\rho))\), with a constant \(c\) depending only on \(c_1,\alpha,m\).

To prove (8) we use the fact that, for any \(\xi\in \overset{0}{W}{}_m^1(K(\rho))\), almost everywhere the identity

\[ \xi(x)=\frac{1}{\varkappa_n}\int_{K(\rho)} \frac{\partial \xi(y)}{\partial y_i} \frac{\partial}{\partial y_i} \frac{1}{|x-y|^{n-2}}\,dy \]

holds, where \(\dfrac{\varkappa_n}{n-2}\) is the surface area of the unit sphere in \(n\)-dimensional space. In view of this and of the conditions of the lemma,

\[ \begin{aligned} \int_{K(\rho)} b^m \xi^2 dx &\leq c\int b^m(x) \left(\int |x-y|^{1-n}|\nabla \xi|\,dy\right)^2 dx \\ &\leq c\int b^m(x) \left(\int |x-y|^{-n+m-\alpha/2}|\nabla \xi(y)|^2 b^{m-2}(y)\,dy\right) \\ &\quad \times \left(\int |x-y|^{-n-m+2+\alpha/2} b^{2-m}(y)\,dy\right) dx \\ &\leq c_1\rho^{2\alpha/m-\alpha/2} \int b^m(x) \left(\int |x-y|^{-n+m-\alpha/2}|\nabla \xi(y)|^2 b^{m-2}(y)\,dy\right) dx \\ &\leq c_2\rho^{2\alpha/m}\int b^{m-2}(y)|\nabla \xi(y)|^2dy . \end{aligned} \]

It follows from Lemma \(2'\) that Lemma 2 is also valid for \(1<m\leq 2\).

Theorem 1. For a bounded generalized solution \(u(x)\) of equation (1), a uniqueness theorem in the small is valid. More precisely: two bounded-

of generalized solutions \(u'(x)\) and \(u''(x)\), equal to each other on the surface of the ball \(K(\rho)\), coincide in \(K(\rho)\), provided only that the radius \(\rho\) is less than a certain number determined by \(\mu\!\left(\max\limits_{\Omega}|u'|, |u''|\right)\) and \(\nu\!\left(\max\limits_{\Omega}|u'|, |u''|\right)\), entering into (2) and (3).

For any bounded \(\eta(x)\in \overset{\circ}{W}{}^{1}_{m}(K(\rho))\) we have

\[ 0=I(u',\eta)-I(u'',\eta)=\int_{0}^{1}\frac{d}{dt}I(u^{t},\eta)\,dt, \]

where \(u^{t}=(1-t)u''+tu'\). Differentiating with respect to \(t\) and putting \(\eta=u'-u''\), we arrive at an equality from which, using (3), we obtain

\[ \int_{K(\rho)}(1+|\nabla u'|+|\nabla u''|)^{m-2}|\nabla\eta|^{2}dx \leq c\int_{K(\rho)}(1+|\nabla u'|+|\nabla u''|)^{m}\eta^{2}dx . \tag{9} \]

Hence, on the basis of Lemma 2 for \(m\geq 2\) and Lemma \(2'\) for \(1<m\leq 2\), we conclude that \(\eta(x)\) is identically zero.

Theorem 2. If conditions (2) and (3) are satisfied, then any bounded generalized solution \(u(x)\) of equation (1) has generalized second derivatives and satisfies equation (1) almost everywhere. For it,

\[ \int_{\Omega'} \left[ |\nabla u|^{m+2} + (1+|\nabla u|)^{m-2} \sum_{i,j}u_{x_i x_j}^{2} \right]dx <c\subset \Omega', \tag{10} \]

where \(\Omega'\) is an arbitrary strictly interior subdomain of \(\Omega\). If \(S\) and \(\varphi=u|_{S}\) are twice continuously differentiable, then (10) is also true for \(\Omega'=\Omega\).

Take in (4) \(\eta(x)=\bigl(u_{(k)}(x)\xi^{2}(x)\bigr)_{(\bar k)}\), where \(\xi(x)\) is a twice continuously differentiable nonnegative cutoff function, equal to zero outside \(K(\rho)\), and the symbols \((\ )_{(k)}\) and \((\ )_{(\bar k)}\) denote right and left difference quotients with respect to \(x_k\). After this we transfer \((\ )_{(\bar k)}\) to the first factor and carry out estimates, using (3). This will lead us to the inequality

\[ \int_{\Omega}\sum_{i} B^{m-2}u_{x_i(k)}^{2}\xi^{2}dx \leq c\int_{\Omega}\left[B^{m-2}u_{(k)}^{2}|\nabla\xi|^{2} + B^{m}u_{(k)}^{2}\xi^{2}\right]dx, \tag{11} \]

where \(B(x)=1+|\nabla u(x)|+|\nabla u(x+\Delta x_k)|\). Since \(u(x+\Delta x_k)\) is, just as \(u(x)\), a bounded generalized solution of an equation of type (1), Lemmas 2 and \(2'\) are valid for them. From these lemmas and (11) there follows the uniform boundedness of the integrals

\[ \int_{\Omega}\left(B^{m-2}\sum_i u_{x_i(k)}^{2}+B^{m}u_{(k)}^{2}\right)\xi^{2}dx<c, \]

on the basis of which we are convinced of the validity of all assertions of the theorem for \(\Omega'\subset\Omega\). The proof of (10) for \(\Omega'=\Omega\) is carried out in the same way as in (⁵), by straightening the boundary and reducing the boundary values to zero.

The facts established here for equations (1) make it possible to strengthen the results proved by us earlier \((^{1,2})\) concerning bounded generalized solutions of the variational problem of determining the minimum of the functional

\[ J(u)=\int_{\Omega}F(x,u,u_x)\,dx_1dx_n,\qquad u|_{S}=\varphi . \tag{12} \]

The Euler equation for \(J\) is a special case of equations of the form (1), and for it conditions (2), (3) follow from the so-called “natural” assumptions on \(F\). Namely, from papers \((^{1,2})\) and the propositions proved here there follows:

Theorem 3. Any bounded function \(u(x)\) from \(W_m^1(\Omega)\), for which

\[ \delta J(u)=\int_{\Omega}\bigl(F_{u_{x_i}}(x,u,u_x)\,\eta_{x_i}+F_u\eta\bigr)\,dx=0 \]

for every bounded

\[ \eta(x)\in \overset{\circ}{W}{}_m^1(\Omega), \]

belongs to \(C_{k,\alpha}(\Omega)\) \((k\geqslant 3,\ \alpha>0)\), if \(F(x,u,p_i)\), as a function of all its arguments, belongs to \(C_{k,\alpha}\) and satisfies only the “natural” assumptions formulated in \((^{1,2})\).

A survey of works on the questions discussed here is given in \((^2)\). The results of the present note, as far as we know, are new even for \(n=2\). Theorem 3 was proved in Morrey’s work \((^6)\), which has only just appeared, for \(n=m=2\).

Leningrad State University
named after A. A. Zhdanov

Received
20 XII 1960

CITED LITERATURE

\(^1\) O. A. Ladyzhenskaya, N. N. Ural’tseva, DAN, 135, No. 6 (1960).
\(^2\) O. A. Ladyzhenskaya, N. N. Ural’tseva, Uspekhi Mat. Nauk, 16, issue 1 (1961).
\(^3\) V. I. Smirnov, A Course of Higher Mathematics, 5, 1959.
\(^4\) C. B. Morrey, Math. Zs., 72, Heft 2 (1959).
\(^5\) O. A. Ladyzhenskaya, The Mixed Problem for a Hyperbolic Equation, 1953.
\(^6\) C. B. Morrey, Ann. d. Scuola Norm. Sup. di Pisa, ser. III, 14, Fasc. I (1960).