MATHEMATICS

O. A. LADYZHENSKAYA and N. N. URALTSEVA

ON A VARIATIONAL PROBLEM AND QUASILINEAR ELLIPTIC EQUATIONS WITH MANY INDEPENDENT VARIABLES

(Presented by Academician V. I. Smirnov on 10 VI 1960)

We shall present results* obtained by us concerning solutions of the variational problem of finding

[
\inf I(u)=\inf \int_{\Omega} F(x,u,u_{x_k})\,dx,\qquad x=(x_1,\ldots,x_n)
\tag{1}
]

under the condition

[
u\big|_S=\varphi(s),
\tag{2}
]

solutions of the first boundary-value problem for quasilinear elliptic equations of general form

[
L_1(u)\equiv a_{ij}(x,u,u_{x_k})u_{x_i x_j}+a(x,u,u_{x_k})=0
\tag{3}
]

and elliptic equations “self-adjoint in the principal part,”

[
M_1(u)\equiv \frac{\partial}{\partial x_i}\bigl(a_i(x,u,u_{x_k})\bigr)+a(x,u,u_{x_k})=0.
\tag{4}
]

The main theorems stated below are new also for (n=2).

Notation. (E_k) is (k)-dimensional Euclidean space; (\Omega) is a bounded domain, and (\Omega') is its strictly interior subdomain; (S) is the boundary of the domain (\Omega); (C_{l,\alpha}(\Omega)), (W_m^l(\Omega)) are the usually defined classes of functions ((^{1,2})), (O_l(\Omega)) is the class of functions (u(x)), (x\in\Omega), for which derivatives of order (l-1) have first differentials, while derivatives up to order (l) inclusive are bounded on every compact part of (\Omega). We shall say that the domain (\Omega) satisfies condition (A) if there exist constants (a>0) and (\theta) in ((0,1)) such that for any ball (K(\rho)) with center on (S) and radius (\rho\le a) the inequality (\operatorname{mes}[K(\rho)\cap\Omega]\le(1-\theta)\operatorname{mes}K(\rho)) holds.

§ 1. A priori estimates. As S. N. Bernstein showed ((^3)), the following restrictions on the coefficients (a_{ij},a) of the quasilinear elliptic equations (3) are necessary in order that there may exist an a priori estimate of (\max_\Omega |u_{x_i}|) in terms of (\max_\Omega |u|) in an arbitrary domain (\Omega):

[
\nu(|u|)(p^2+1)^{m/2}\le a_{ij}(x,u,p_k)p_i p_j\le \mu(|u|)(p^2+1)^{m/2};
\tag{5}
]

[
|a(x,u,p_k)|\le \mu(|u|)(p^2+1)^{m/2},\qquad
p=\left(\sum_{k=1}^{n}p_k^2\right)^{1/2},
\tag{6}
]

* Part of these results was reported in the autumn of 1959 at V. I. Smirnov’s seminar in Leningrad and in a survey lecture at a meeting of the Leningrad Mathematical Society, and also in December 1959 at I. G. Petrovsky’s seminar in Moscow.

where (\mu(|u|)) here and below denotes a positive monotonically increasing function, (\nu(|u|)) a positive monotonically decreasing function ((|u|)), and (m) is some number greater than 1. It is assumed here that each differentiation of the functions (a_{ij}(x,u,p_k)), (a(x,u,p_k)), (a_i(x,u,p_k)), (F(x,u,p_k)) with respect to (p_k) lowers their order of growth with respect to (p) by at least 1, while differentiations with respect to (x_k) and (u) do not increase them. We shall call these conditions (B). We shall say that the equation is uniformly elliptic if, for (\sum \xi_i^2=1),

[
\nu(|u|)(p^2+1)^{m/2-1} \leq a_{ij}(x,u,p_k)\xi_i\xi_j
\leq \mu(|u|)(p^2+1)^{m/2-1}.
\tag{7}
]

Theorem 1. Let (u(x)) be a solution of equation (3), belonging to the class (O_3(\Omega)\cap C_1(\overline{\Omega})) and satisfying condition (2), and let
(a_{ij}(x,u,p_k),\ a(x,u,p_k)\in O_1(\Omega\times E_1\times E_n)), and suppose that conditions (B) and (7) are fulfilled for them. Then
(\max_{\Omega}|u_{x_i}|) is estimated in terms of (\max_{\Omega}|u|) and (|\varphi|{C).}(S)}), if the oscillation of (u(x)) in (\Omega) is small(^*) and the boundary (S) belongs to (C_{2,0

Theorem 2. Suppose the conditions of Theorem 1 are fulfilled, except for the assumptions on (S) and (\varphi). Then for any (\Omega'\subset \Omega), (\max_{\Omega'}|u_{x_i}|) is estimated in terms of (\max_{\Omega}|u|).

Theorem 3. Suppose the conditions of Theorem 1 are fulfilled, except for the assumption on the smallness of the oscillation of (u(x)) in (\Omega), and suppose that there exists a sufficiently large (R>0) such that, for (p\geq R,\ |u|\leq M), the quantities

[
\frac{1}{p^{m-2}}\left|\frac{\partial a_{ij}(x,u,p_k)}{\partial u}\right|,
\qquad
\frac{1}{p^m}\frac{\partial a(x,u,p_k)}{\partial u}
\tag{8}
]

do not exceed a certain number (\varepsilon>0), determined by the data of the problem. Then, for the solution (u) of problem (3), (2), (\max_{\Omega}|u_{x_i}|) can be estimated in terms of (\max_{\Omega}|u|) and (|\varphi|{C).}(S)

Suppose that for equations (4) conditions (B) and (7) are fulfilled, where (m-1) is the order of growth of (a_i(x,u,p_k)) with respect to (p). We single out from them an important class of quasilinear equations for which, for large (p), one must have

[
a_i(x,u,p_k)p_i \geq \nu(|u|)p^m,\qquad p\gg 1.
\tag{9}
]

Let us note that, for linear equations and for equations (3) not containing (u_{x_i}) in (a_{ij}), condition (9) is a consequence of the ellipticity condition. For the Euler equation it is a consequence of the natural assumptions, indicated below, concerning (F).

Theorem 4. If for (4) only the conditions just enumerated are fulfilled and
(a_i(x,u,p_k)\in O_2(\Omega\times E_1\times E_n)),
(a(x,u,p_k)\in O_1(\Omega\times E_1\times E_n)), then for any solution (u) of equation (4) from (O_3(\Omega)) the norm
(|u|{C), (S\in O_2), then the norm}(\Omega')}), with some (\alpha>0), can be estimated in terms of (\max_{\Omega}|u|). If, moreover, (\varphi\in C_{2,0
(|u|{C|u|) and (|\varphi|}(\Omega)}) is estimated in terms of (\max_{\Omega{C).}

Theorem 5. Suppose all the assumptions of Theorem 4 are fulfilled, except for inequality (9). Then, if the functions
(\partial^2 a_i(x,u,p_k)/\partial p_j\partial u),
(\partial^2 a_i(x,u,p_k)/\partial u^2),
(\partial a(x,u,p_k)/\partial u) have, with respect to (p), orders of growth
(m-2-\varepsilon,\ m-1-\varepsilon,\ m-\varepsilon) ((\varepsilon>0)), respectively, then the norm
(|u|{C), (S\in O_2), then the norm}(\Omega')}) is estimated in terms of (\max_{\Omega}|u|). If (\varphi\in C_{2,0
(|u|{C|u|) and (|\varphi|}(\Omega)}) is estimated in terms of (\max_{\Omega{C).}

(^*) That is, if (\operatorname{osc}{u;\Omega}) is less than a certain number (which we do not give here), determined by the constants entering into the conditions of the problem.

§ 2. Existence theorems. Consider

[
M_\tau(u)\equiv \tau M_1[u]+(1-\tau)M_0(u)=0,\qquad
u|_S=\tau\varphi(s),
\tag{10}
]

where

[
M_0(u)=\frac{\partial}{\partial x_i}F^0_{u_{x_i}}-F^0_u,\qquad
F^0(x,u,u_{x_k})=\left(\sum_i u_{x_i}^2+1\right)^{m/2}+u^2.
]

Theorem 6. Suppose that: 1) for (M_1(u)) the conditions of Theorem 4 or 5 are satisfied; 2) (a_i(x,u,p_k)), (a(x,u,p_k)), as functions of all their arguments, belong to (C_{2,\alpha}), (C_{1,\alpha}), respectively; 3) (S\in C_{2,\alpha}), (\varphi\in C_{2,\alpha}). Then problem (10) has at least one solution (u(x,\tau)) for all (\tau\in[0,1]), if for all possible solutions (\max |u(x,\tau)|) is uniformly bounded. All solutions lie in the space

[
C_{2,\alpha}^{\Omega}(\overline{\Omega})\cap C_{3,\alpha}(\Omega).
]

For (n=2) the following theorem is also valid:

Theorem 7. Let for equation (3), with (n=2), the conditions (Б) and (7) be satisfied, and without loss of generality let us assume (m=2). Suppose that instead of (6) the stronger restriction

[
|a(x,u,p_k)|\leq \mu(|u|)(p^2+1)^{1-\varepsilon},\qquad \varepsilon>0
]

is satisfied. Then the problem

[
L_\tau(u)\equiv \tau L_1(u)+(1-\tau)(\Delta u-u)=0,\qquad u|_S=\tau\varphi(s)
]

has at least one solution (u(x,\tau)) in (C_{2,\alpha}(\overline{\Omega})\cap C_{3,\alpha}(\Omega)) for all (\tau\in[0,1]), provided only that, for all possible such solutions (u(x,\tau)), the moduli are uniformly bounded. The functions (a_{ij}), (a) must belong to (C_{1,\alpha}), (\varphi\in C_{2,\alpha}), (S\in C_{2,\alpha}), and (\Omega) is homeomorphic to a disk.

§ 3. The variational problem. Let (F(x,u,p_k)) have growth order (m>1) with respect to (p), and let each differentiation with respect to (p_k) lower the growth order in (p) by at least 1, while differentiations with respect to (x_k) and (u) do not increase it. Suppose, further,*

[
F(x,u,p_k)\geq \nu_1(|u|)p^m;
]

[
F_{p_i p_j}(x,u,p_k)\xi_i\xi_j
\geq \nu_2(|u|)(p^2+1)^{(m-2)/2}\sum_i \xi_i^2;
\tag{11}
]

[
F_{p_i}(x,u,p_k)p_i\geq \nu_3(|u|)p^m,\qquad p\gg 1.
]

Then, as is not difficult to see, for the Euler equation for (I(u)) Theorem 4 is valid if (F\in C_3), and Theorem 6 if (F\in C_{3,\alpha}). One of the fundamental questions is that of the conditions on (F) under which every generalized solution of the variational problem possesses certain differential properties.

We have proved the following theorems:

Theorem 8. Let (u) be a generalized solution from (W_m^1(\Omega)) of the “conditional” variational problem (1), (2), i.e. the problem to which the condition is added that all comparison functions do not exceed some constant (M\geq \max_S |u|). It will belong to (C_{0,\alpha}(\Omega)), if (F\in C_1) and the conditions

[
\mu(|u|)p^m\geq F_{p_i}(x,u,p_k)p_i\geq \nu(|u|)p^m,\qquad p\gg 1,
]

[
|F_u(x,u,p_k)|\leq \mu(|u|)p^m.
\tag{12}
]

are satisfied.

[
\text{* If }F\text{ is representable in the form }F(x,u,p_k)=F'(x,u,p_k)+F''(x,u,p_k),
]
where (F') is a positive homogeneous function of (p_k) of order (m), and (\frac{1}{p^m}F''\to0) as (p\to\infty), then (11) are consequences of the positivity and convexity of (F) and (F').)

Under the same conditions on (F), every bounded function (u) from (W_m^1(\Omega)) for which (\delta I(u)=0) belongs to (C_{0,\alpha}(\Omega)). If, moreover, (\Omega) satisfies condition (A) and (\varphi\in C_1), then (u\in C_{0,\alpha}(\overline{\Omega})).

Theorem 9. Under the conditions on (F) formulated at the beginning of the paragraph, every bounded generalized solution (u(x)) of the variational problem (1), (2) from the class (W_m^1(\Omega)) belongs to (C_{k,\alpha}(\Omega)), if (F\in C_{k,\alpha}), (k\geqslant 3), and if
[
\Delta I(u)=I(u+\eta)-I(u)>0
]
for every sufficiently small local variation (\eta(x)). If, moreover, (S\in C_{l,\alpha}), (\varphi\in C_{l,\alpha}), (2\leqslant l\leqslant k), then
[
u\in C_{l,\alpha}(\overline{\Omega})\cap C_{k,\alpha}(\Omega).
]

For (m=2) the condition (\Delta I(u)>0) may be omitted.

Finally, we give two lemmas which generalize De Giorgi’s lemma ({}^{4}).

Let (u\in W_m^1(\Omega)), (m>1). Denote by (A_{k,\rho}) the set of points of the ball (K(\rho)) of radius (\rho), lying entirely in (\Omega), for which (u(x)>k), and by (B_{k,\rho}) the set of points of (K(\rho)) for which (u<k). We shall say that the function (u) belongs to the class (\mathfrak{B}m(\Omega;M;\gamma,\delta)) if (u\in W_m^1(\Omega)), (|u|\leqslant M), and for every (K(\rho)\subset\Omega) the estimate
[
\int} |\operatorname{grad}u|^m\,dx \leqslant \gamma \rho^{\,n-m
\tag{13}
]
holds.

In addition, for those (k_1) for which
[
\max_{A_{k_1,\rho}} |u(x)-k_1|\leqslant \delta,
]
for any (\sigma) from ((0,1)) the inequality
[
\int_{A_{k_1,\rho-\sigma\rho}} |\operatorname{grad}u|^m\,dx
\leqslant
\gamma\,\operatorname{mes} A_{k_1,\rho}
\left{
\frac{1}{(\sigma\rho)^m}\max_{A_{k_1,\rho}} |u(x)-k_1|^m+1
\right},
\tag{14}
]
holds; and for (k_2) for which
[
\max_{B_{k_2,\rho}} |k_2-u(x)|\leqslant \delta,
]
the inequality
[
\int_{B_{k_2,\rho-\sigma\rho}} |\operatorname{grad}u|^m\,dx
\leqslant
\gamma\,\operatorname{mes} B_{k_2,\rho}
\left{
\frac{1}{(\sigma\rho)^m}\max_{B_{k_2,\rho}} |k_2-u(x)|^m+1
\right}.
\tag{15}
]
holds.

Here (M,\gamma,\delta), and (m) are fixed positive numbers, with (1<m\leqslant n).

Lemma 1. Let (u\in\mathfrak{B}_m(\Omega;M;\gamma,\delta)); let (x_0) be an arbitrary interior point of (\Omega); and let (\rho_0) be the distance from (x_0) to the boundary (S). Then for every ball (K(\rho)) with center at (x_0) and radius (\rho\leqslant\rho_0) the estimate
[
\operatorname{osc}(u,K(\rho))\leqslant C\rho_0^{-\alpha}\rho^\alpha
]
holds, where (C>0) and some (\alpha\in(0,1)) are constants for the given class.

Denote by (\mathfrak{B}_m^{0}(\Omega;M;\gamma,\delta)) the class of functions (u) from (\mathfrak{B}_m(\Omega;M;\gamma,\delta)) satisfying the following requirements: a) (u\in W_m^1(\Omega)) and (u\equiv 0) outside (\Omega); b) if the ball (K(\rho)) intersects the boundary (S), then inequality (13) is preserved, while inequalities (14) and (15) hold not for all the indicated values (k_1) and (k_2), but for (k_1\geqslant 0) and (k_2\leqslant 0), respectively.

Lemma 2. Let (u\in\mathfrak{B}_m^{0}(\Omega;M;\gamma,\delta)), and let (\Omega) have property (A). Then for every (K(\rho))
[
\operatorname{osc}{u;K(\rho)}\leqslant C\rho^\alpha
]
with constants (C) and (\alpha>0) determined for the given class.

Leningrad State University
named after A. A. Zhdanov

Received
2 VI 1960

REFERENCES

1. K. Miranda, Equations with Partial Derivatives of Elliptic Type, IL, 1957.
2. S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, L., 1950.
3. S. N. Bernstein, UMN, vol. 8 (1941).
4. E. de Giorgi, Mem. d. Acad. Sc. Torino, s. 3, t. 3° (1957).