MATHEMATICS

S. N. KRUZHKOV

ON SOME PROPERTIES OF SOLUTIONS OF ELLIPTIC EQUATIONS

(Presented by Academician L. S. Pontryagin on 22 XI 1962)

In this paper estimates are obtained for the modulus of continuity, Harnack’s inequality, and a Liouville-type theorem for solutions of a certain class of elliptic equations admitting degeneracies. For the investigation we use the method proposed by Moser in ((^{1,2})). We note that related questions are studied in ((^{1-11})) and others.

1. The following two lemmas are consequences of embedding theorems (see ((^{12-14})) and others).

Lemma 1. Let, in the ball (K_\rho{|x|\leq \rho}), (x=(x_1,\ldots,x_n)), the functions (\mu(x)\geq \lambda(x)\geq 0), (\mu\in L_s), (\lambda^{-1}\in L_t), with (1/s+1/t\leq 2/n); let
[
\int_{K_\rho}(\lambda u_x^2+\mu u^2)\,dx<\infty,\qquad u_x=\operatorname{grad}u.
]
Then
[
\left(\rho^{-n}\int_{K_\rho}\mu |u|^{2k}\,dx\right)^{1/k}
\leq cM^{1/k}(\rho)P(\rho)\left(\rho^{2-n}\int_{K_\rho}\lambda u_x^2\,dx+\rho^{-n}\int_N \mu u^2\,dx\right),
\tag{1}
]
where
[
M(\rho)=\left(\rho^{-n}\int_{K_\rho}\mu^s\,dx\right)^{1/s},\qquad
P(\rho)=\left(\rho^{-n}\int_{K_\rho}\lambda^{-t}\,dx\right)^{1/t},
]
[
1\leq k=1+\frac{(2/n-1/s-1/t)}{(1+1/t-2/n)}\leq \frac{n}{n-2},
]
the set (N\subseteq K_\rho), (\operatorname{mes}N\geq c_0\rho^n), and the constant (c) depends only on (c_0) and (n) (everywhere we assume (n\geq 3)).

In what follows, (\lambda(x)) and (\mu(x)) are bounds for the eigenvalues of the matrix (a(x)) (see (3), (4)). Denote by (W_2^1(a,\Omega)) the closure of the space (C^\infty(\Omega)) in the norm
[
|u|=\left(\int_\Omega [(u_x,au_x)+\mu u^2]\,dx\right)^{1/2},
]
by (\overset{\circ}{W}{}2^1(a,\Omega)) the closure of the space (C_0^\infty(\Omega)) in the norm
[
|u|=\left(\int\Omega (u_x,au_x)\,dx\right)^{1/2},
]
and by (c) constants depending on (n).

Lemma 2. Let, in (K_\rho), the function (h(x)\in L_p), with (1/p+1/t\leq 2/n). Then for (u(x)\in \overset{\circ}{W}{}2^1(a,K\rho)) the inequality
[
\int_{K_\rho}|u^2h|\,dx
\leq c\rho^2P(\rho)\left(\rho^{-n}\int_{K_\rho}|h|^p\,dx\right)^{1/p}\int_{K_\rho}\lambda u_x^2\,dx
\tag{2}
]
is valid.

We shall first consider solutions of elliptic equations of the form
[
Lu=\sum_{i,j=1}^n \frac{\partial}{\partial x_i}\left(a_{ij}(x)\frac{\partial u}{\partial x_j}\right)
+\sum_{i=1}^n b_i(x)\frac{\partial u}{\partial x_i}=0,\qquad
b=(b_1,\ldots,b_n);
\tag{3}
]
[
a=|a_{ij}|,\qquad \lambda(x)\xi^2\leq (\xi,a\xi)\leq \mu(x)\xi^2,\qquad a=a';
\tag{4}
]
everywhere in what follows we assume that in some ball (K_R) the functions (\lambda(x)) and (\mu(x)) satisfy the conditions of Lemma 1 and that ((b^2/\lambda)\in L_p(K_R)), with (1/s+1/t<2/n) (then (k>1) in (1)) and (1/p+1/t\leq 2/n). Depending on whether the local behavior of a solution is being considered or a Liouville-type theorem is being established, we shall assume, with respect to (b_i(x)), respectively, that the following condition A or B is fulfilled:

A. For (r \leqslant r_0): 1)
[
c \sum_{i=1}^{m} r^2 P(r)\left[r^{-n}\int_{K_r}(b_i^2/\lambda)^p\,dx\right]^{1/p}\leqslant \frac14,
]
where (0\leqslant m\leqslant n);

2)
[
\sum_{i=m+1}^{n}\frac{\partial b_i}{\partial x_i}\geqslant 0,\qquad
\sum_{i=m+1}^{n} b_i x_i\leqslant 0,\qquad
\frac{\partial b_i}{\partial x_i}\in L_p(K_r),\quad i\geqslant m+1,\qquad
\frac1p+\frac1t\leqslant \frac2n
]
(cf. (11)).

B. (1/p+1/t=2/n), and for (r\geqslant R_0): 1)
[
c r^2 \sum_{i=1}^{m} P(r)\left[r^{-n}\int_{K_r}(b_i^2/\lambda)^p\,dx\right]^{1/p}\leqslant \frac14,
]
where (0\leqslant m\leqslant n); 2) the same as in A 2).

Definition 1. A function (u(x)\in W_2^1(a,\Omega)) is called a generalized solution of equation (3) in the domain (\Omega), if for every function (\varphi(x)\in \overset{0}{W}{}2^1(a,\Omega))
[
\int\varphi(b,u_x)\,dx.}(\varphi_x, au_x)\,dx=\int_{\Omega
\tag{5}
]

Definition 2. A function (v(x)\in W_2^1(a,\Omega)) is called a generalized subsolution of equation (3) in (\Omega), if for every function (\varphi(x)\in \overset{0}{W}{}2^1(a,\Omega)), (\varphi(x)\geqslant 0),
[
\int\varphi(b,v_x)\,dx.}(\varphi_x, av_x)\,dx\leqslant \int_{\Omega
\tag{6}
]

Lemma 3. Let (v(x)) be a nonnegative subsolution in (K_{\rho+\sigma}), and suppose that either condition A or condition B is satisfied. Then
[
\int_{K_\rho}\lambda v_x^2\,dx\leqslant \frac{c}{\sigma^2}\int_{K_{\rho+\sigma}}\mu v^2\,dx.
\tag{7}
]

For the proof one must put in (6) (\varphi=\eta^4(r)v(x)), where (\eta=1) for (0\leqslant r\leqslant \rho), (\eta=0) for (r\geqslant \rho+\sigma), and (\eta(r)) is linear for (\rho\leqslant r\leqslant \rho+\sigma). Combining (1) for (u=v), (N=K_\rho), and (7), we have
[
\left(\rho^{-n}\int_{K_\rho}\mu v^{2k}\,dx\right)^{1/k}
\leqslant
cM^{1/k}(\rho)P(\rho)\left(1+\frac{\rho^2}{\sigma^2}\right)(\rho+\sigma)^{-n}
\int_{K_{\rho+\sigma}}\mu v^2\,dx.
\tag{8}
]

Since everywhere below (r\leqslant \rho<\rho+\sigma\leqslant 4r), it follows that (M(\rho)\leqslant cM), (P(\rho)\leqslant cP), where (M=M(4r)), (P=P(4r)).

Lemma 4. Let (v(x)) be a nonnegative subsolution in (K_{4r}); then
[
\operatorname*{vrai\,max}{K_r} v^2(x)
\leqslant
c' M^{1/(k-1)}P^{k/(k-1)}r^{-n}\int\mu v^2\,dx,}
\qquad c'=c'(n,s,t).
\tag{9}
]

Proof. Noting that (f(v)=v^m) is a nonnegative subsolution for (m\geqslant 1), put (\rho_\nu=r(1+2^{-\nu})), (\nu=0,1,\ldots), (\sigma_\nu=\rho_{\nu-1}-\rho_\nu=r\cdot 2^{-\nu}),
[
q_\nu=\left(\rho_\nu^{-n}\int_{K_{\rho_\nu}}\mu v^{2k^\nu}\,dx\right)^{1/k^\nu}
=
\left[\left(\rho_\nu^{-n}\int_{K_{\rho_\nu}}\mu\left(v^{k^{\nu-1}}\right)^{2k}\,dx\right)^{1/k}\right]^{1/k^{\nu-1}}.
]

By virtue of inequality (8) for (v^{k^{\nu-1}}), (\nu\geqslant 1),
[
q_\nu\leqslant c^{(\nu+1)/k^{\nu-1}}M^{1/k^\nu}P^{1/k^{\nu-1}}q_{\nu-1},
\qquad \nu=1,2,\ldots,
]
whence (9) follows, if one takes into account that
[
\lim_{\nu\to\infty} q_\nu\geqslant \operatorname*{vrai\,max}_{K_r} v^2(x).
]

Lemma 5. Let (u(x)) be a nonnegative solution in (K_{4r}), and suppose
[
\operatorname{mes} N{x\in K_{2r}; u(x)\geqslant 1}\geqslant cr^n;
]
then for (x\in K_r)
[
u(x)\geqslant \gamma(n,s,t,Q)=\exp\left[-c'Q^{(2k-1)/2(k-1)}\right],
\quad \text{where } Q=MP.
\tag{10}
]

For the proof, consider (w=g(u+\varepsilon)), where (g(u)=\ln^{+}u^{-1}); similarly to how Lemma 3 was proved, we establish the inequality:
[
\int_{K_{2r}} \lambda w_x^2\,dx \leq
]
[
\leq cr^{-2}\int_{K_{4r}}\mu\,dx \leq cr^{n-2}M.
]
If we apply Lemma 1 to (w) with (k=1), noting that (w(x)=0) for (x\in N), we obtain the estimate
[
r^{-n}\int_{K_{2r}}\mu w^2\,dx \leq cM^2P.
]
Since (w=g(u+\varepsilon)\geq 0) and (g''\geq 0), (w(x)) is a nonnegative subsolution. To complete the proof it remains to apply Lemma 4 to (w) and let (\varepsilon) tend to (0).

Lemma 6. Let (u(x)) be a solution in the ball (K_{8r}). Then
[
\operatorname{osc}_{K_r} u \leq \eta\, \operatorname{K u,}
\qquad \text{where } \eta=1-\gamma/2.
]

By Lemma 4, applied to (|u|), (u(x)) is bounded in (K_{4r}); we may assume that (M_0=\operatorname*{vrai\,max}(\pm u)) in (K_{4r}). The proof of Lemma 6 follows from applying Lemma 5 to that one of the functions (1\pm (u/M_0)) which is (\geq 1) on a set (N\subset K_{2r}) with (\operatorname{mes}N\geq \tfrac12\operatorname{mes}K_{2r}).

2. Estimate of the modulus of continuity of the solution.

Theorem 1. Let (u(x)) be a generalized solution of equation (3) in (K_{r_0}), and let condition A be satisfied; let (\lambda(x)=\mu(x)/\mu_0) and (Q(r)\leq \varkappa) for (0<r\leq r_0). Then, for (r\leq r_0/2),
[
\operatorname{osc}_{K_r} u \leq 8\left(r/r_0\right)^{\alpha}\operatorname{K u,}
]
where (\alpha) depends only on (n,s,t,\varkappa).

Theorem 1 follows easily from Lemma 6.

Remark 1. It is obvious that if the hypotheses of Theorem 1 are satisfied uniformly in an (r_0)-neighborhood of any point (x\in\Omega'\subset\Omega), (d(\Omega',\partial\Omega)\geq\delta>0), then (u(x)) satisfies the Hölder condition in (\Omega'). In particular, it suffices to require uniform ellipticity ((\mu(x)=\mu_1,\ s=t=\infty)) and (b_i\in L_q(\Omega)), (q\geq n) (cf. (5)); for (q=n), condition A 1) is satisfied by virtue of the absolute continuity property of the Lebesgue integral: for (\varepsilon>0) there exists (r_0>0) such that, for (r\leq r_0),
[
\int_{K(x_0,r)} |b|^n\,dx \leq \varepsilon,
]
where (K(x_0,r)={x;\ |x-x_0|\leq r}\subset\Omega).

Remark 2. If (\lambda(x)\not\equiv \mu(x)/\mu_0), but (Q(r)) does not grow too strongly as (r\to 0), then one can also estimate the modulus of continuity of the solution, namely: let
[
c'Q^{(2k-1)/2(k-1)} \leq \ln \psi(r)
]
(see (10)), and let (\psi(r)) not decrease as (r\to 0). Then, by Lemma 6,
[
\operatorname{osc}_{K_r} u \leq \left(1-\frac{1}{2\psi(r)}\right)\operatorname{K u;}
]
it is not hard to see that from the condition
[
\int_0 \frac{dr}{r\psi(r)}=\infty
]
there follows the continuity of (u(x)) at the origin.

3. Harnack inequality.

Theorem 2. Let (u(x)) be a positive solution of equation (3) in (K_{3r_0}), (\lambda(x)=\mu(x)/\mu_0), and for (x_0\in K_{r_0}), (0<r\leq 2r_0),
[
Q(x_0,r)=\mu_0\left(r^{-n}\int_{K(x_0,r)}\mu^s\,dx\right)^{1/s}
\left(r^{-n}\int_{K(x_0,r)}\mu^{-t}\,dx\right)^{1/t}\leq \varkappa<\infty,
]
[
c\mu_0^2r^2
\left[r^{-n}\int_{K(x_0,r)}(b^2/\mu)^p\,dx\right]^{1/p}
\left[r^{-n}\int_{K(x_0,r)}\mu^{-t}\,dx\right]^{1/t}
\leq \frac14.
]
Then in (K_{r_0/2})
[
\max u \leq c_1\min u,
]
where (c_1) depends only on (n,s,t,\varkappa).

For Theorem 2 there is a remark analogous to Remark 1. In the case when (\mu(x)=\mu_1=\mathrm{const}) and (b_i\in L_q), (q>n), Theorem 2 was established by L. P. Kuptsov.

4. A Liouville-type theorem

Theorem 3. Let (u(x)) be a generalized solution in the whole space (i.e., (5) is satisfied in every ball); let condition B be satisfied and
[
\lim_{r\to\infty} Q(r)=\chi<\infty .
]
Then, for
[
aR_1)
]
that (u=\mathrm{const}).

Proof. Let (\chi_1>\chi),
[
[1-\tfrac12\gamma(n,s,t,\chi_1)]=\eta_1<4^{-a},\qquad
Q(r)\le \chi_1 \quad \text{for } r>R_2(\chi_1)\ge \max(R_1,R_0).
]
Fix (r_0>R_2) and put (R=4^m r_0), where (m) is an arbitrary natural number. By Lemma 6,
[
\operatorname{osc}{K u}
\le
\eta_1^m \operatorname{K_R} u
\le
2A r_0^a(\eta_1 4^a)^m;\qquad
\eta_1 4^a<1 .
]
Letting (m\to\infty), we obtain that (u=\mathrm{const}) in any ball (K).

5. Behavior of the solution in a neighborhood of an isolated singular point

Definition 3. A function
[
u(x)\in W_2^1(a,K_R\setminus K_\rho),\qquad 0<\rho<R,
]
is called a generalized solution of equation (3) in (K_R^0), where (K_R^0) is the ball (K_R) with its center removed, if the integral identity (5) is satisfied for every
[
\varphi(x)\in \overset{0}{W}{}{}_2^1(a,K_R)
]
that is equal to zero in some neighborhood of the origin.

Theorem 4. Let (u(x)) be a generalized solution in (K_{r_0}^0), (\lambda(x)=\mu(x)/\mu_0), condition A be satisfied, and
[
\int_{K_\rho}\mu^2\,dx\le \tilde c\,\rho^2,\qquad 0<\rho\le r_0;
]
let (Q(r)\le \chi<\infty), and for (n=2) let (\mu\le\mu_1), while for (n=3) let (\mu\in L_3) in (K_{r_0}). Then (u(x)) is bounded in (K_{r_0/2}), and for (r\le r_0/2)
[
\operatorname{osc}_{K_r} u
\le
8(r/r_0)^a\operatorname{K u,}
\qquad
a=\alpha(n,s,t,\chi).
]

For (n=2), the example
[
a_{ij}=\delta_{ij}\ln^2 r,\qquad b_i=0,\qquad u=1/\ln r
]
shows that the assertion of Theorem 4 will be false if the condition (\mu(x)\le\mu_1) is replaced by the condition (\mu(x)\in L_p), where (p) is an arbitrary number.

6. Equations of a more general form

By introducing new variables (see ((15))), Theorems 1 and 2 are carried over to the case of the equation
[
Lu+c(x)u=f(x),
]
and, for (f\ne0), for positive solutions an inequality of the form (see ((8)))
[
\max u\le c_1(\min u+c_2)
]
is valid. This device, applicable also to equations of higher order, is given here for the case of elliptic equations of the second order: let
[
\mathcal L u+\sum_{i=1}^n b_i\frac{\partial u}{\partial x_i}+cu=f,
]
where
[
\mathcal L u=\sum_{i,j=1}^n
\frac{\partial}{\partial x_i}\left(a_{ij}\frac{\partial u}{\partial x_j}\right)
]
or
[
\mathcal L u=\sum_{i,j=1}^n
a_{ij}\frac{\partial^2u}{\partial x_i\partial x_j};
]
then the function (w=yu+z) satisfies the equation
[
\mathcal L w+\sum_{i=1}^n D_iw+k_0\frac{\partial^2 w}{\partial y^2}
+\frac{\partial^2 w}{\partial z^2}
+yc\,\frac{\partial w}{\partial y}
=
yf\,\frac{\partial w}{\partial z},
\tag{11}
]
where
[
D_iw=b_i\partial w/\partial x_i=y b_i\partial^2 w/\partial x_i\partial y,
]
and (k_0>0) is a constant. Equation (11), in turn, can be reduced to an equation containing only highest-order derivatives.

Moscow State University
named after M. V. Lomonosov

Received
22 XI 1962

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