S. N. KRUZHKOV

A PRIORI ESTIMATES FOR GENERALIZED SOLUTIONS OF ELLIPTIC AND PARABOLIC EQUATIONS OF SECOND ORDER

(Presented by Academician L. S. Pontryagin on 27 XII 1962)

In this paper estimates are established for the maximum of the modulus, the modulus of continuity, and a Liouville-type theorem for solutions of elliptic and parabolic equations of the form

\[ Lu=\sum_{i,j=1}^{n}\frac{\partial}{\partial x_i} \left(a_{ij}(x)\frac{\partial u}{\partial x_j}\right) +\sum_{i=1}^{n} b_i(x)\frac{\partial u}{\partial x_i} +c(x)u+f(x)=0, \tag{1} \]

\[ \sum_{i,j=1}^{n}\frac{\partial}{\partial x_i} \left(a_{ij}(t,x)\frac{\partial u}{\partial x_j}\right) +\sum_{i=1}^{n} b_i(t,x)\frac{\partial u}{\partial x_i} +c(t,x)u+f(t,x)=\frac{\partial u}{\partial t}; \tag{2} \]

\[ x=(x_1,\ldots,x_n);\quad a=\|a_{ij}\|,\quad \lambda^{-1}|\xi|^2\leq(\xi,a\xi),\quad |a_{ij}|\leq\lambda . \tag{3} \]

Regarding the coefficients \(b_i, c\), and \(f\), it is assumed that they belong to the corresponding spaces \(L_p\), where the summability exponents \(p\) are greater than or equal to certain “limiting” exponents depending on \(n\) (the question of estimates at the limiting exponents is also considered here). The paper develops methods used in \((^{1-4})\). Analogous questions were studied in papers \((^{1-10})\), etc. We note that in the case where the summability exponents \(p\) are greater than the “limiting” ones, Theorem 2 was proved in paper \((^8)\), while a result close to Theorem 4B, under stronger a priori assumptions on the solution \(u(t,x)\), was established in paper \((^{10})\).

1. We shall denote by \(\Omega\) a domain in \(E_n(x_1,\ldots,x_n)\), by \(K_r\) the ball \(\{|x|\leq r\}\), and by \(Q\) and \(Q_r^h\) respectively the cylinders \(\{\Omega\times [T_1,T_2]\}\) and \(\{K_r\times[-h,0]\}\).

A generalized solution of equation (1) (or (2)) in \(\Omega\) (in \(Q\)) is a function \(u(x)\in W_2^1(\Omega)\) \(\bigl(u(t,x)\in W_2^1(Q)\bigr)\) satisfying the following identity for every \(\varphi(x)\in \overset{\circ}{C}{}^\infty(\Omega)\) \(\bigl(\varphi(t,x)\in C^\infty(Q),\ \varphi(t,x)\in \overset{\circ}{C}{}^\infty(\Omega)\) for \(T_1\leq t\leq T_2\bigr)\):

\[ \int_{\Omega}(\varphi_x,au_x)\,dx = \int_{\Omega}\varphi\bigl[(b,u_x)+cu+f\bigr]\,dx \tag{4} \]

or

\[ \iint_Q\bigl[\varphi u_t+(\varphi_x,au_x)\bigr]\,dx\,dt = \iint_Q \varphi\bigl[(b,u_x)+cu+f\bigr]\,dx\,dt, \tag{5} \]

where \(u_x=\operatorname{grad}_x u,\ \varphi_x=\operatorname{grad}_x\varphi,\ u_t=\partial u/\partial t,\ b=(b_1,\ldots,b_n)\).

If the function \(u(x)\) (or \(u(t,x)\)) satisfies (4) (or (5)), where instead of the sign \(=\) there stands the sign \(\leq\) and \(\varphi(x)\geq 0\) (or \(\varphi(t,x)\geq 0\)), then \(u(x)\) (or \(u(t,x)\)) is called a generalized subsolution in \(\Omega\) (or in \(Q\)).

In the proofs the following known inequality is used:

Lemma 1. Let \(u(x)\in W_2^1\) in \(K_\rho\), \(N\subseteq K_\rho\), and \(\operatorname{mes} N\geq c_0\,\operatorname{mes} K_\rho\). Then

\[ \left(\rho^{-n}\int_K u^{2k}\,dx\right)^{1/k} \leq c\left[ \rho^{2-n}\int_{K_\rho} u_x^2\,dx + \left(\rho^{-n}\int_N u^{2q}\,dx\right)^{1/q} \right], \]

where \(1\leq k\leq n/(n-2)\), \(q\geq 1\), the constant \(c\) depends on \(n\) and \(c_0\); if \(u(x)\in \overset{\circ}{W}{}_2^1\) in \(K_\rho\), then the second term on the right-hand side may be omitted (for \(n=2\) the number \(k\) may be arbitrary \(\geq 1\), but the constant \(c\) also depends on \(k\)).

In what follows, by \(c\) we denote constants depending on \(n\) and \(\lambda\), and also on the numbers \(p\) and \(s\), which will be specified below.

2. Equations of elliptic type.

Theorem 1. Let \(v(x)\) be a nonnegative subsolution of equation (1) in the ball \(K_{3r}\); let
\[ \|c(x)\|_{L_p(K_{3r})}+\|f(x)\|_{L_p(K_{3r})}=c_1,\qquad p>n/2, \]
\[ b_i\in L_s(K_{3r}),\qquad s=n\ \text{for } n\ge 3,\quad s>2\ \text{for } n=2, \]
and let
\[ cc_1 r^{2\gamma}\le 1 \]
and
\[ cr^{1-n/s}\|b_i(x)\|_{L_s(K_{3r})}\le 1, \]
where \(\gamma=1-n/2p\). Then
\[ \operatorname*{vrai\,max}_{K_r} v^2 \le c\left(r^{-n}\int_{K_{3r}} v^2\,dx+1\right). \]

If \(p=n/2\) for \(n\ge 3\), then \(v(x)\) is summable with any power in \(K_r\) (for \(n=2\) this follows directly from the embedding theorem (see Lemma 1)).

Let \(w=v^m\), \(m\ge 1\); substituting the corresponding test functions into the integral inequality for \(v(x)\) and applying Lemma 1, we obtain the following estimate for \(k>q=p/(p-1)\):
\[ \left(\rho^{-n}\int_{K_\rho} w^{2k}\,dx\right)^{1/k} \le c\left[\left(m^3+\frac{\rho^2}{\sigma^2}\right) \left((\rho+\sigma)^{-n}\int_{K_{\rho+\sigma}} w^{2q}\,dx\right)^{1/q} +m^{-2m}\right]. \]

On the basis of this inequality one carries out an iteration process, which also completes the proof of Theorem 1.

Theorem 2. If \(u(x)\) is a generalized solution of equation (1) in \(K_{3r}\) and the hypotheses of Theorem 1 on its coefficients are satisfied (\(p>n/2\)), then for \(\rho\le r\)
\[ \operatorname*{osc}_{K_\rho} u\le c_2\rho^\alpha, \]
where \(\alpha>0\) depends only on \(n,\lambda,\gamma\), while \(c_2\) depends on \(n,\lambda,\gamma,r,\|u\|_{L_2(K_{3r})}\).

By virtue of Theorem 1, for \(v=|u|\), which is a subsolution for an equation of the form (1), the function \(u(x)\) is bounded in \(K_r\) (hence one may assume that \(c(x)=0\) in equation (1)). Let
\[ M=\operatorname*{vrai\,max}_{K_{4\rho}} \pm u, \qquad 4\rho\le r_0(n,\lambda,\gamma)<r; \]
denote by \(w(x)\) that one of the functions
\[ \ln^+\bigl[1\pm(u/M)+\varepsilon\bigr] \]
which is equal to 0 on the set \(N\subset K_{3\rho}\) and
\[ \operatorname{mes}N\ge \tfrac12 \operatorname{mes}K_{3\rho} \]
(the number \(\varepsilon>0\) depends only on \(n\) and \(\lambda\)). Applying Theorem 1 to \(w\), we establish the dilemma: either
\[ \operatorname*{osc}_{K_\rho} u\le 2\rho^\gamma, \]
or
\[ \operatorname*{osc}_{K_\rho} u\le \eta\,\operatorname*{osc}_{K_{4\rho}} u, \]
where \(0<\eta<1\), from which Theorem 2 follows easily.

Remark. Theorems 1 and 2 are also valid for more general equations:
\[ Lu=\sum_{i=1}^{n}\frac{\partial}{\partial x_i}f_i(x), \]
where \(f_i\in L_{2p}\), \(p>n/2\). Let us also note that in the proof of Theorems 1 and 2, for \(p>(n+1)/2\) one may apply the device consisting in the introduction of new variables (see [4], item 6).

3. Equations of parabolic type.

Theorems 3A and 3B. Let \(v(t,x)\ge 0\) be a nonnegative generalized subsolution of equation (2) in \(Q_{2r}^{4r^2}\), and let one of the following conditions A or B be satisfied:

A. The functions
\[ b_i(t,x)\in L_s\bigl(Q_{2r}^{4r^2}\bigr), \qquad s=n+2\ \text{for } n\ge 3,\quad s>4\ \text{for } n=2, \]
\[ \|c(t,x)\|_{L_p(Q_{2r}^{4r^2})} +\|f(t,x)\|_{L_p(Q_{2r}^{4r^2})} =c_3<\infty, \qquad p>\frac n2+1, \]
and
\[ cr^{1-(n+2)/s}\|b_i(t,x)\|_{L_s(Q_{2r}^{4r^2})}\le 1 \]
and
\[ cc_3 r^{2\gamma}\le 1, \]
where
\[ \gamma=1-\frac{n+2}{2p}. \]

B. For \(-4r^2\le t\le 0\), the functions
\[ b_i(t,x)\in L_s(K_{2r}), \qquad s=n\ \text{for } n\ge 3,\quad s>2\ \text{for } n=2, \]
\[ \left(\|c(t,x)\|_{L_p(K_{2r})} +\|f(t,x)\|_{L_p(K_{2r})}\right)\le c_4<\infty, \qquad p>n/2, \]
and
\[ cr^{1-n/s}\|b_i(t,x)\|_{L_p(K_{2r})}\le 1, \qquad cc_4 r^{2\gamma}\le 1, \]
where
\[ \gamma=1-\frac{n}{2p}. \]

Then
\[ \operatorname*{vrai\,max}_{Q_r^{r^2}} v^2(t,x) \le c\left[ r^{-n-2}\iint_{Q_{2r}^{4r^2}} v^2\,dx\,dt+1 \right]. \]

If, for \(n \ge 3\), in condition A or B respectively \(p=(n+2)/2\), or \(p=n/2\), then \(v(t,x)\) is summable in \(Q_r^{r^2}\) with any power.

Under the condition that \(b_i(t,x)=c(t,x)=f(t,x)=0\), this theorem was established in paper \((^3)\).

Theorems 4A and 4B. Let \(u(t,x)\) be a generalized solution of equation (2) in \(Q_{2r}^{4r^2}\). Then, under assumption A or B on the coefficients of equation (2) (see Theorems 3A and 3B), for \(p \le r\)

\[ \operatorname*{osc}_{Q_\rho^{\rho^2}} u \le c_5 \rho^\alpha, \]

where \(\alpha>0\) depends only on \(n,\lambda,\gamma\), and \(c_5\) on \(n,\lambda,\gamma,r,\|u\|_{L_2(Q_{2r}^{4r^2})}\).

We shall consider here the case where in equation (2) \(c=f=0\); in this case the following Liouville-type theorem will also be established:

Theorem 5. Let \(u(t,x)\) be a generalized solution of equation (2) (with \(c=f=0\)) in the half-space \(\{t\le 0\}\); let

\[ \lambda^{-1}|\xi|^2 \le (\xi,a,\xi), \qquad |a_{ij}|\le \mu \]

and, for \(n\ge 3\),

\[ \text{a) } \quad (\lambda+\lambda^2) \left( \iint_{\{t\le 0\}} |b_i|^{n+2}\,dx\,dt \right)^{2/(n+2)} \le c(n) \]

or

\[ \text{b) } \quad \operatorname*{vrai\,max}_{-\infty<t\le 0}\lambda^2 \left( \int_{E_n} |b_i(t,x)|^n\,dx \right)^{2/n} \le c(n), \]

and for \(n=2\), \(b_i=0\).* Then there exists an \(\alpha_0\), depending only on \(n,\lambda,\mu\), such that from the condition

\[ |u(t,x)|\le H(r^\alpha+1) \quad \text{for } (t,x)\in Q_r^{r^2}, \qquad 0<\alpha<\alpha_0 \]

(\(H\) is a certain constant) it follows that \(u=\mathrm{const}\).

Let \(v(t,x)\) be a nonnegative subsolution in \(Q_{2r}^{4r^2}\). By means of the iterative process (see \((^3)\)) we establish the estimate **

\[ \operatorname*{vrai\,max}_{Q_r^{r^2}} v^2(t,x) \le c r^{-n-2}\iint_{Q_{2r}^{4r^2}} v^2\,dx\,dt . \tag{6} \]

Lemma 2. Let \(u(t,x)\ge 0\) be a generalized solution in \(Q_r^{r^2}\) and

\[ \operatorname{mes}\{(t,x)\in Q_r^{r^2};\ u(t,x)\ge 1\} \ge \frac12 \operatorname{mes} Q_r^{r^2}. \]

Then there exist numbers \(\theta\) and \(c\), \(0<\theta<1\), depending only on \(n,\lambda,\mu\), such that

\[ u(t,x)\ge c>0 \quad \text{in } Q_{\theta r}^{\theta^2 r^2}. \]

We first prove the following assertion: there exist numbers \(\alpha,\beta\), and \(h\), depending only on \(n,\lambda,\mu\) \((0<\alpha,\beta,h<1)\), such that for \(-\alpha r^2\le t\le 0\)

\[ \operatorname{mes} N_t\{x\in K_{\beta r};\ u(t,x)\ge h\} \ge \frac14 \operatorname{mes} K_{\beta r}. \]

Let

\[ \mu(t)=\operatorname{mes}\{x\in K_r;\ u(t,x)\ge 1\} \]

and \(0<\alpha<\frac12\); it is obvious that there exists

\[ \tau\in[-r^2,-\alpha r^2] \]

such that

\[ \mu(\tau)\ge (1/2-\alpha)(1-\alpha)^{-1}\operatorname{mes}K_r. \]

Let

\[ v=f(u)=\ln^+ \frac{1}{u+h}, \qquad 0<h<\frac12; \]

put, in the integral identity for \(u(t,x)\),

\[ \varphi=f'(u)\eta^2(y)\xi(t), \]

where \(y=|x|\), \(\eta(y)=1\) for \(0\le y\le \beta r<r\), \(\eta(y)=0\) for \(y\ge r\), and \(\eta(y)\) is linear for \(\beta r\le y\le r\); \(\xi(t)=1\) for \(\tau\le t\le s\in[-\alpha r^2,0]\), \(\xi(t)=0\) for \(t<\tau\) and \(t>s\). We obtain the estimate

\[ \int_{K_{\beta r}} v(s,x)\,dx +\frac{1}{4\lambda}\int_\tau^s \int_{K_{\beta r}} v_x^2\,dx\,dt \le c\beta^{-n}(1-\beta)^{-1}\operatorname{mes}K_{\beta r} +\int_{K_r} v(\tau,x)\,dx; \]

hence

\[ \ln\frac{1}{2h}\operatorname{mes}[K_{\beta r}\setminus N_s] \le c\beta^{-n}(1-\beta)^{-1}\operatorname{mes}K_{\beta r} +\ln\frac{1}{h}\bigl(\operatorname{mes}K_r-\mu(\tau)\bigr) \le \]

\[ \le \left[ c\beta^{-n}(1-\beta)^{-1} +\ln\frac{1}{h}\,2^{-1}\beta^{-n}(1-\alpha)^{-1} \right]\operatorname{mes}K_{\beta r}. \]

We choose first \(\alpha\) and \(\beta\) so that

\[ \frac12(1-\alpha)\beta^n \le \frac23, \]

and then \(h\) so small that

\[ \operatorname{mes}[K_{\beta r}\setminus N_s] \le \frac34 \operatorname{mes}K_{\beta r} \]

for \(-\alpha r^2\le s\le 0\), whence the assertion being proved follows.

* Other conditions, analogous to those under which Theorem 3 is proved in paper \((^4)\), may also be imposed on the coefficients \(b_i\) in this theorem.

** Here and below the constants \(c\) depend on \(n,\lambda\), and \(\mu\).

Consider now \(w=g(u)=\ln^{+}\dfrac{h}{u+\varepsilon}\), where the positive number \(\varepsilon\ll h/2\) will be chosen below; just as above, we obtain the estimate

\[ \int_{-\alpha r^{2}}^{0}\int_{K_{3r}} w_x^{2}\,dx\,dt \le c\,\operatorname{mes}K_{3r} +c\int_{K_r} w(-\alpha r^{2},x)\,dx \le cr^n\ln\frac{3h}{\varepsilon}. \tag{7} \]

Applying Lemma 1 to \(w(t,x)\) for \(-\alpha r^2\le t\le 0\) (\(k=1\)), noting that \(w(t,x)=0\) for \(x\in N_t\), \(\operatorname{mes}N_t\ge \frac14\operatorname{mes}K_{3r}\), and taking (7) into account, we find that

\[ \iint_{Q_{3r}^{\alpha r^2}} w^2\,dx\,dt \le cr^{n+2}\ln\frac{3h}{\varepsilon}. \]

Since \(w(t,x)\) is a nonnegative subsolution, by estimate (6), for some \(\theta\), \(0<\theta=\theta(n,\lambda,\mu)<1\), for \((t,x)\in Q_{\theta r}^{\theta^2r^2}\),

\[ w^2(t,x)\le c\ln\frac{3h}{\varepsilon}. \]

At any point \((t,x)\in Q_{\theta r}^{\theta^2r^2}\) either \(u(t,x)\ge h-\varepsilon\ge h/2\), or \(u<h-\varepsilon\); in the second case, for sufficiently small \(\varepsilon=\varepsilon(n,\lambda,\mu)\),

\[ \ln^2\frac{h}{u+\varepsilon}\le c\ln\frac{3h}{\varepsilon} <\left(\ln\frac{h}{\sqrt{\varepsilon}}\right)^2, \]

whence \(u\ge \sqrt{\varepsilon}-\varepsilon=\sqrt{\varepsilon}(1-\sqrt{\varepsilon})\).

For the proof of Theorems 4B and 5, note that the generalized solution \(u(t,x)\) is bounded in \(Q_r^{r^2}\), and put \(M=\operatorname{vrai\,max}\pm u(t,x)\) in \(Q_r^{r^2}\); applying Lemma 2 to that one of the functions \(1\pm u/M\) which is \(\ge1\) on a set of measure \(\ge \frac12\operatorname{mes}Q_r^{r^2}\), we find that

\[ \operatorname*{osc}_{Q_{\theta r}^{\theta^2r^2}} u \le \eta \operatorname*{osc}_{Q_r^{r^2}} u, \qquad \eta=\eta(n,\lambda,\mu)<1, \]

from which Theorems 4B and 5 follow.

1. The following examples show that in Theorems 1, 2, 3B, and 4B one cannot decrease the exponents \(p\) and \(s\) of summability of the coefficients \(b_i,c\), and \(f\).

Example 1. \(u=(-\ln r)^\alpha\), \(r=|x|\), \(0<\alpha<1/3\),

\[ \Delta u=\alpha r^{-2}(-\ln r)^{\alpha-2} [\alpha-1+(n-2)\ln r]=f =\alpha(r\ln r)^{-2}[\alpha-1+(n-2)\ln r]u=cu \]

\[ =\sum_{i=1}^{n}[\alpha-1+(n-2)\ln r]\,r^{-2}(\ln r)^{-1}x_i\frac{\partial u}{\partial x_i} =\sum_{i=1}^{n} b_i\frac{\partial u}{\partial x_i}; \]

here \(p=n/2\), \(s<n\) for \(n\ge3\), \(s=2\) for \(n=2\); the solution is unbounded.

Example 2. \(u=r^\alpha\), \(0<-\alpha<n/2-1\) (\(n\ge3\)),

\[ \Delta u=\alpha r^{\alpha-2}(\alpha-2+n)=f=cu=\sum_{i=1}^{n} b_i\frac{\partial u}{\partial x_i}; \]

here \(p<n/2\), \(s<n\); the solution is summable only to a certain power.

The following example shows that Theorems 2 and 4B for \(p=n/2\), \(s<n\) (\(n\ge3\)), \(s=2\) (\(n=2\)) do not hold even under the additional assumption that the solution is bounded.

Example 3. \(u=1/\ln r\),

\[ \Delta u=r^{-2}\ln^{-3}r\,[2+(2-n)\ln r]=f=cu =\sum_{i=1}^{n} b_i\frac{\partial u}{\partial x_i} \]

(see Example 1).

Analogous examples can also be constructed for Theorems 3A and 4A.

Moscow State University
named after M. V. Lomonosov

Received
27 XII 1962

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