Mathematics

A. V. Ivanov

A Priori Estimates for Solutions of Linear Second-Order Equations of Elliptic and Parabolic Types

(Presented by Academician A. D. Aleksandrov on 17 X 1964)

In this paper integral and pointwise estimates are formulated, obtained by the author for classes of functions satisfying inequalities of a special form and including solutions of various second-order differential equations. These results are used in obtaining a priori estimates for solutions of linear second-order equations of elliptic and parabolic types. For linear equations of parabolic type an existence and uniqueness theorem is established for a certain class of generalized solutions. Results close to certain special cases 1) and 4) of Theorem 3 are contained in \((^6)\). Theorem 3, case 4), and Theorem 4 were proved in \((^7)\), but for a narrower class of generalized solutions. In the paper the methods created in \((^{1,2})\) are developed.

Let \(\Omega\) denote a certain domain in the space \(E_n\), and let \(Q\) denote the cylinder \(\Omega \times [t_1,t_2]\). By \(K_\rho = K_\rho(x_0)\) we shall denote the ball
\[ |x-x_0|\leq \rho,\quad |x-x_0|=\left(\sum_{i=1}^{n}(x_i-x_{0i})^2\right)^{1/2}, \]
and by \(Q_{\rho,\tau}=Q_{\rho,\tau}(x_0,t_0)\) the cylinder
\[ K_\rho(x_0)\times [t_0-\tau,t_0]. \]
If a certain set \(D\) and a certain function \(v(P)\), defined and measurable in \(D\), are given, then by \(\{D,v>s\}\) is denoted the set of those points \(P\) in \(D\) where \(v(P)>s\). We shall also denote by \(\|v\|_{m,D}\) the expression
\[ \|v\|_{m,D}=\left(\operatorname{mes}^{-1}D\cdot \int_D |v|^m dP\right)^{1/m}. \]

Let, as usual, the function \(v_+(P)\) denote the positive part of the function \(v(P)\), i.e.
\[ v_+(P)=\frac12\bigl(|v(P)|+v(P)\bigr). \]
By \(W_m^1(\Omega)\) and \(W_m^1(Q)\) are denoted the well-known spaces of S. L. Sobolev. We introduce the spaces \(V_m^{1,0}(Q)\) and \(\overset{\circ}{V}{}_m^{1,0}(Q)\). Let the norm \(\|u\|_{V_m^{1,0}(Q)}\) be defined for all functions \(u(x,t)\) continuously differentiable in the cylinder \(\overline Q\) by
\[ \|u\|_{V_m^{1,0}(Q)} = \max_{t\in[t_1,t_2]} \left(\int_\Omega |u|^m dx\right)^{1/m} + \left(\iint_Q |u_x|^m dx\,dt\right)^{1/m}. \tag{1} \]

By the space \(V_m^{1,0}(Q)\) \([\overset{\circ}{V}{}_m^{1,0}(Q)]\) we shall mean the closure, with respect to the norm (1), of the set of all smooth functions in \(\overline Q\) (the set of all smooth functions equal to zero near the lateral surface \(S_{\mathrm{lat}}\) of the cylinder \(Q\)). We note that functions from \(V_m^{1,0}(Q)\) have generalized derivatives \(u_{x_i}\in L_m(Q)\) and are continuous in \(t\) in \([t_1,t_2]\) as elements of \(L_m(\Omega)\).

O. A. Ladyzhenskaya and N. N. Ural’tseva, in connection with the study of properties of solutions of second-order elliptic equations in \((^{2e})\) (see also \((^{2a-2d})\)), considered the class of functions \(v(x)\in W_m^1(K_r)\), \(1<m\leq n\), satisfying, for all \(k\geq k'\), \(\sigma\in(0,1)\), \(\rho\in[r/2,r]\), the inequalities

of the form

\[ \int_{\{K_{\rho-\sigma\rho},\, v>k\}} |v_x|^m\,dx \leq \gamma_1(\sigma\rho)^{-m} \int_{\{K_\rho,\, v>k\}} (v-k)^m\,dx + \gamma_2 k^\alpha \rho^{-n\varepsilon} \operatorname{mes}^{1-m/n+\varepsilon}\{K_\rho,\, v>k\}, \tag{2} \]

where \(k'\geq 0\), \(\gamma_1\geq 0\), \(\gamma_2\geq 0\), \(0\leq \alpha<m+m\varepsilon\), and \(\varepsilon>0\). For functions of such a class, an estimate of the maximum of the modulus has been obtained. Here we consider inequalities (2) also for nonpositive values of \(\varepsilon\), and obtain estimates in weaker norms than \(\|u\|=\max |u|\).

Lemma 1. Let the numbers \(r>0\), \(1<m\leq n\), and the numbers \(\varepsilon,\alpha\geq 0\), \(\gamma_1\geq 0\), \(\gamma_2\geq 0\), \(k'\geq 0\) be fixed. Suppose that the function \(v(x)\in W_m^1(K_r)\) and, for all \(k\geq k'\), \(\sigma\in(0,1)\), \(\rho\in[r/2,r]\), satisfies inequality (2).

1) Let \(-(n-m)/m<\varepsilon<0\), \(0\leq \alpha\leq m\). Then

\[ r^{-n}\operatorname{mes}\{K_{r/2},\, v>s\} \leq c_1 s^{-(m-\alpha)/|\varepsilon|} \bigl(\|v_+\|_{m,K_r}+1\bigr)^{(m-\alpha)/|\varepsilon|}; \]

\[ s\geq s_0=c_0^{(1)}\bigl(\|v_+\|_{m,K_r}+1\bigr). \tag{3} \]

2) Let \(\varepsilon=0\), \(0\leq \alpha\leq m\). There exists a number \(\delta>0\), determined only by the numbers \(n,m\), such that, under the condition \(\gamma_2\leq \delta e^{-p}\), and for any \(p>1\), the inequality

\[ r^{-n}\operatorname{mes}\{K_{r/2},\, v>s\} \leq c_2 s^{-p}\bigl(\|v_+\|_{m,K_r}+1\bigr)^p; \]

\[ s\geq s_0=c_0^{(2)}\bigl(\|v_+\|_{m,K_r}+1\bigr). \tag{4} \]

holds.

3) Let \(\varepsilon=0\), \(0\leq \alpha<m\). Then

\[ r^{-n}\operatorname{mes}\{K_{r/2},\, v>s\} \leq c_3 \exp\left\{-c_4\bigl(\|v_+\|_{m,K_r}+1\bigr)^{-(m-\alpha)/m}s^{(m-\alpha)/m}\right\}, \]

\[ s\geq s_0=c_0^{(3)}\bigl(\|v_+\|_{m,K_r}+1\bigr). \tag{5} \]

In (3)—(5), the constants \(c_0^{(i)}\) and \(c_j\) \((j\ne 2)\) are determined only by the numbers \(n,m,\alpha,\varepsilon,\gamma_1,\gamma_2,k'\), while the constant \(c_2\) is determined by the same numbers and by the number \(p\).

In the study of parabolic equations, in \((2^e)\) (see also \((2a\text{–}2d)\)) there was introduced and studied a class of functions \(v(x,t)\) from \(W_m^1(Q_{r,r^m})\), \(1<m\leq n\) (let \(Q_{r,r^m}=Q_{r,r^m}(0,0)\)), satisfying, for all \(k\geq k'\), \(\sigma\in(0,1)\), \(\rho\in[r/2,r]\), and for almost all \(t\in[-r^m,0]\), inequalities of the form

\[ \frac{\partial}{\partial t} \int_{\{K_{\rho-\sigma\rho},\, v(t)>k\}} (v-k)^m\,dx + \nu \int_{\{K_{\rho-\sigma\rho},\, v(t)>k\}} |v_x|^m\,dx \leq \]

\[ \leq \gamma_1(\sigma\rho)^{-m} \int_{\{K_\rho,\, v(t)>k\}} (v-k)^m\,dx + \gamma_2 k^\alpha \rho^{-n\varepsilon} \operatorname{mes}^{1-m/n+\varepsilon}\{K_\rho,\, v(t)>k\}, \]

where \(\nu>0\), \(\gamma_1\geq 0\), \(\gamma_2\geq 0\), \(k'>0\), \(0\leq \alpha<m+m\varepsilon\), \(\varepsilon>0\).

For the study of parabolic equations whose coefficients are subject to other conditions, we introduce and study (see \((^8)\)) the class of functions \(v(x,t)\in V_m^{1,0}(Q_{r,r^m})\), satisfying, for all \(k\geq k'\), \(\sigma\in(0,1)\), \(\theta\in(0,1)\), \(\rho\in[r/2,r]\), \(\tau\in[r^m/2,r^m]\), inequalities of the form

\[ \max_{t\in[-\tau,0]} \int_{\{K_{\rho-\sigma\rho},\, v(t)>k\}} (v-k)^m\,dx + \iint_{\{Q_{\rho-\sigma\rho,\tau-\theta\tau},\, v>k\}} |v_x|^m\,dx\,dt \leq \tag{6} \]

\[ \leq \gamma_1\left[(\sigma\rho)^{-m}+(\theta\tau)^{-1}\right] \iint_{\{Q_{\rho,\tau},\, v>k\}} (v-k)^m\,dx\,dt + \]

\[ + \gamma_2 k^\alpha \rho^{-n\varepsilon}\tau^{-\varepsilon} \operatorname{mes}^{1-m/(m+n)+\varepsilon}\{Q_{\rho,\tau},\, v>k\}. \]

Lemma 2. Let the numbers \(r>0\), \(m\geq 1-n/2+\sqrt{n^2+4}/2\), and the numbers \(\varepsilon,\alpha\geq 0\), \(\gamma_1\geq 0\), \(\gamma_2\geq 0\), \(k'\geq 0\) be fixed. Suppose that the function

\(v(x,t) \in V_m^{1,0}(Q_r,r^m)\), and for all \(k \geq k'\), \(\sigma \in (0,1)\), \(\theta \in (0,1)\), \(\rho \in [r/2,r]\) and \(\tau \in [r^m/2,r^m]\) satisfies inequality (6).

1) Let \(-n/(m+n) < \varepsilon < 0,\ 0 \leq \alpha \leq m\). Then

\[ r^{-(m+n)} \operatorname{mes}\{Q_{r/2,r^m/2},\, v>s\} \leq c_1 s^{-(m-\alpha)/|\varepsilon|} \left(\|v_+\|_{m,Q_{r,r^m}}+1\right)^{(m-\alpha)/|\varepsilon|}; \]

\[ s \geq s_0=c_0^{(1)}\left(\|v_+\|_{m,Q_{r,r^m}}+1\right). \tag{7} \]

2) Let \(\varepsilon=0,\ 0 \leq \alpha \leq m\). There exists a number \(\delta>0\), determined by the numbers \(n,m\), such that under the condition \(\gamma_2 \leq \delta e^{-p}\), for any \(p>1\) the inequality

\[ r^{-(m+n)} \operatorname{mes}\{Q_{r/2,r^m/2},\, v>s\} \leq c_2 s^{-p}\left(\|v_+\|_{m,Q_{r,r^m}}+1\right)^p; \]

\[ s \geq s_0=c_\theta^{(2)}\left(\|v_+\|_{m,Q_{r,r^m}}+1\right). \tag{8} \]

holds.

3) Let \(\varepsilon=0,\ 0 \leq \alpha<m\). Then

\[ r^{-(m+n)} \operatorname{mes}\{Q_{r/2,r^m/2},\, v>s\} \leq c_3 \exp\left\{-c_4\left(\|v_+\|_{m,Q_{r,r^m}}+1\right)^{-(m-\alpha)/m}s^{(m-\alpha)/m}\right\}, \]

\[ s \geq s_0=c_0^{(3)}\left(\|v_+\|_{m,Q_{r,r^m}}+1\right). \tag{9} \]

4) Let \(\varepsilon>0,\ 0 \leq \alpha \leq m\). Then

\[ \operatorname*{vrai\,max}_{\{Q_{r/2,r^m/2},\,v>0\}} v \leq c_5\left(\|v_+\|_{m,Q_{r,r^m}}+1\right). \tag{10} \]

In inequalities (7)—(10) the constants \(c_0^{(i)}, c_j(j\ne 2)\) depend on \(n,m,\varepsilon,\alpha,\gamma_1,\gamma_2,k'\), and the constant \(c_2\) on \(n,m,\varepsilon,\alpha,\gamma_1,\gamma_2,k',p\).

1. Consider in the cylinder \(Q_T=\Omega \times [0,T]\) the equation

\[ \frac{\partial u}{\partial t} - \sum_{i=1}^n \frac{\partial}{\partial x_i} \left( \sum_{j=1}^n a_{ij}\frac{\partial u}{\partial x_j} +a_i u+f_i \right) + \sum_{i=1}^n b_i\frac{\partial u}{\partial x_i} +cu+g=0,\qquad n\geq 1, \tag{11} \]

where

\[ \nu \sum_{i=1}^n \xi_i^2 \leq \sum_{i,j=1}^n a_{ij}(x,t)\xi_i\xi_j \leq \mu \sum_{i=1}^n \xi_i^2,\qquad \nu,\mu>0;\qquad a_i \in L_{n+2}(Q_T); \]

\[ b_i \in L_{n+2}(Q_T);\qquad c \in L_{(n+2)/2}(Q_T);\qquad f_i \in L_2(Q_T);\qquad g \in L_{2(n+2)/(n+4)}(Q_T). \]

A generalized solution (g.s.) of equation (11) of the class \(V_2^{1,0}(Q_T)\) \((\dot V_2^{1,0}(Q_T))\) is a function \(u(x,t)\in V_2^{1,0}(Q_T)\) \((\dot V_2^{1,0}(Q_T))\) satisfying, for all \(t_1\) and \(t_2\) from the interval \([0,T]\) and \(\Phi(x,t)\in \dot W_2^1(\Omega\times [t_1,t_2])\), the equality

\[ \left.\int_\Omega u\Phi\,dx\right|_{t_1}^{t_2} + \int_{t_1}^{t_2}\int_\Omega \left[ -u\Phi_t + (a_{ij}u_{x_j}+a_i u+f_i)\Phi_{x_i} + (b_i u_{x_i}+cu+g)\Phi \right]\,dx\,dt =0. \]

Theorem 1. Every g.s. of equation (11) of the class \(\dot V_2^{1,0}(Q)\) satisfies the inequality

\[ \|u(x,t)\|_{V_2^{1,0}(Q_T)} \leq c\left[ \sum_{i=1}^n \|f_i(x,t)\|_{L_2(Q_T)} + \|g\|_{L_{2(n+2)/(n+4)}(Q_T)} + \|u(x,0)\|_{L_2(\Omega)} \right], \]

where \(c=c(\nu^{-1},T/\tau)\); \(\tau\) is determined from the equality

\[ \sup_{\substack{0\leq t_1\leq T-\tau\\ i=1,\ldots,n}} \|a_i^2+b_i^2+c\|_{L_{(n+2)/2}(\Omega\times [t_1,t_1+\tau])} = \nu\min(1,\nu)\delta,\qquad \delta=\delta(n)>0. \]

Theorem 2. For every \(\varphi(x)\in L_2(\Omega)\) there exists a unique g.s. of equation (11) of the class \(\dot V_2^{1,0}(Q_T)\), equal to \(\varphi(x)\) for \(t=0\).

If the g.s. \(u(x,t)\) of equation (11) of the class \(V_2^{1,0}(Q_T)\) satisfies the condition
\(\|u(x,t+h)-u(x,t)\|_{L_2(Q_T)}=o(h^{1/2})\), then it will be called a g.s. of the class \(V_2^{1,1/2}(Q_T)\).

Theorem 3. Let \(u(x,t)\) be an arbitrary generalized solution of equation (11) from the class \(V_2^{1,1/2}(Q_T)\). There exists a quantity \(d\) (depending only on \(n\) and \(\nu\) in cases 1), 3), and 4) singled out below, and only on \(n,\nu,p\) in case 2)) such that, if the constant \(r_0>0\) is determined from the equality

\[ \sup_{\substack{Q_{r_0,r_0^2}\subset Q_T\\ i=1,\ldots,n}} \left\|a_i^2+b_i^2+c\right\|_{L_{(n+2)/2}(Q_{r_0,r_0^2})}=d, \]

then in every cylinder \(Q_{r,r^2}\), \(r\le r_0\), the following estimates hold for the solution \(u\):

1) Let \(a_i\in L_{n+2}(Q_T)\), \(b_i\in L_{n+2}(Q_T)\), \(c\in L_{(n+2)/2}(Q_T)\), \(f_i\in L_q(Q_T)\), \(2<q<n+2\), \(g\in L_p(Q_T)\), \(2(n+2)/(n+4)<p<(n+2)/2\). Then, for any \(\sigma>0\),

\[ \|u\|_{p_*-\sigma,Q_{r/2,r^2/2}}\le c_1\bigl(\|u\|_{2,Q_{r,r^2}}+1\bigr); \qquad p_*=\min\left[ \frac{q(n+2)}{n+2-q},\, \frac{p(n+2)}{n+2-2p} \right]. \]

2) Let \(a_i\in L_{n+2}(Q_T)\), \(b_i\in L_{n+2}(Q_T)\), \(c\in L_{(n+2)/2}(Q_T)\), \(f_i\in L_{n+2}(Q_T)\), \(g\in L_{(n+2)/2}(Q_T)\). Then, for any \(p>1\),

\[ \|u\|_{p,Q_{r/2,r^2/2}}\le c_2\bigl(\|u\|_{2,Q_{r,r^2}}+1\bigr). \]

3) Let \(a_i\in L_q(Q_T)\), \(b_i\in L_{n+2}(Q_T)\), \(c\in L_{q/2}(Q_T)\), \(f_i\in L_{n+2}(Q_T)\), \(g\in L_{(n+2)/2}(Q_T)\), \(q>n+2\). Then

\[ r^{-(n+2)} \iint_{Q_{r/2,r^2/2}} \exp\left\{c_3\bigl(\|u\|_{2,Q_{r,r^2}}+1\bigr)^{-1}|u(x,t)|\right\}\,dx\,dt \le c_4 . \]

4) Let \(a_i\in L_q(Q_T)\), \(b_i\in L_{n+2}(Q_T)\), \(c\in L_{q/2}(Q_T)\), \(f_i\in L_q(Q_T)\), \(g\in L_{q/2}(Q_T)\), \(q>n+2\). Then

\[ \operatorname{vrai\,max}_{Q_{r/2,r^2/2}} |u| \le c_5\bigl(\|u\|_{2,Q_{r,r^2}}+1\bigr). \]

In the inequalities given above, the constants \(c_i\) are determined by the numbers \(n,\nu,\mu\) and by the norms \(\|\cdot\|_{L_p(Q_{r,r^2})}\) of the coefficients \(a_i,c,f_i\) and \(g\) with those exponents \(p\) which are indicated in the hypotheses of these inequalities. In addition, the constant \(c_1\) also depends on \(\sigma\) and \(r_0\), and the constant \(c_2\) on \(p\).

Theorem 4. Under the assumptions of Theorem 3, 4), the estimate

\[ |u|_{C_{\alpha,\alpha/2}(Q_{r/2,r^2/2})} \le c\left(\operatorname{vrai\,max}_{Q_{r,r^2}} |u|,\, n,\,\nu,\,\mu\right), \qquad \alpha>0, \]

is valid, where

\[ |u|_{C_{\alpha,\alpha/2}(Q_{r/2,r^2/2})} = \operatorname{vrai\,max}_{(x,t),(x',t')\in Q_{r/2,r^2/2}} \frac{|u(x,t)-u(x',t')|} {|x-x'|^\alpha+|t-t'|^{\alpha/2}}, \]

and the constant \(c\) also depends on the norms
\(\|a_i,f_i\|_{L_q(Q_{r,r^2})}\), \(\|c,g\|_{L_{q/2}(Q_{r,r^2})}\), \(q>n+2\).

1. With the aid of Lemma 1, results analogous to Theorem 3, cases 1), 2), and 3), are obtained for generalized solutions from the class \(W_2^1(\Omega)\) of uniformly elliptic equations. Results close to some elliptic analogues of Theorem 3, cases 1) and 2), were obtained in \((3^{a},\,4,\,5)\). A result close to the special case of the theorem corresponding to Theorem 3, case 3), was obtained in \((3^{b})\).

In conclusion, the author expresses sincere gratitude to O. A. Ladyzhenskaya and N. N. Ural’tseva for their attention to this work.

Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
5 X 1964

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