UDC 517.946.9

MATHEMATICS

M. D. IVANOVICH

ESTIMATES OF SOLUTIONS OF GENERAL BOUNDARY-VALUE PROBLEMS FOR PARABOLIC SYSTEMS IN (C_{l+\alpha(r)}(Q))

(Presented by Academician I. G. Petrovskii on 24 X 1967)

In ((^1)) Schauder-type estimates were obtained for solutions of general boundary-value problems for general parabolic systems. For systems parabolic in the sense of I. G. Petrovskii, in ((^2)) Schauder-type estimates were obtained inside the domain in ([\cdot]_{i+\alpha(r)}) ((\alpha(r)) is the refined Hölder exponent) under the condition that

[
A r^{\alpha(r)}=\int_0^r t^{\alpha(t)-1}\,dt<\infty .
\tag{*}
]

S. N. Kruzhkov in ((^3)) showed that condition ((*)) is minimal under which Schauder-type estimates are possible.

In the present paper we obtain, under the minimal condition ((*)), estimates for solutions of general parabolic systems inside the domain and for solutions of the Cauchy problem. For solutions of general boundary-value problems the estimates are obtained under the condition that (\alpha(r)\in \mathcal A_l), where by (\mathcal A_l) we denote the class of refined Hölder exponents for which (A^l r^{\alpha(r)}<\infty).

Consider in a domain (\mathcal D) of the ((n+1))-dimensional space ((x_1,x_2,\ldots,x_n,t)=(x,t)) the system of differential equations

[
\mathcal L(x,t;D_x,D_t)u(x,t)=f(x,t),
\tag{1}
]

where (u=(u_1,u_2,\ldots,u_m)); (f=(f_1,f_2,\ldots,f_m)); (\mathcal L(x,t;D_x,D_t)) is a square matrix with elements (l_{kj}(x,t;D_x,D_t)).

System (1) is called parabolic if there exist integers (s_k,t_j) ((k,j=1,2,\ldots,m)) such that:

1) The degree of the polynomial (l_{kj}(a,t;i\xi\lambda,p\lambda^{2b})) with respect to the variable (\lambda) at each point ((x,t)\in\mathcal D) does not exceed (s_k+t_j), and if (s_k+t_j<0), then (l_{kj}=0). Let (\mathcal L^0) be the principal part of the matrix (\mathcal L).

2) There exists a constant (\delta>0) such that the roots of the polynomial (L(x,t;i\xi,p)=\det|\mathcal L^0(x,t;i\xi,p)|) with respect to the variable (p), for any real (\xi), satisfy the inequality

[
\operatorname{Re} p_s \le -\delta|\xi|^{2b}
]

for all points ((x,t)\in\mathcal D).

Let

[
\sum (s_i+t_i)=2br,\qquad r>0,\qquad \max_i s_i=0.
]

By (C_{l+\alpha(r)}(\mathcal D)) we shall denote the space of functions (v(x,t)) for which the norm is finite

[
|v|{l+\alpha(r)}^{\mathcal D}=\sum[v]}^{lj^{\mathcal D}+[v],}^{\mathcal D
]

where

[
[v]^{\mathscr D}{j}=
\sum\left|D_t^\mu D_x^\nu v(x,t)\right|,}\sup_{\mathscr D
]

[
[v]^{\mathscr D}{l+\alpha(r)}=[v]^{\mathscr D}}+[v]^{\mathscr D}_{l+\alpha(r),t
\equiv
]

[
\equiv
\sum_{(2\mu+\nu)=l}\sup_{(x,t),(x',t)\in\mathscr D}
\frac{\left|D_t^\mu D_x^\nu v(x,t)-D_t^\mu D_x^\nu v(x',t)\right|}
{\left|x-x'\right|^{\alpha(|x-x'|)}}
+
]

[
+
\sum_{0<l-2b\mu-\nu<2b}\sup_{(x,t),(x,t')\in\mathscr D}
\frac{\left|D_t^\mu D_x^\nu v(x,t)-D_t^\mu D_x^\nu v(x,t')\right|}
{\left|t-t'\right|^{(l-2b\mu-\nu+\alpha(|t-t'|^{1/2b}))/2b}}.
]

For (\alpha(r)\in\mathcal A_1) set

[
B\alpha(r)=\frac{1}{\ln r}\ln\frac{Ar^{\alpha(r)}}{(Ar^{\alpha(r)})(1)}.
]

Theorem 1. Let (u(x,t)=(u_1(x,t),\ldots,u_m(x,t))) be a solution of system (1) in a bounded domain (\mathscr D), and let the derivative (D_t^\mu D_x^\nu u_j(x,t)), (2b\mu+\nu=t_j), be continuous in (\mathscr D). Let the coefficients of the operators (l_{k_j}(x,t;D_xD_t)) be bounded by a constant (K_1) in the norms (|\cdot|^{\mathscr D}_{l-s_k+\alpha(r)}), (l\geq 0), (\alpha(r)\in\mathcal A_1). Let

[
M=\sum_i\left(|f_i|^{\mathscr D}{\,l-s_i+\alpha(r)}+|u_i|^{\mathscr D}\right)<\infty .
]

Then the estimate holds

[
|u_j|^{\mathscr D'}_{t_j+l+B\alpha(r)}\leq KM,\qquad j=1,2,\ldots,m,
]

where (\overline{\mathscr D}'\subset\mathscr D), and the constant (K) depends on (\mathscr D'), (K_1), (n), (m), (\delta), (s_1,\ldots,s_m), (t_1,\ldots,t_m), the diameter of the domain (\mathscr D), and the function (\alpha(r)).

Consider in (\mathscr D_{n+1}^{(T)}=E_n\times[0,T]), (T<\infty), the Cauchy problem for system (1)

[
\mathscr L(x,t;D_x,D_t)u(x,t)=f(x,t),\qquad
\mathscr E(x;D_x,D_t)u(x,t)\big|_{t=0}=\varphi(x), \tag{2}
]

where (\mathscr E(x;D_x,D_t)) is a matrix with elements (C_{\gamma j}(x,D_x,D_t)) ((\gamma=1,2,\ldots,r,\ j=1,2,\ldots,m)). Suppose that there exist integers (\rho_\gamma) such that the degree of the polynomial (C_{\gamma j}(x,i\xi\lambda,p\lambda^{2b})) in (\lambda) does not exceed (\rho_\gamma+t_j), and if (\rho_\gamma+t_j<0), then (C_{\gamma j}=0). Let (\mathscr E) satisfy the complementarity condition (see (1), § 1).

Theorem 2. Let the coefficients of the operators (l_{kj}), (C_{\gamma j}) be bounded by a constant (K_2) in the norms (|\cdot|^{\mathscr D_{n+1}^{(T)}}{l-s_k+\alpha(r)}), (|\cdot|^{E_n}), respectively, (l\geq 0), (\alpha(r)\in\mathcal A_1).

Then, for arbitrary

[
f_k(x,t)\in C_{l-s_k+\alpha(r)}(\mathscr D_{n+1}^{(T)}),\qquad
\varphi_\gamma(x)\in C_{l-\rho_\gamma+\alpha(r)}(E_n),
]

the Cauchy problem (2) has a unique solution with

[
u_j(x,t)\in C_{l+t_j+B\alpha(r)}(\mathscr D_{n+1}^{(T)})
]

and

[
\sum_{j=1}^{m}|u_j|^{\mathscr D_{n+1}^{(T)}}{l+t_j+B\alpha(r)}
\leq
K\left(
\sum}^{m}|f_j|^{\mathscr D_{n+1}^{(T)}{l-s_j+\alpha(r)}
+
\sum}^{r}|\varphi_j|^{E_n{l-\rho\gamma+\alpha(r)}
\right),
]

where the constant (\gamma) depends on (\delta), (K_2), (n), (\rho_\gamma), (s_i), (t_j), (Ar^{\alpha(r)}), and the constant in the complementarity condition.

Let (Q=\Omega\times[0,T]), (T<\infty), where (\Omega) is a bounded domain with boundary (S\in C_{l+t_{\max}+\alpha(r)}); (\Gamma=S\times[0,T]).

Consider in (Q) the problem

[
\mathscr L u=\mathscr L(x,t;D_xD_t)u(x,t):=f(x,t),
]

[
\mathscr E(x,D_xD_t)u(x,t)\big|_{t=0}=\varphi(x), \tag{3}
]

[
\mathscr B(x,t;D_xD_t)u(x,t)\big|_{\Gamma}=\Phi(x,t),
]

where (\mathfrak B(x,t;D_x,D_t)) is a matrix with elements (B_{qj}(x,t;D_x,D_t)) ((q=1,\ldots,br)). Suppose that there exist integers (\sigma_q) such that the degree of the polynomial (B_{qj}(x,t;i\xi\lambda,p\lambda^{2b})) in (\lambda) does not exceed (\sigma_q+t_j). Suppose that the matrices (\mathfrak B) and (\mathfrak E) satisfy the complementarity conditions (see ((^1)), § 1).

Let (x\in S) be an arbitrary point. Introduce a local coordinate system with center at the point (x). Suppose that in this neighborhood the coefficients of the operators (l_{kj}) in the local coordinates do not depend on the normal coordinate for (0\le t\le T).

Theorem 3. Let the coefficients of the operators (l_{kj}, C_{\gamma j}, B_{qj}) be bounded by the constant (K_3) in the norms

[
|\cdot|^{Q}{l-s_k+\alpha(r)},\qquad
|\cdot|^{\Omega},\qquad
|\cdot|^{\Gamma}_{l-\sigma_q+B\alpha(r)}
]

respectively, (l\ge \sigma_0=\max(0,\sigma_1,\ldots,\sigma_{br})), (\alpha(r)\in A_4).

If (f_j\in C_{l-s_j+\alpha(r)}(Q)), (\varphi_j\in C_{l-\rho_\gamma+\alpha(r)}(\Omega)), (\Phi_q\in C_{l-\sigma_q+B\alpha(r)}(\Gamma)) and the compatibility condition of order (l) is fulfilled (see ((^1)), § 14), then problem (3) has a unique solution
[
u(x,t)=(u_1(x,t),\ u_2(x,t),\ldots,u_m(x,t))
]
with
[
u_j(x,t)\in C_{t_j+l+B^2\alpha(r)}(Q)
]
and

[
\sum_{j=1}^{m}|u_j|^{Q}{l+t_j+B^2\alpha(r)}
\le
K\left(
\sum}^{m}|f_j|^{Q{l-s_j+B\alpha(r)}
+
\sum}^{r}|\varphi_j|^{Q{l-\rho\gamma+\alpha(r)}
+
\sum_{q=1}^{br}|\Phi_q|^{\Gamma}_{l-\sigma_q+B\alpha(r)}
\right),
]

where the constant (K) depends on (\delta, K_3, n, s_i, t_j, \rho_\gamma, \sigma_q, Ar^{\alpha(r)}) and the constants of the complementarity condition.

The proofs of these theorems are based on the results obtained in ((^1)) for problems with constant coefficients (2), (3), in which only the principal parts of the matrices (\mathfrak L,\mathfrak B,\mathfrak E) enter, and on the following estimates for the basic potentials (for notation see ((^1)), § 11) for (l\ge 0), (\alpha(r)\in A_1):

[
[(\Gamma*f)]^{E_{n+1}^{(T)}}{l+2br+B\alpha(r)}
\le
C[f]^{E,}^{(T)}}_{l+\alpha(r)
]

[
[(\Gamma*{1}\varphi)]^{\mathfrak D}^{(T)}{l+2b(r-1)+B\alpha(r)}
\le
C[\varphi]^{E_n},
]

[
[((K^{Q}{jq}*}\Phi)]^{\widetilde{\mathfrak D{n+1}^{(T)}}
\le
C|\Phi|^{\widetilde E_n^{(T)}}_{l+\alpha(r)}
]

for (2br-s_j+\sigma_q+2bQ>0;\ 0<t0) in (E_{n+1}), (E_n) is the space (x_n=0) in (E_{n+1}).

Remark. Since in the proof only the fact is used that the function (r^{\alpha(r)}) has the properties of a modulus of continuity and of a continuous derivative for (r>0), one may take as the refined Hölder exponent the function

[
\alpha(r)=\frac{1}{\ln r}\ln\frac{\omega(r)}{\omega(1)},
]

where (\omega(r)) is a smooth modulus of continuity, which need not satisfy conditions 1), 2) of the paper ((^2)).

Fundamental solutions and solutions of the Cauchy problem for parabolic systems in the sense of Petrovskii under analogous conditions were considered in ((^4)).

The author expresses his deep gratitude to T. D. Venttsel.

Moscow State University
named after M. V. Lomonosov

Received
24 X 1967

REFERENCES

(^1) V. A. Solonnikov, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 33, 3 (1965).
(^2) M. D. Ivanovich, DAN, 175, No. 5 (1967).
(^3) S. N. Kruzhkov, Matem. zametki, 2, No. 5 (1967).
(^4) E. I. Matijchuk, S. D. Eidelman, DAN, 165, No. 3 (1965).