Abstract
Full Text
UDC 517.946+532.501.32
MATHEMATICS
V. I. NALIMOV
A PRIORI ESTIMATES OF SOLUTIONS OF ELLIPTIC EQUATIONS IN THE CLASS OF ANALYTIC FUNCTIONS AND THEIR APPLICATIONS TO THE CAUCHY–POISSON PROBLEM
(Presented by Academician M. A. Lavrent'ev on January 6, 1969)
In this paper estimates are established, up to the boundary, for solutions of elliptic equations in the class of functions analytic in the tangential directions. As an application, the Cauchy–Poisson problem on the motion of a fluid with a free surface is considered in a rigorous formulation. After the estimates have been established in the class of analytic functions, the existence and uniqueness theorem for the solution of the Cauchy–Poisson problem follows almost immediately from the works of L. V. Ovsyannikov ((^{1,2})), Leray and Ohya ((^3)), which generalize the Cauchy–Kovalevskaya theorem.
1. Norms and their properties. We shall use the notation
(\beta=(\beta_0,\beta_1,\ldots,\beta_n)), (|\beta|=\beta_0+\beta_1+\cdots+\beta_n), (x=(x_0,x_1,\ldots,x_n)),
(D^\beta=\partial^{|\beta|}/\partial x_0^{\beta_0}\cdots \partial x_n^{\beta_n}).
Let (y=(y_1,\ldots,y_m)); let (Y\subset R^m) be an open set; let (\Omega\subset R^{n+1}) be either an open set, or a hyperplane, or the union of an open set and several hyperplanes. For functions (f(x):\Omega\to R) and (F(x,y):\Omega\times Y\to R) define
[
|f,\Omega|{\rho,k+\alpha}
=
\sum}^{\infty}\frac{\rho^l}{l!
\max_{|\beta|=l,\ \beta_0=0}
|D^\beta f|{C,}(\Omega)
]
[
|F,\Omega\times Y,\nu,\theta|{\rho,k}
=
C(1+\nu_1+\cdots+\nu_m)^{k+1}
\sum}\frac{\rho^l\theta^\sigma}{l!\sigma!
\max_{|\beta|=l,\ \beta_0=0}
\times
]
[
\times
|D_x^\beta D_y^\sigma F|{C.}(\Omega\times Y)
]
Here (\nu=(\nu_1,\ldots,\nu_m)), (\theta=(\theta_1,\ldots,\theta_m)), (\theta^\sigma=\theta_1^{\sigma_1}\cdots \theta_m^{\sigma_m}), (\sigma!=\sigma_1!\cdots\sigma_m!). The constant (C=C(k)). The norm of a differential operator is defined as the sum of the norms of its coefficients.
Let us formulate the basic properties of the introduced norms.
If (f,g:\Omega\to R), then
[
|D_j f,\Omega|{\rho,k+\alpha}
\ll
\frac{\partial}{\partial\rho}|f,\Omega|,
\qquad
0<j\le n,
\tag{1}
]
[
|fg,\Omega|{\rho,k+\alpha}
\ll
|f,\Omega|.}|g,\Omega|_{\rho,k+\alpha
\tag{2}
]
For a differential operator (a(x,D)) define
[
|[a]f,\Omega|{\rho,k,\alpha}
=
\sum}^{\infty}\frac{\rho^l}{l!
\max_{|\beta|=l,\ \beta_0=0}
|([aD^\beta-D^\beta a]f)|{C.}(\Omega)
]
If the dimension of (\Omega) is equal to (n+1), then
[
|[a]f,\Omega|{\rho,k+\alpha}
\ll
[|a,\Omega|-|a,\Omega|{0,k+\alpha}]
|f,\Omega|;
\tag{3}
]
(m) is the order of the operator (a(x,D)).
Let (F(x,y):\Omega\times Y\to R), (V(x):\Omega\to Y). Then
[
|F\circ V,\Omega|{\rho,k+\alpha}
\ll
|F,\Omega\times Y,|V,\Omega|,|V,\Omega|{\rho,k+\alpha}-|V,\Omega|,}|_{\rho,k
\tag{4}
]
where (F\circ V=F(x,V(x))), and (|V,\Omega|{\rho,k+\alpha}) is the vector with coordinates (|V_j,\Omega|).
Inequalities (1)–(3) follow from the definitions and Leibniz’ formula. The proof of (4) is omitted for lack of space.
In what follows we shall say: the series
[
F(t,\tau,\rho,\theta)=\sum_{s,\sigma}\frac{\rho^s\theta^\sigma}{s!\sigma!}\,F_{s\sigma}(t,\tau)
]
as a function of (t,\rho,\tau=(\tau_1,\ldots,\tau_N)), (\theta=(\theta_1,\ldots,\theta_N)) belongs to the space (\Gamma), if it converges and is continuous in (t,\tau) in a neighborhood of the point (t=\rho=0,\ \tau=\theta=0).
If (\Phi(t,\rho)=(\Phi_1,\ldots,\Phi_N)), then by (F(\Phi)) we shall denote the function
(F(t,\Phi(t,0),\Phi(t,\rho)-\Phi(t,0))). It is clear that (F(\Phi)\in\Gamma), if (F,\Phi\in\Gamma).
The function (f(t,x):{0,T}\times\Omega\to R) belongs to the space (B^p_{k+\alpha}(\Omega)), if
(|D_t^j f;\Omega|{\rho,k+\alpha}\in\Gamma) for (j\le p). The operator (L(t,x,D)\in B^p(\Omega)). Similarly, (F(t,x,y)\in B_k^p(\Omega,Y)), if}(\Omega)), if its coefficients (a_\beta(t,x)\in B^p_{k+\alpha
(|D_t^jF,\Omega\times Y,\tau,\theta|{\rho,k}\in\Gamma) for (j\le p).
(L(t,x,y,D)\in B_k^p(\Omega,Y)), if (a\beta(t,x,y)\in B_k^p(\Omega,Y)).
From inequalities (1), (2), and (4) we have:
[
D_x^\beta D_t^q:B^p_{k+\alpha}(\Omega)\to B^{p-q}_{k-\beta_0+\alpha}(\Omega),\qquad
q\le p,\quad \beta_0\le k.
]
[
f\cdot g\in B^p_{k+\alpha}(\Omega),\quad \text{if } f,g\in B^p_{k+\alpha}(\Omega).
]
[
F\circ V\in B^p_{k+\alpha}(\Omega),\quad
\text{if } f\in B^p_{k+\alpha}(\Omega),\quad F\in B_k^p(\Omega\times Y),
]
[
f(t,x):{0,T}\times\Omega\to Y.
]
2. Estimates of solutions of elliptic equations. Introduce the notation:
(X={x=(x_0,x_1,\ldots,x_n):0\le x_0\le 1,\ -\infty<x_1,\ldots,x_n<\infty});
(S_j={x:x_0=j,\ -\infty<x_1,\ldots,x_n<\infty}), (j=0,1);
(\overline X=X\cup S_0\cup S_1).
In the layer (X) the problem to be considered is
[
L(x,D)u=\sum_{|\beta|\le m}a_\beta(x)D^\beta u=f,\qquad x\in X;
]
[
B_0(x,D)u=\sum_{|\beta|\le l_0}a_\beta^0(x)D^\beta u=\varphi_0,\qquad x\in S_0;
\tag{5}
]
[
B_1(x,D)u=\sum_{|\beta|\le l_1}a_\beta^1(x)D^\beta u=\varphi_1,\qquad x\in S_1.
]
It is assumed that the coefficients of the operator (L) and (B_j) are infinitely differentiable and bounded, (L) is uniformly elliptic, and (B_j) satisfy the complementing condition (see (4)).
Theorem 1. Let the conditions listed above be fulfilled. Then, for
(k\ge \max(m,l_0,l_1)), the inequality
[
|u,\overline X|{\rho,k+\alpha}\le
A(\rho)\left[
|Lu,\overline X|
+\sum_{j=0}^{1}|B_j u,S_j|{\rho,k-l_j+\alpha}
+|u|
\right],
\tag{6}
]
holds, where
[
A(\rho)=C\left{1-C\left[
\rho+|L,\overline X|{\rho,k-m+\alpha}-|L,\overline X|
+\sum_{j=0}^{1}\bigl(|B_j,S_j|{\rho,k-l_j+\alpha}-|B_j,S_j|\bigr)
\right]\right}^{-1}
]
and the constant (C) depends on (n,\alpha); the ellipticity constant of the operator (L); the constants characterizing fulfillment of the complementing condition, and the norms of the coefficients of the operators (L) and (B_j) in the spaces (C_{k-m+\alpha}(\overline X)) and (C_{k-l_j+\alpha}(S_j)), respectively.
The proof is carried out in the same way as in (5) when deriving estimates
solutions of the Dirichlet problem. It is based on a Schauder-type estimate
[
|V|{C}(\bar X)
\leq
C\left[
|LV|{C}(\bar X)
+
\sum_{j=0}^{1}|B_jV|{C}(S_j)
+
|V|_{C_0(\bar X)}
\right],
\tag{7}
]
which follows from the results of [4]. Putting (V=D^\beta u,\ \beta_0=0), we obtain (6) from (3) and (7).
Corollary. If (m=2,\ B_0=D_0+\sum_{j=1}^{n} a_j^0D_j,\ a_{0\ldots 0}\leq 0) and (B_1\equiv 1), then the term (|u|_{C_0(\bar X)}) in the right-hand side of inequality (6) may be omitted.
The required assertion is given by the following
Lemma 1. Let the operator (L) be uniformly elliptic; (a_{0\ldots 0}\leq 0;\ m=2), the coefficients of the operators (L) and
[
B_0=D_0+\sum_{j=1}^{n} a_j^0(x)D_j
]
be bounded and continuous, and let (B_1\equiv 1). If (u\in C_2(\bar X)), then
[
|u(x)|\leq C\left[|Lu|{C_0(\bar X)}+|B_0u|\right],}+|u|_{C_0(S_1)
]
where the constant (C) depends on the ellipticity constant of the operator (L) and on an exact upper bound for the moduli of the coefficients of the operators (L) and (B_0).
Proof. Introduce
[
g(x)=\sum_{j=1}^{n}\ln\left[(x_0+1)^2+x_j^2\right];\qquad
\lambda=\sup_{x\in\bar X}(|(L-a_{0\ldots 0})g|,\ |B_0g|);
]
[
u_\varepsilon(x)=u(x)+(H+\lambda\varepsilon)e^{\alpha x_0}-\varepsilon g(x);\qquad
H=|Lu|{C_0(\bar X)}+|B_0u|.
]
Choose (\alpha>1) so that (e^{-\alpha x_0}Le^{\alpha x_0}>1). Then (Lu_\varepsilon>0,\ Bu_\varepsilon>0), and, by the boundedness of (u), (u_\varepsilon<0) for sufficiently large (R^2=x_1^2+\cdots+x_n^2). The required result follows from the well-known maximum principle for bounded domains.
3. Waves on the surface of a liquid. Consider potential motion in a field of external forces in the domain
[
\Omega_t={x=(x_0,x_1,x_2):\ h(x_1,x_2)<x_0<\zeta(t,x_1,x_2),\ -\infty<x_1,x_2<\infty},
]
where (x_0=\zeta(t,x_1,x_2)) is the free surface, and (x_0=h(x_1,x_2)) is the bottom.
It is known that (\zeta) and the potential (u) satisfy the system of equations
[
\begin{aligned}
\Delta u&=0,\qquad x\in\Omega_t;\
du/dN&=0,\qquad x_0=h(x_1,x_2);\
\partial u/\partial t&=-\tfrac12|\nabla u|^2+F(t,x),\qquad x_0=\zeta(t,x_1,x_2);\
\partial\zeta/\partial t&=-\nabla u\cdot\nabla\zeta+\partial u/\partial x_0,\qquad x_0=\zeta(t,x_1,x_2).\
\zeta(0,x_1,x_2)&=\zeta_0(x_1,x_2);\quad
u(0,\zeta_0(x_1,x_2),x_1,x_2)=u_0(x_1,x_2).
\end{aligned}
]
Here (N) is the normal to the surface (x_0=h(x_1,x_2)).
Theorem 2. Let (h,\zeta_0,u_0\in B_{2+\alpha}^0(\mathbb R^2)), (F(t,\zeta_0(x_1,x_2)+y,x_1,x_2)\in B_2^0(X,{|y|<\delta})), and (0<\delta_0\leq \zeta_0-h\leq d<\infty). Then, for sufficiently small times (t), the system written above has a unique solution ((\zeta,u)), continuously differentiable in (t) and analytic in (x).
Proof. Make the change of variables
[
x'=(x_0',x_1',x_2');\qquad
x_0'=\frac{x_0-h(x_1,x_2)}{\zeta(t,x_1,x_2)-h(x_1,x_2)};
]
[
x_1'=x_1;\qquad x_2'=x_2;
]
[
\zeta'(t,x_1',x_2')=\zeta(t,x_1',x_2')-\zeta_0(x_1',x_2');
]
[
u'(t,x')=u(t,h+(\zeta-h)x_0',x_1',x_2')
-u_0'(h+(\zeta-h)x_0',x_1',x_2').
]
Here (u_0'(x_0,x_1,x_2)) is the solution of the problem
[
\begin{aligned}
\Delta u_0'&=0,\qquad x\in\Omega_0;\
du_0'/dN&=0,\qquad x_0=h(x_1,x_2);\
u_0'&=u_0,\qquad x_0=\zeta(x_1,x_2).
\end{aligned}
]
Rewrite the original system, omitting primes and denoting any function (f), depending on (t,x,\varphi) and on derivatives of (\varphi) of order not higher than (q), by (f(D^q\varphi)):
[
L(D^2\zeta,D)u
=
\sum_{0<|\beta|\leq 2} a_\beta(D^2\zeta)D^\beta u
=
L(D^2\zeta,D)u_0,\qquad x\in X;
]
[
B(D\xi,D)u=\left[D_0+\sum_{j=1}^{2}a_j(D\xi)D_j\right]u=B(D\xi,D)u_0,\qquad x\in S_0;
]
[
\begin{aligned}
\partial \xi/\partial t&=a(D\xi,Du), && x\in S_1;\
\partial u/\partial t&=b(D\xi,Du), && x\in S_1;\
u(0,x)&=0, && x\in X;\
\xi(0,x)&=0, && x\in S_1,
\end{aligned}
]
where
[
L(0,D)u_0=0,\ x\in X;\qquad B(0,D)u_0=0,\ x\in S_0;\qquad u_0=u_0(x_1,x_2),\ x\in S_1.
]
From the substitution it is clear that
[
\begin{gathered}
L(y,D)\in B^0(\bar X,Y),\qquad B(y,D)\in B_1^0(S_0,Y),\
a(y)\in B_2^0(S_1,Y),\qquad b(y)\in B_2^0(S_1,Y),
\end{gathered}
]
where (Y) is a ball with center at the origin whose radius is less than (\delta_0).
According to Theorem 1, (u_0\in B_{2+\alpha}^0(\bar X)).
In order to find (u), it is sufficient to know (\xi) and (v=u|{S_1}). Let (\xi,v\in B^0(S_1)); (|\xi,S_1|{\rho,2+\alpha}=\psi(t,\rho)), (\psi(0,\rho)=0); (|v,S_1|=\Phi(t,\rho)), (\Phi(0,\rho)=0).
Introduce
[
P(\xi,v)=\partial u/\partial x_0|_{S_1},
]
where (u) is the solution of the problem
[
\begin{aligned}
L(0,D)u&=[L(0,D)-L(D^2\xi,D)]u+L(D^2\xi,D)u_0, && x\in X;\
B(0,D)u&=[B(0,D)-B(D\xi,D)]u+B(D\xi,D)u_0, && x\in S_0;\
u&=v, && x\in S_1.
\end{aligned}
]
According to Lemma 1, for sufficiently small (t) there exists a unique (u\in C_{2+\alpha}(\bar X)). Applying inequality (4),
[
|L(0,D)-L(D^2\xi,D),\bar X|_{\rho,\alpha}\ll A_1(\psi)\psi\in\Gamma;
]
[
|B(0,D)-B(D\xi,D),S_0|_{\rho,1+\alpha}\ll A_2(\psi)\psi\in\Gamma;
]
[
|B(D\xi,D)u_0,S_0|{\rho,1+\alpha}\ll A_3(\psi)\psi\in\Gamma;\qquad
|L(D^2\xi,D)u_0,\bar X|\ll A_4(\psi)\psi\in\Gamma,
]
from the a priori estimate (6) and the corollary to Theorem 1 we obtain:
[
|u,\bar X|_{\rho,2+\alpha}\ll F_1(\psi,\Phi)\in\Gamma,\qquad F_1(0,0)=0,
]
i.e.
[
u\in B_{2+\alpha}^0(\bar X);\qquad u(0,x)=0,\ x\in X.
]
Since (|f,S_1|{\rho,k+\alpha}\ll \mathrm{const}\,(1+\partial/\partial\rho)|f,S_1|), it follows that
[
|P(\xi,v),S_1|_{\rho,2+\alpha}\ll (1+\partial/\partial\rho)F(\psi,\Phi);\qquad F\in\Gamma,\quad F(0,0)=0.
]
Thus the original problem is reduced to finding ((\xi,v)) from the system
[
\begin{aligned}
\partial\xi/\partial t&=a(D\xi,Dv,P(\xi,v)), && x\in R^2;\
\partial v/\partial t&=b(D\xi,Dv,P(\xi,v)), && x\in R^2;\
\xi(0,x)&=v(0,x)=0, && x\in R^2,
\end{aligned}
]
where (a\in B_2^0(R_2^2,Y)), (b\in B_2^0(R_2^2,Y)), and for the operator (P(\xi,v)) the estimate
[
|P(\xi,v),R^2|{\rho,2+\alpha}\ll (1+\partial/\partial\rho)F(|\xi,R^2|),\qquad F(0,0)=0},\ |v,R^2|_{\rho,2+\alpha
]
is valid.
The theorem on the existence and uniqueness of the solution ((\xi,v)) follows almost immediately from the results of works ((1–3)). It can be proved, like the Cauchy–Kovalevskaya theorem, by successive approximations.
In conclusion, the author takes this opportunity to express gratitude to L. V. Ovsyannikov and A. B. Shabat for their constant attention and support during the work.
Institute of Hydrodynamics
Siberian Branch of the Academy of Sciences of the USSR
Novosibirsk
Received
22 V 1968
REFERENCES
- L. V. Ovsyannikov, DAN, 163, No. 4 (1965).
- L. V. Ovsyannikov, Fluid Dynamics Trans., 3, Warszawa (1967).
- J. Leray, Y. Ohya, Math. Ann., 170, No. 3, 167 (1967).
- S. Agmon, A. Douglis, L. Nirenberg, Estimates of solutions of elliptic equations near the boundary, Moscow, 1962.
- V. I. Nalimov, DAN, 183, No. 1 (1968).