Abstract
Full Text
UDC 517.9
MATHEMATICS
L. A. BAGIROV
BOUNDARY-VALUE PROBLEMS FOR ELLIPTIC AND PARABOLIC EQUATIONS IN UNBOUNDED DOMAINS
(Presented by Academician I. G. Petrovskii, August 15, 1969)
I. Let (G) be the exterior of a bounded domain in (R^n). The boundary of (G) is a smooth ((n-1))-dimensional surface (\Gamma). We shall consider the boundary-value problem
[
A(x,D)u=\sum_{|\alpha|\leq 2l} a_\alpha(x)D^\alpha u=f(x),\quad x\in \bar G;
\tag{1}
]
[
B_j(x,D)u=\sum_{|\beta|\leq m_j} b_{j\beta}(x)D^\beta u=g_j(x),\quad x\in \Gamma,
\tag{2}
]
where
[
j=1,\ldots,l;\quad \alpha=(\alpha_1,\ldots,\alpha_k),\quad |\alpha|=\sum_{k=1}^{n}\alpha_k,
]
[
D^\alpha=D_1^{\alpha_1}\cdots D_n^{\alpha_n},\quad
D_k^{\alpha_k}=\frac{1}{(-i)^{\alpha_k}}\frac{\partial^{\alpha_k}}{\partial x_k^{\alpha_k}}.
]
We assume that the following conditions are satisfied:
-
The operator (A) is uniformly elliptic in (\bar G).
-
At every point (x\in\Gamma) the Shapiro–Lopatinskii condition is satisfied.
-
The limit (\lim_{|x|\to\infty} a_\alpha(a)) exists for all (\alpha), (|\alpha|\leq 2l).
-
The symbol of the limiting operator
[
a(\infty,\xi)=\sum_{|\alpha|\leq 2l} a_\alpha(\infty)\xi^\alpha
]
has no zeros for all (\xi\in R^n). -
The coefficients (a_\alpha(x)) and (b_{j\beta}(x)) are infinitely smooth functions. The spaces (H^k(G)) and (H^{k-1/2}(\Gamma)) are the usual Sobolev–Slobodetskii spaces.
Theorem 1. Suppose that conditions 1–5 are satisfied. Then:
a) for any (u\in H^k(G)) the inequality
[
|u|k(G)\leq C\left{|f|(G)+
\sum_{j=1}^{l}|g_j|{k-m_j-1/2}(\Gamma)+|u|(G_0)\right},
]
holds, where (k\geq \max(2l,m_j+1/2)), (G_0=G\cap K_R); (K_R={x:\ |x|\leq R});
b) there exists a regularizer for problem (1), (2), i.e., a bounded operator
[
R:H^{k-2l}(G)\times H^{k-m_j-1/2}(\Gamma)\to H^k(G)
]
such that (LR\Phi=\Phi+T\Phi), where (L=(A,B_j)), (\Phi=(f,g_j)), and (T) is a completely continuous operator in the direct product
[
H^{k-2l}(G)\times H^{k-m_j-1/2}(\Gamma).
]
The space (H_a^k(G)) is the space of functions with norm
[
|u|{k,a}(G)=\sum\,dx.}\iint |D^\alpha u|^2 e^{2a|x|
]
Theorem 2. Suppose that conditions 1–5 are satisfied and that the polynomial (a(\infty,\xi+i\tau)) has no real zeros in (\xi) for (|\tau|\leq a). Then:
a) for any function (u(x)\in H_a^k(G)) the estimate is valid
[
|u|{k,a}(G)\leq C\left{|f||g_j|}(G)+\sum_{1}^{l{k-m_j-1/2}(\Gamma)+|u|(G_0)\right};
]
b) there exists a regularizer of the problem (1), (2), acting from the space
(H_a^{k-2l}(G)\times H^{k-m_j-1/2}) into the space (H_a^k(G)).
II. In the same domain with boundary (\Gamma) we consider the elliptic problem with complex parameter
[
A(x,D,q)u(x,q)=\sum_{|\alpha|+\beta\leq 2l} a_{\alpha\beta}(x)q^\beta D^\alpha u=f(x,q),\qquad x\in \bar G;
\tag{3}
]
[
B_j(x,D,q)u(x,q)=\sum_{|\beta|+|\gamma|\leq m_j} q^\gamma b_{j\beta\gamma}(x)D^\beta u=g_j(x,q),\qquad x'\in \Gamma.
\tag{4}
]
The analogous problem in a bounded domain was considered in the work ((^1)). The parameter (q) varies in the angle (Q:\varphi_1\leq \arg q\leq \varphi_2). We assume that the following conditions are satisfied:
-
The coefficients (a_{\alpha\beta}(x)) and (b_{j\beta\gamma}(x)) are infinitely differentiable functions in (\bar G), and the inequalities (|D^\gamma a_{\alpha\beta}|\leq C_{a,\gamma}^{\beta}) hold, where (C_{a,\gamma}^{\beta}) are constants.
-
The operator (A) is elliptic with parameter, and the boundary-value problem (A,B_j) satisfies the Shapiro–Lopatinskii condition with parameter (for the exact definition see Chap. I, § 4 of ((^1))).
Theorem 3. Let conditions 6 and 7 be satisfied and let (k) be an integer (k\geq k_0=\max(2l,m_j+1)). Then there exists a number (q_1>0) such that for (|q|>q_1,\ q\in Q), problem (3), (4) has a unique solution (u(x)\in H^k(G)), if (f(x)\in H^{k-2l}(G)), and (g_j(x)\in H^{k-m_j-1/2}(\Gamma)). Moreover, the inequality holds
[
|u|k(G)+|q|^k|u|_0(G)\leq C\left{|f||f|_0+\right.}+|q|^{k-2l
]
[
\left.
+\sum_{1}^{l}\left(|g_j|_{k-m_j-1/2}(\Gamma)+|q|^{k-m_j-1/2}|g_j|_0(\Gamma)\right)\right}.
]
The constant (C) does not depend on (u(x)) and (q).
Remark. An analogous result is valid for the case of an unbounded domain (G) with unbounded boundary (\Gamma).
III. In the author’s work ((^3)) problems for elliptic equations with unbounded coefficients in the exterior of a bounded domain were considered. Here we shall study the analogous problem with parameter. To this end, assume that problem (3), (4) satisfies the following conditions:
- The inequality holds
[
\left|\sum_{|\alpha|+\beta=2l} a_{\alpha\beta}(x)\xi^\alpha q^\beta\right|\geq \delta(x)(|\xi|^2+|q|^2)^l
]
for any (\xi\in R^n,\ q\in Q={q:\varphi_1\leq \arg q\leq \varphi_2}), (\delta(x)\geq \delta_0>0).
- The function
[
a_{2l}(x,\xi,q)=\sum_{|\alpha|+\beta=2l} a_{\alpha\beta}(x)q^\beta \xi^\alpha
]
for (|x|>N,\ q\in Q,\ \xi\in R^n) leaves free of values the angle of the complex plane
(Q_1={z:\psi_1\leq \arg z\leq \psi_2}).
-
The values of the function (a_{0,0}(x)) for (|x|>N) lie in the angle
(Q_2={z:\psi_1+\pi+\varepsilon\leq \arg z\leq \psi_2+\pi-\varepsilon},\ \varepsilon>0). -
At every point (x'\in \Gamma) the Shapiro–Lopatinskii conditions with parameter are satisfied.
-
Let (x) be any point of (G,\ |x|>N); (H) a cube with edge (h) centered at the point (x); (y) any point of (H). We require that the functions (|a_{\alpha\beta}(x)-)
— (a_{\alpha\beta}(y)|/ \delta(x)) for (|\alpha|+\beta=2l), (|a_{0,0}(x)-a_{0,0}(y)|/|a_{0,0}(x)|) tend to zero as (h\to 0) uniformly in (x), (|x|\ge N), (x\in G).
Introduce the notation (\delta_k(x)=(\delta(x))^{k/2l}), (\gamma_k(x)=(|a_{0,0}(x)|)^{k/2l}).
- Let (|\alpha|+\beta=2l-k), and (|\gamma|=p); then the ratio (|D^\gamma a_{\alpha\beta}(x)|/\delta_{2l-(k+p)}\gamma_{k+p}\to 0) as (|x|\to\infty), if (0\le p\le s-2l) for (0<k<2l) and (0<p\le s-2l) for (k=0,2l).
Conditions 12 and 13 allow certain classes of unbounded functions. Condition 12 is satisfied by polynomials and functions of the type (e^{k|x|}).
For convenience we shall use special norms depending on the parameter (q). For each fixed (q) they are equivalent to the norms (|!|u|!|_k(G)), which were used by the author. The definition of the norm is as follows:
[
\langle u\rangle_k^2(G)=\int!!\int{\delta_k^2(x)|D^k u|^2+|q|^{2k}|u|^2+\gamma_k^2(x)|u|^2}\,dx .
]
Theorem 4. Suppose conditions 8—13 are fulfilled. Then there exists a number (q_1>0) such that, for (|q|>q_1), (q\in Q), there exists a unique solution of problem (3), (4) belonging to the space (H^k(\delta,\gamma,G)), if (f\in H^{k-2l}(\delta,\gamma,G)), and (g_j\in H^{k-m_j-1/2}(\Gamma)). Moreover, the estimate
[
\langle u\rangle_k(G)\le C\left{\langle f\rangle_{k-2l}(G)+\sum_1^l|g_j|_{k-m_j-1/2}(\Gamma)\right}.
]
holds.
The constant (C) does not depend on (u(x)) and (q).
An analogous result is also true in the case of an unbounded boundary.
IV. After studying the elliptic problem with a parameter, it is natural to investigate the general mixed problem for a parabolic equation in a cylinder with unbounded base. Let (G), as before, be the exterior of a bounded domain with boundary (\Gamma), and let (\Omega_T) be the cylinder (\Omega_T=G\times(0,T)) with lateral surface (\Omega_T'=\Gamma\times(0,T)).
Consider the parabolic problem
[
A(x,t,D_x,\partial/\partial t)u(x,t)=f(x,t),\qquad (x,t)\in\Omega_T;
\tag{5}
]
[
B_j(x,t,D_x,\partial/\partial t)u(x,t)\big|_{x=x'}=g_j(x',t),\qquad (x',t)\in\Omega_T';\ j=1,\ldots,l.
\tag{6}
]
For (t=0) we prescribe the initial conditions
[
\partial^k u/\partial t^k\big|_{t=0}=\varphi_k(x)\qquad (k=0,\ldots,\chi-1,\ x\in G),\ \chi=l/b.
\tag{7}
]
The operator (A) has order (2l), and the operators (B_j) have order (m_j). The parabolic weight of the problem is (2b).
Definition. Problem (5), (6) is called normally parabolic if, for each fixed (t=a), (0\le a\le T), the problem with a parameter in the domain (G), obtained from problem (5), (6) by replacing (\partial/\partial t) with (q^{2b}), satisfies all conditions 8—13 in the angle (|\arg q|\le \pi/4b).
We shall use on the surface (\Omega_T') the usual Sobolev—Slobodetskii space (H_{\lambda,\lambda/2b}(\Omega_T')), and in the cylinder (\Omega_T) the space (H_{s,s/2b}(\delta,\gamma,\Omega_T)) with norm
[
\langle u\rangle_{s,s/2b}^2
=
\int_{\Omega_T}!!\int
{\delta_s^2(x)(|D_x^s u|^2+|D_t^{s/2b}u|^2)+\gamma_s^2(x)|u|^2}\,dx\,dt .
]
It is assumed that (s), (s/2b) are integers.
The discussion is carried out by the method of the Laplace transform, analogously to how this was done in paper ((^1)).
Theorem 5. Let problem (5), (6), (7) be a normally parabolic problem. Suppose also that the usual compatibility conditions for the right-hand sides are fulfilled. Then, if (f(t,t)\in H_{\mu,\mu/2b}(\delta,\gamma,\Omega_T)); (g_j(x',t)\in H_{\lambda_j,\lambda_j/2b}(\Omega_T'));
(\varphi_k(x) \in H_{s-2bk-b}(G),\ k=0,\ldots,\varkappa-1), where (\mu=s-2l,\ \lambda_j=s-m_j-\frac12), then there exists a unique solution of the problem (u(x,t)\in H_{s,s/2b}(\delta,\gamma,\Omega_T)). Moreover, the a priori estimate holds
[
\langle u\rangle_{s,s/2b}\leq C\left{\langle f\rangle_{\mu,\mu/2b}+
\sum_1^l |g_j|{\lambda_j,\lambda_j/2b}+
\sum_1^{\varkappa-1}|\varphi_k|\right}.
]
The constant (C) does not depend on (u(x,t)).
V. Now in the same domain (G) with boundary (\Gamma) consider problem (1), (2). We shall assume that conditions 1, 2, and 5 are satisfied. Conditions 3 and 4 will be replaced by the following.
Condition 9°. The coefficients of equation (1) and their derivatives have the form
[
D^\beta a_\alpha(x)=a_{\alpha\beta}^0(\omega)r^{|\alpha|-|\beta|-2l}+a_{\alpha\beta}^1(x);\quad r=|x|.
]
Here (\omega=(\omega_1,\ldots,\omega_{n-1})) is an infinitely differentiable coordinate system on the unit sphere, and
[
a_{\alpha\beta}^1(x)=O(r^{|\alpha|-|\beta|-2l-\delta})
]
as (r\to\infty); (\delta>0).
Consider in (R^n) the elliptic equation (\sum a_\alpha^0(\omega)D^\alpha u=f). Make the change of variables (x\to(r,\omega)), and then the change (r=e^{-t}). It is well known that after these changes we obtain the equation
[
A^0(\omega,D_\omega,D_t)v(t,\omega)=e^{-2lt}f(t,\omega)
\tag{8}
]
in the cylinder (\Omega=S\times(-\infty,+\infty)), where (S) is the unit sphere (|x|=1). After the Fourier transform in (t) we obtain an elliptic equation with a parameter on the sphere:
[
A^0(\omega,D_\omega,\lambda)\tilde v(\lambda,\omega)=\tilde f_1(\lambda,\omega).
\tag{9}
]
It is known that the inverse operator (R(\lambda)) for equation (9) is a meromorphic function of (\lambda), and in each strip (|\operatorname{Im}\lambda|