UDC 517.946

MATHEMATICS

Ya. A. ROITBERG

A HOMEOMORPHISM THEOREM FOR GENERAL ELLIPTIC BOUNDARY-VALUE PROBLEMS WITH BOUNDARY CONDITIONS THAT ARE NOT NORMAL

(Presented by Academician I. N. Vekua, 26 IX 1969)

Homeomorphism theorems for elliptic boundary-value problems were established in papers ((^1\text{--}^{11})). In all these papers it is assumed that the boundary conditions are normal. Meanwhile, it is well known that the Noether property of elliptic problems holds without the assumption of normality of the boundary expressions. Therefore the problem naturally arises of establishing homeomorphism theorems without the assumption of normality of the boundary conditions. This problem was formulated in conversations with the author by Yu. M. Berezanskii and M. I. Vishik; it was also posed by Madzhennes in ((^5)). The present paper is devoted to the solution of the indicated problem; it is a continuation of ((^{12})), whose basic notation and results are used here. We also note paper ((^{13})), in which a homeomorphism theorem is established (without the assumption of normality of the boundary expressions) in spaces conjugate to the Hölder spaces.

1. In a bounded domain (G) with boundary (\Gamma), consider a properly elliptic differential expression (L=L(x,D)) of order (2m) with complex coefficients, and on (\Gamma) a system of (m) expressions (B_j(x,D)), in general pseudodifferential in the tangential directions and differential in the directions normal to (\Gamma), of orders (m_j \leqslant 2m-1), which cover (L). For simplicity it is assumed that both the surface (\Gamma) and all the expressions are infinitely smooth. It is well known that for every real (s \geqslant 0) the operator

[
A_s:u\to (Lu,B_1u|\Gamma,\ldots,B_mu|\Gamma)\quad (u\in W_2^{2m+s}(G))
\tag{1}
]

is Noetherian

[
\text{from } W_2^{2m+s}(G)\text{ into } W_2^s(G)\dotplus \sum_{j=1}^{m} W_2^{2m+s-m_j-1/2}(\Gamma)\equiv K_s(G),
\tag{2}
]

i.e. (\mathfrak N={u\in W_2^{2m+s}(G): A_su=0}) is finite-dimensional (and does not depend on (s)), the range (\mathfrak R(A_s)) of the operator (A_s) is closed in (K_s(G)) and has finite codimension. Moreover, there exists a finite-dimensional space (\mathfrak N^+\subset C^\infty(\bar G)\dotplus C^{\infty,m}(\Gamma)) independent of (s\geqslant 0) ((C^{\infty,r}(\Gamma)=C^\infty(\Gamma)\dotplus\cdots\dotplus C^\infty(\Gamma))) such that (F=(f,\varphi_1,\ldots,\varphi_m)\in K_s(G)) belongs to (\mathfrak R(A_s)) if and only if

[
[F,V]\equiv (f,v)+\sum_{j=1}^{m}\langle \varphi_j,v_j\rangle=0
\quad (V=(v,v_1,\ldots,v_m)\in \mathfrak N^+)
\tag{3}
]

(((\cdot,\cdot)) and (\langle\cdot,\cdot\rangle) denote the scalar product respectively in (L_2(G)) and (L_2(\Gamma))).

As in ((^{6-10,\ 12,\ 14})), denote by (\widetilde W_2^l(G)=\widetilde W_{2,2m}^l(G)) ((l) an arbitrary integer) the completion of the set (C^\infty(\bar G)) with respect to the norm

[
|||u|||l=\left(|u|^2}+\sum_{j=1}^{2m}|D_\nu^{j-1}u|^2_{W_2^{\,l-j+1/2}(\Gamma)}\right)^{1/2
\tag{4}
]

[
\left(D_\nu=\frac1{\nu}\frac{\partial}{\partial \nu},\ \nu=\nu(x)\text{ is the unit vector of the inward normal to }\Gamma\text{ at the point }x;\right.
]
the spaces (W_2^l(G)) and (W_2^{-l}(G)) (\bigl(W_2^{l-1/2}(\Gamma)) and (W_2^{-(l-1/2)}(\Gamma)\bigr)) are mutually conjugate with respect to ((\cdot,\cdot)) ((\langle\cdot,\cdot\rangle)). If (l\ge 2m), then the norms (|||u|||l) and (|u|) are equivalent and (\widetilde W_2^l(G)=W_2^l(G)); for (l<2m) these norms are not equivalent.

Since the norm (4) is the norm of the direct sum
[
W_2^l(G)\dotplus \sum_{j=1}^{2m} W_2^{\,l-j+1/2}(\Gamma),
]
in essence (\widetilde W_2^l(G)) consists of elements of the form (u=(u_0,u_1,\ldots,u_{2m})), where (u_0\in W_2^l(G)), (u_j=D_\nu^{j-1}u_0|\Gamma), if (l-j\ge 0); when (l-j<0), (u_j) is an arbitrary element of the space (W_2^{\,l-j+1/2}(\Gamma)). If (t) is not an integer, (l<t<l+1), then define (W_2^t(G)) by complex interpolation between the spaces (\widetilde W_2^l(G)) and (\widetilde W_2^{l+1}(G)). For arbitrary real (t), the norm (|u|) is equivalent to the norm (|u|{W_2^t(G)}+|Lu|)). Below (u|_G) is the first component of an element (u\in\widetilde W_2^t(G)).}(G)}) ((^{14

For each real (t), the closure by continuity (A_t) of the mapping
[
u\to (Lu,B_1u|\Gamma,\ldots,B_mu|\Gamma)\qquad (u\in C^\infty(\bar G))
]
acts continuously from all of (\widetilde W_2^{2m+t}(G)) into
[
K_t(G)=W_2^t(G)\dotplus\sum_{j=1}^{m}W_2^{\,2m+t-m_j-1/2}(\Gamma).
]

We now formulate the main result of this paper.

Theorem 1. In order that the problem (A_tu=F\in K_t(G)) have a solution (u\in W_2^{2m+t}(G)), it is necessary and sufficient that the element (F=(f,\varphi_1,\ldots,\varphi_m)) satisfy relations (3). The restriction (\widetilde A_t) of the operator (A_t) to the subspace
[
\widetilde P\,\widetilde W_2^{2m+t}(G)
={u\in\widetilde W_2^{2m+t}(G):(u|_G,\mathfrak N)=0}
]
of the space (\widetilde W_2^{2m+t}(G)) realizes a homeomorphism
[
\widetilde P\,\widetilde W_2^{2m+t}(G)\to \widetilde Q^{+}K_t(G),
]
where
[
\widetilde Q^{+}K_t(G)={F\in K_t(G):[F,\mathfrak N^{+}]=0}
]
is a subspace of (K_t(G)).

2. To prove Theorem 1, we first, using the Green formula derived in ((^{12})), describe the set (\mathfrak N^{+}) more concretely. Denoting by
[
Bu=(B_1u,\ldots,B_mu),\qquad
Cu=(C_1u,\ldots,C_mu),
]
[
B'v=(B'1v,\ldots,B'_mv),\qquad
C'v=(C'_1v,\ldots,C'_mv),
]
[
\xi^n=(u^n|\Gamma,\ldots,D_\nu^{2m-1}u^n|_\Gamma),
]
we write Green’s formula (23) from ((^{12})) in the form

[
(Lu,v)+\langle Bu,C'v\rangle_{L_2^m(\Gamma)}
=(u,L^{+}v)+\langle Cu,B'v\rangle_{L_2^m(\Gamma)}
+\langle \xi^n,Tv\rangle_{L_2^{2m}(\Gamma)}
\tag{5}
]

[
(u,v\in W_2^{2m}(G)).
]

Let (\hat r) and (r) be operators assigning to each (2m)-dimensional vector (\eta=(\eta_1,\ldots,\eta_{2m})), respectively, the vectors
[
\hat r\eta=(\eta_1,\ldots,\eta_m)
\quad\text{and}\quad
r\eta=(\eta_{m+1},\ldots,\eta_{2m}).
]
Represent the space (\mathfrak N_\Gamma^{+}) in the form
[
\mathfrak N_\Gamma^{+}
=\mathfrak N_\Gamma^{+1}\oplus\mathfrak N_\Gamma^{+2}\oplus\mathfrak N_\Gamma^{+3},
]
where
[
\mathfrak N_\Gamma^{+1}={\eta\in\mathfrak N_\Gamma^{+}:r\eta=0},\qquad
\mathfrak N_\Gamma^{+2}={\eta\in\mathfrak N_\Gamma^{+}:\hat r\eta=0};
]
if (\eta\ne 0)

belongs to (\mathfrak N_\Gamma^{+3}), then, since (r\hat\eta\ne0), also (r\eta\ne0). Let
[
W_2^{2m+s}(\mathrm{pr})={u\in W_2^{2m+s}(G): Bu|\Gamma=0}\quad (s\ge0),
]
[
W_2^{2m}(\mathrm{pr})^+={v\in W_2^{2m}(G):(Lu,v)=(u,L^+v)\ (u\in W_2^{2m}(\mathrm{pr}))}.
]
From Green’s formula it follows that
[
W_2^{2m}(\mathrm{pr})^+={v\in W_2^{2m}(G):\langle Tv,\mathfrak N\Gamma\rangle_{L_2^{2m}(\Gamma)}=0,\quad
B'v|\Gamma\in r\mathfrak N\Gamma^{+3}}. \tag{6}
]

We also put
[
W_2^{2m+s}(\mathrm{pr})^+=W_2^{2m}(\mathrm{pr})^+\cap W_2^{2m+s}(G)\quad (s\ge0).
]

Consider problems with homogeneous boundary conditions
[
Lu=f\in W_2^s(G),\quad u\in W_2^{2m+s}(\mathrm{pr})\quad (s\ge0); \tag{7}
]
[
L^+v=g\in W_2^s(G),\quad v\in W_2^{2m+s}(\mathrm{pr})^+\quad (s\ge0). \tag{8}
]

Lemma 1. The space (\mathfrak N^+) of solutions of problem (8) with (g=0) is finite-dimensional and does not depend on (s\ge0). For solvability of problem (7) it is necessary and sufficient that ((f,\mathfrak N^+)=0), and for solvability of problem (8) it is necessary and sufficient that ((g,\mathfrak N)=0).

Now consider problems with nonhomogeneous boundary conditions:
[
Lu=f,\quad Bu|\Gamma=\varphi\quad (\varphi=(\varphi_1,\ldots,\varphi_m)); \tag{9}
]
[
L^+v=g,\quad B'v|\Gamma-\psi\in r\mathfrak N_\Gamma^{+3},\quad
\langle Tv,\mathfrak N_\Gamma\rangle_{L_2^{2m}(\Gamma)}=0 \tag{10}
]
[
(\psi=(\psi_1,\ldots,\psi_m)).
]

For any real (t), for each vector
[
\varphi\in\sum_{j=1}^m W_2^{2m+t-m_j-1/2}(\Gamma)
]
we construct a vector (\tilde\varphi\in r\mathfrak N_\Gamma^{+3}) such that the vector ((\varphi,\tilde\varphi)) is orthogonal in (L_2^{2m}(\Gamma)) to (\mathfrak N_\Gamma^{+3}). It is clear that by these conditions (\tilde\varphi) is uniquely determined by the vector (\varphi); if (e_1,\ldots,e_q) is a basis in (\mathfrak N_\Gamma^{+3}) such that (re_1,\ldots,re_q) forms an orthonormal (with respect to (L_2^m(\Gamma))) basis in (r\mathfrak N_\Gamma^{+3}), then
[
\tilde\varphi=-\sum_{j=1}^q \langle \varphi,\hat r e_j\rangle_{L_2^m(\Gamma)}\,re_j. \tag{11}
]

Theorem 2. In order that problem (9) with (F=(f,\varphi)\in K_s(G)) ((s\ge0)) have a solution (u\in W_2^{2m+s}(G)), it is necessary and sufficient that
[
\langle\varphi,r\mathfrak N_\Gamma^{+1}\rangle_{L_2^m(\Gamma)}=0; \tag{12}
]
[
(f,v)+\langle\varphi,C'v\rangle_{L_2^m(\Gamma)}
-\langle\tilde\varphi,B'jv\rangle=0\quad (v\in\mathfrak N^+).
]

In order that problem (10) with
[
(g,\psi)\in W_2^s(G)+\sum_{j=1}^m W_2^{2m+s-m'j-1/2}(\Gamma)\equiv K'_s(G)
]
[
(s\ge0)
]
have a solution (v\in W_2^{2m+s}(G)), it is necessary and sufficient that
[
\langle\psi,r\mathfrak N\Gamma^{+2}\rangle_{L_2^m(\Gamma)}=0,\quad
(g,u)+\langle\psi,Cu\rangle_{L_2^m(\Gamma)}=0\quad (u\in\mathfrak N). \tag{13}
]

The necessity of this theorem follows from Green’s formula (5) and the considerations of Sec. 2 of [12]. Sufficiency is proved by reducing nonhomogeneous problems to homogeneous ones and using Lemma 1. We note also that conditions (12) (taking (11) into account) make explicit the solvability conditions (3) for problem (9).

Consider also the problem

[
Lu(x)=f(x)\quad (x\in G),\qquad Bu|{\Gamma}=\varphi,\quad
\langle Cu,\mathfrak r\mathfrak N=0.}^{+}\rangle_{L_2^m(\Gamma)
\tag{14}
]

Theorem 3. In order that problem (14) with (F=(f,\varphi)\in K_s(G)) ((s\geq 0)) have a solution (u\in W_2^{2m+s}(G)), it is necessary and sufficient that

[
\langle \varphi,\mathfrak r\mathfrak N_{\Gamma}^{+}\rangle_{L_2^m(\Gamma)}=0,\qquad
(f,v)+\langle \varphi,C'v\rangle_{L_2^m(\Gamma)}=0\quad (v\in \mathfrak N^{+}).
\tag{15}
]

From Theorems 2, 3 and Green’s formula (5), by means of the method of M. I. Vishik—S. L. Sobolev ((^{15})), it follows that in order that problem (14) with (F\in K_s(G)) ((s\leq 0)) have a solution (u\in \widetilde W_2^{(2m+s)}(G)), it is necessary and sufficient that relations (15) be satisfied. Hence it already follows easily that, in order that problem (9) with (F\in K_s(G)) ((s\leq 0)) have a solution (u\in \widetilde W_2^{2m+s}(G)), it is necessary and sufficient that relations (12) be satisfied, i.e. Theorem 1 follows.

In conclusion the author expresses deep gratitude to Yu. M. Berezanskii for discussion of the results.

Chernigov State Pedagogical Institute
named after T. G. Shevchenko

Received
1 IX 1969

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