MATHEMATICS

Yu. M. Berezanskii, S. G. Krein, Ya. A. Roitberg

A THEOREM ON HOMEOMORPHISMS AND LOCAL INCREASE OF SMOOTHNESS UP TO THE BOUNDARY OF SOLUTIONS OF ELLIPTIC EQUATIONS

(Presented by Academician I. G. Petrovskii on 21 VIII 1962)

In this note a theorem is proved on homeomorphisms realized by an elliptic operator, for the case of general homogeneous boundary conditions, and its application is given to the local increase of smoothness up to the boundary of generalized solutions that are elements of spaces with negative norm.

Theorems on homeomorphisms for spaces of functions of increased smoothness follow from coercivity inequalities, which for general boundary conditions were obtained by M. Shekhter ((^{2-4})). For the case of the first boundary-value problem, a collection of theorems on homeomorphisms, also pertaining to spaces of functions of decreased smoothness, was obtained by Lions (see ((^{1,5}))) and developed in ((^{6,7})). Below, for the general case, results of the same type are obtained, and thereby a complete collection of homeomorphisms is established. The proof is also based on the interpolation theorem and on general considerations expressed in a report by one of the authors at the Fourth Conference on Functional Analysis (Baku, 1959).

In considering the question of smoothness of generalized solutions, the complete collection of homeomorphisms is used.

1. Let (H_0) be a Hilbert space and let (D) be an unbounded self-adjoint positive operator with domain (\mathcal D(D)) such that (|u|0 \leqslant |Du|_0). The domain of definition (\mathcal D(D^\alpha)) ((\alpha>0)) is a Hilbert space (H\alpha) with respect to the scalar product
   [
   (u,v)\alpha=(D^\alpha u,D^\alpha v)_0.
   ]
   We take this space as positive; the corresponding negative space (H) may be obtained by completing (H_0) with respect to the scalar product
   [
   (f,g){-\alpha}=(D^{-\alpha}f,D^{-\alpha}g)
   ]
   ((f,g\in H_0)); (H\alpha) and (H_{-\alpha}) are mutually conjugate with respect to the scalar product ((\cdot,\cdot)0). The family ({H\alpha}) ((-\infty<\alpha<\infty)) is called a Hilbert scale of spaces.

The following interpolation theorem holds:

Let ({H_\alpha}) and ({H'\alpha}) be two Hilbert scales and let (B) be a linear operator, continuously acting from (H) to (H'{\alpha_1}) and from (H) to (H'{\beta_1}). Then (B) acts continuously from any (H),}) to (H'_{\alpha_1(\mu)
[
\alpha_i(\mu)=(1-\mu)\alpha_i+\mu\beta_i
]
((i=0,1)), (0\leqslant\mu\leqslant1). This theorem was obtained independently by Lions ((^8)) and by one of the authors ((^9)). We shall now give one application of this theorem.

Consider (H_0) and a fixed space (H_l) ((l>0)). Let (N) and (N^) be finite-dimensional subspaces of (H_l), and let
[
H'_0=H_0\ominus N^, \qquad H''0=H_0\ominus N
]
be their orthogonal complements in (H_0); let (B) be a continuous operator, with respect to the metric of (H_0), acting from all of (H'_0) onto a dense part of (H''_0); let (B^) be the operator adjoint to (B), acting from (H''_0) to (H'_0). We shall additionally assume that the ranges of the operators (B) and (B^) are contained in (H_l). It is convenient to denote the closures of these ranges in the metric of (H_s) ((0\leqslant s\leqslant l)), respectively, by (H_s(\mathrm{gr})) and (H_s(\mathrm{gr})^+). Let (H)^+) be the corresponding conjugate spaces with respect to ((\cdot,\cdot)_0).}(\mathrm{gr})) and (H_{-s}(\mathrm{gr

Theorem 1. If the inequalities

[
|Bf|_l \leq C_1|f|_0 \quad (f\in H'_0), \qquad
|B^{*}g|_l \leq C_2|g|_0 \quad (g\in H''_1),
\tag{1}
]

hold, then for (\alpha\in[0,l]) the estimate

[
|Bf|\alpha \leq C_3|f|}(\mathrm{rp})^{+}
\quad (f\in H'_0)
\tag{2}
]

is valid.

Let us outline the proof, first restricting ourselves to the case in which there is no defect: (N=N^{}=0,\ H'_0=H''_0=H_0). The second inequality in (1) permits one to regard (B^{}) as a continuous operator from (H_0) into (H_l); denote by (\widetilde B) the adjoint operator to it, acting from (H_{-l}) into (H_0). The operator (\widetilde B) is an extension of (B), and (|\widetilde B f|0\leq C_2|f|). Hence, from the first inequality in (1), using the interpolation theorem we obtain the estimate
[
|Bf|\alpha \leq C_1^{\alpha/l}C_2^{1-\alpha/l}|f|
\quad (f\in H_0).
]
But, as is known, the space adjoint to the subspace (H_s(\mathrm{rp})^{+}) of the space (H_s) is isometric to the quotient space (H_{-s}) by its subspace (V_{-s}), consisting of elements annihilating (H_s(\mathrm{rp})^{+});
[
|\varphi|{H}(\mathrm{rp})^{+}
=
\inf_{\psi\in V_{-s}}|\varphi+\psi|_{-s}.
]

And since the operator (\widetilde B) annihilates elements of (V_{-s}), it follows that
[
|Bf|\alpha
=
|\widetilde B(f+\psi)|\alpha
\leq
C_1^{\alpha/l}C_2^{1-\alpha/l}|f+\psi|{\alpha-l}
\quad (\psi\in V).
]
Passing here to the (\inf), we find (2). In the case where a defect is present, the operator (B^{*}) may be regarded as acting from (H''0) into (H_l). The adjoint operator (\widetilde B) will act from (H) into (H''0\subset H_0), and all the preceding arguments can be repeated; the interpolation theorem should be applied to the operator (\widetilde B\widetilde P), where (\widetilde P) is the extension to (H) of the operator (P) of orthogonal projection of (H_0) onto (H'_0).

2. We describe the general scheme for applying Theorem 1 to differential operators. Let (H_0=L_2(G)), where (G) is a bounded domain in (n)-dimensional space, and let (\pi_l) ((l>0) an integer) be the Sobolev space (W_2^l(G)). It can be shown that the Hilbert scale (H_\alpha) ((0\leq\alpha\leq l)) consists of the Sobolev spaces (W_2^\alpha(G)) for integral (\alpha) and of the Aronszajn–Slobodetskii spaces for fractional (\alpha); moreover the norms in (H_\alpha) are equivalent to the corresponding norms in (W_2^\alpha(G)).

Consider in (H_0=L_2(G)) an operator (A), generated by an elliptic differential expression (\mathcal L) with sufficiently smooth coefficients and by a certain system of boundary conditions. Let (W_2^l(\mathrm{rp})) be the totality of all functions in (W_2^l(G)) satisfying these conditions; (Au=\mathcal Lu,\ u\in D(A)=W_2^l(\mathrm{rp})). We shall assume that (A^{}) is constructed analogously from the formally adjoint expression (\mathcal L^{+}) and the conditions ((\mathrm{rp})^{+}). Suppose the following assertions hold: 1) the subspaces (N) and (N^{}) of solutions of the equations (Au=0) and (A^{}v=0), respectively, are finite-dimensional. Denote
[
H_l(\mathrm{rp})=W_2^l(\mathrm{rp})\cap H''_0,\qquad
H_l(\mathrm{rp})^{+}=W_2^l(\mathrm{rp})^{+}\cap H'_0,
]
where, as before,
[
H'_0=H_0\ominus N,\qquad H''_0=H_0\ominus N^{};
]
2) the inequalities
[
|u|_l\leq C_1|Au|_0 \quad (u\in H_l(\mathrm{rp})),\qquad
|v|_l\leq C_2|A^{}v|_0 \quad (v\in H_l(\mathrm{rp})^{+})
]
hold; 3) the ranges of (A) on (H_l(\mathrm{rp})) and of (A^{}) on (H_l(\mathrm{rp})^{+}) coincide respectively with (H'_0) and (H''_0).

If we denote by (B) the operator inverse to the restriction of (A) to (H_l(\mathrm{rp})), then the hypotheses of Theorem 1 will be satisfied for it, and from that theorem, for (\alpha\in[0,l]), one obtains the inequality
[
|Au|{H}(\mathrm{rp})^{+}
\geq C|u|{l-\alpha}
\quad (u\in H_l(\mathrm{rp})).
]
In many cases, when (\alpha=k\in[0,l]) is an integer, the converse inequality is also valid; then
[
C|u|\leq |Au|{H}(\mathrm{rp})^{+}
\leq C_1|u|_{l-k}
\quad (u\in H_l(\mathrm{rp}),\ k=0,\ldots,l).
\tag{3}
]

This inequality makes it possible to extend, by continuity, the operator (A) to an operator (A_{-k}) that carries out a homeomorphic mapping of (H_{l-k}(\mathrm{gr})) onto all of (H_{-k}(\mathrm{gr})^{+}) ((k=0,\ldots,l)). Precisely such a situation occurs for the operator considered in the next section.

1. We shall now formulate the exact assertion that can be obtained on the basis of the considerations of Section 2. In doing so we shall make essential use of the results of the papers ({}^{(2-4)}) of M. Schechter. Thus, in a bounded domain (G) with boundary (\Gamma), consider a properly elliptic ({}^{(2)}) differential expression of order (l=2m) with complex coefficients

[
\mathcal{L}u=\sum_{|\mu|\leq 2m} a_\mu(x)D^\mu u
]

[
\left(\mu=(\mu_1,\ldots,\mu_n),\ |\mu|=\mu_1+\cdots+\mu_n,\quad
D^\mu=\left(\frac{1}{i}\frac{\partial}{\partial x_1}\right)^{\mu_1}\cdots
\left(\frac{1}{i}\frac{\partial}{\partial x_n}\right)^{\mu_n}\right).
]

On (\Gamma) there are given (m) differential expressions

[
B_k=\sum_{|\mu|\leq m_k} b_{k\mu}(x)D^\mu
]

[
(k=1,\ldots,m;\ m_k\leq 2m-1),
]

which determine the boundary conditions. Now (W_2^l(\mathrm{gr})) consists of the functions (u\in W_2^l(G)) for which (B_k u=0) ((k=1,\ldots,m)).

Theorem 2. Let the operators (B_k) be normal and cover (\mathcal{L}) in the sense of ({}^{(2,3)}). Consider the operator (A\colon Au=\mathcal{L}u) ((u\in H_{2m}(\mathrm{gr}))). 1) If the boundary (\Gamma) is of class (C^{4m+s}), the coefficients (a_\mu(x)\in C^{2m+\max(|\mu|,s)}(G\cup\Gamma)), (b_{k\mu}(x), b^+_{k\mu}(x)\in C^{2m+s-1}(\Gamma)), then, for an integer (s\geq 0), the restriction of the operator (A) to

[
\dot H_{2m+s}(\mathrm{gr})=H_{2m}(\mathrm{gr})\cap W_2^{2m+s}(G)
]

establishes a homeomorphism (H_{2m+s}(\mathrm{gr})\to H_s^{+}=W_2^s\cap H_0'). 2) If the preceding smoothness conditions are satisfied for (s=0), then the operator (A) admits an extension (A_{-s}) on (H_{2m-s}(\mathrm{gr})), carrying out a homeomorphism (H_{2m-s}(\mathrm{gr})\to H_{-s}(\mathrm{gr})^{+}) ((0\leq s\leq 2m,\ s) integer). 3) Let the smoothness conditions be the same as in 1); then the operator (A) admits an extension to the space (\dot H_{-s}), conjugate to (H_s^{+}=W_2^s(G)\cap H_0'); this extension carries out a homeomorphism (\dot H_{-s}\to H_{-2m-s}(\mathrm{gr})^{+}).

Let us explain that the homeomorphism in the case of the first pair of spaces follows directly from ({}^{(3,4)}), and in the case of the second pair from the construction of Section 2; moreover the second of the inequalities (3) is established by means of integration by parts. For the third pair it is derived from the first homeomorphism by passing to conjugate spaces.

Above, the principal difficulty consisted in establishing the homeomorphism for the second pair. For zero boundary conditions this fact is, in essence, known ({}^{(5-7)}). We note that in the latter case, and even in the more general case when the boundary operators (B_k) have order (\leq m-1), the homeomorphism can also be established with the aid of the Aronszajn lemma. In this case the energy inequalities are proved in a stronger form—with boundary norms.

1. Theorem 2 on homeomorphisms can be applied to the local increase of smoothness up to the boundary of generalized solutions of elliptic equations that are elements of spaces with negative norm. For simplicity we shall speak about zero boundary conditions and strongly elliptic expressions. Below all indices in the notation of spaces are integers.

Let us give a definition. Let (G) be a bounded domain, on whose boundary (\Gamma) lies a piece (\Gamma_0); let (\varphi\in W_2^s(G)), (s<0). We shall say that (\varphi) belongs to (W_2^t), with (t>s), inside (G) up to the piece (\Gamma_0) (we shall write this as (\varphi\in W_{2,\mathrm{loc}}^t(G,\Gamma_0))), if for every subdomain (G'\subset G) having common boundary with (G) only inside the piece (\Gamma_0), there exists (\psi_{G'}\in W_2^t(G)) such that

[
(\varphi,u)0=(\psi,u)_0
]

for all (u\in C^\infty(G\cup\Gamma)) that additionally vanish in a neighborhood of (G\setminus G'). It holds that

Theorem 3. Let an expression (\mathscr L) of order (l=2m) be strongly elliptic, let its coefficients (a_\mu \in C^{|\mu|+p}(G\cup \Gamma_0)), and let (\Gamma_0) be of class (C^{l+p}) ((p \geqslant 0)). Suppose that a function (\varphi \in W_2^s(G)) ((s=\ldots,-1,0,1,\ldots)) satisfies the equation (\mathscr L u=f) inside (G) up to the piece (\Gamma_0), where zero boundary conditions are prescribed (\left(D^\alpha u\big|{\Gamma_0}=0,\ |\alpha|\leqslant m-1\right)). Here (f\in W(G,\Gamma_0)); satisfaction of the equation means that}}^{s-l

[
(\varphi,\mathscr L^+v)_0=(f,v)_0
\tag{4}
]

for all (v\in W_2^{2m+\max(-s,0)}(G)), (D^\alpha v\big|{\Gamma_0}=0) ((|\alpha|\leqslant m-1)), and additionally vanishing in neighborhoods of the set (\Gamma\setminus\Gamma_0). Suppose that (f\in W^t(G,\Gamma_0)), where (t>s-l) is of arbitrary sign. Then: 1) for (p}