Elliptic Problems with Additional Unknowns in Boundary Conditions and Generalized Sobolev Spaces

: In generalized inner product Sobolev spaces we investigate elliptic differential problems with additional unknown functions or distributions in boundary conditions. These spaces are parametrized with a function OR-varying at inﬁnity. This characterizes the regularity of distributions more ﬁnely than the number parameter used for the Sobolev spaces. We prove that these problems induce Fredholm bounded operators on appropriate pairs of the above spaces. Investigating generalized solutions to the problems, we prove theorems on their regularity and a priori estimates in these spaces. As an application, we ﬁnd new sufﬁcient conditions under which components of these solutions have continuous classical derivatives of given orders. We assume that the orders of boundary differential operators may be equal to or greater than the order of the relevant elliptic equation.


Introduction
This work is a contribution to the theory of elliptic boundary-value problems in generalized Sobolev spaces founded recently by Mikhailets and Murach [1][2][3][4][5][6][7][8] and developed in [9][10][11][12][13][14][15][16]. These spaces are parametrized with a general enough function of frequency variables (which are dual to spatial variables with respect to the Fourier transform). It characterizes the regularity of distributions more finely than the number parameter used for the Sobolev spaces or other classical distribution spaces. Thus, the order of regularity of generalized Sobolev spaces is a function, not a number. We apply these spaces to elliptic differential problems with additional unknown functions or distributions in boundary conditions. Such problems were introduced by Lawruk [17][18][19] and appear naturally as formally adjoint problems to nonregular elliptic problems with respect to a relevant Green formula. Important examples of such problems occur, e.g., in hydrodynamics and the theory of elasticity [20][21][22]. Since these problems belong to the Boutet de Monvel algebra, the theorems on their solvability in Sobolev spaces of large enough orders are contained in the results by Boutet de Monvel [23], Rempel and Schulze (Chapter 4, [24]), Grubb [25,26]. Such a theorem is also proved in (Section 23, Subsection 4, [27]) within Eskin and Vishik's theory of elliptic pseudodifferential boundary problems. The case of Sobolev spaces of arbitrary orders was investigated by Kozlov, Maz'ya, and Rossmann (Chapters 3 and 4, [28]), I. Roitberg [29,30], Y. Roitberg (Chapter 2, [31]), and A. Kozhevnikov [32] in the framework of special spaces introduced by Y. Roitberg [33,34].
In contrast to the works just mentioned, we study these problems in Hilbert distribution spaces that form the extended Sobolev scale investigated in [35,36] and (Section 2.4, [8]). The regularity orders of such spaces are arbitrary OR-varying (O-regularly varying) functions at infinity. It is remarkable that this scale consists of all Hilbert spaces that are interpolation ones between inner product Sobolev spaces, which allows the use of the interpolation (with function parameter) between Hilbert spaces in proofs. Unlike the nearest articles [37][38][39][40][41] to the present research, we do note impose any restrictions on the orders of the boundary differential operators involved in the problems and do not require that the regularity orders of the generalized Sobolev spaces being used satisfy any additional (unessential) conditions. The results obtained in this paper are partly announced in [42] (without proofs).
This paper consists of eight sections. Section 1 is Introduction. Section 2 gives the statement of the elliptic problem under investigation. Section 3 presents and discusses generalized Sobolev spaces being used. The main results are formulated in Section 4. They consist of the Fredholm property of bounded operators induced by the problem on appropriate pairs of generalized Sobolev spaces, relevant isomorphisms between some subspaces of finite codimension, conditions for local (up to the boundary) regularity of generalized solutions to the problem, and their a priori estimate in these spaces. The case of the homogeneous elliptic equation is separately considered at the end of this section. Section 5 is devoted to the method of interpolation (with function parameter) between Hilbert spaces and discusses some of its properties used in our proofs. The proofs are given in Section 6. Section 7 is devoted to applications of the extended Sobolev scale to the investigations of classical smoothness of the generalized solutions. We find new sufficient conditions under which components of these solutions have continuous classical derivatives of given orders. Among them are conditions for generalized solutions to be classical. The final Section 8 contains concluding remarks.

Statement of the Problem
Let Ω be a bounded domain in the Euclidean space R n , with n ≥ 2, and let Γ denote the boundary of Ω. Suppose that Γ is an infinitely smooth closed manifold of dimension n − 1, with the C ∞ -structure on Γ being induced by R n . Let ν denote the field of the unit inward normal vectors to Γ.
Choose integers q ≥ 1, κ ≥ 1, m 1 , . . . , m q , and r 1 , . . . , r κ arbitrarily. We consider the following boundary-value problem in Ω: B j u + κ ∑ k=1 C j,k v k = g j on Γ, j = 1, . . . , q + κ, (2) Here, the unknowns are the distribution u in Ω and κ distributions v 1 , . . . , v κ on Γ. We suppose that A := A(x, D) is a linear partial differential operator (PDO) on Ω := Ω ∪ Γ; each B j := B j (x, D) is a linear boundary PDO on Γ, and every C j,k := C j,k (x, D τ ) is a linear tangent PDO on Γ. Their orders satisfy the conditions ord A = 2q, ord B j ≤ m j , and ord C j,k ≤ m j + r k , and their coefficients are infinitely smooth complex-valued functions of x ∈ Ω or x ∈ Γ respectively. (Of course, a PDO of negative order is assumed to equal zero identically.) We consider complex-valued functions and distributions and use corresponding complex function or distribution spaces.
This assumption is natural; indeed, if m + r k < 0 for some k, all the operators C 1,k ,. . . , C q+κ,k will equal zero identically, i.e., the unknown distribution v k will be absent in the boundary conditions (2). Note that the m ≥ 2q case is possible.
, and C • j,k (x, τ) denote the principal symbols of the PDOs A(x, D), B j (x, D), and C j,k (x, D τ ) respectively, the last two PDOs being considered as that of the formal orders m j and m j + r k respectively. Thus, A • (x, ξ) and B • j (x, ξ) are homogeneous polynomials in ξ ∈ C n of order 2q and m j respectively, and C • j,k (x, τ) is a homogeneous polynomial of order m j + r k in τ, where τ is a tangent vector to the boundary Γ at the point x. Defining the principal symbols, we consider the principal parts of the PDOs as polynomials with respect to D := i∂/∂x , where = 1, . . . , n, and then replace each differential operator D with the -th component ξ of the vector ξ.
The boundary-value problem (1), (2) is called elliptic in Ω if the following three conditions are satisfied: is properly elliptic at every point x ∈ Γ; i.e., for an arbitrary tangent vector τ = 0 to Γ at x, the polynomial A • (x, τ + ν(x)ζ) in ζ ∈ C has q roots with positive imaginary part and q roots with negative imaginary part (of course, these roots are counted with regard for their multiplicity). (iii) The boundary conditions (2) cover A(x, D) at every point x ∈ Γ. This means that, for each vector τ = 0 from condition (ii), the boundary-value problem has only the trivial (zero) solution. Here, the function θ ∈ C ∞ ([0, ∞)) and numbers λ 1 , . . . , λ κ ∈ C are unknown. In addition, are differential operators with respect to D t := i∂/∂t. We obtain them putting ζ := D t in the polynomials A • (x, τ + ν(x)ζ) and B • j (x, τ + ν(x)ζ) in ζ, respectively. As is known (Chapter 2, Sections 1.1 and 1.2, [43]), condition (ii) follows from condition (i) in the n ≥ 3 case and also in the case where n = 2 and where all the leading coefficients of A(x, D) are real-valued. If κ = 0, condition (iii) is equivalent to the Lopatinskii condition for classical elliptic problems.
Examples of elliptic problems of the form (1), (2) are given in (Subsection 3.1.5, [28]). We supplement them with the following boundary-value problem: Here, we arbitrarily choose integers p ≥ 0 and ≥ 1 and real-valued functions a, b ∈ C ∞ (Γ) such that |a(x)| + |b(x)| = 0 for every x ∈ Γ. As usual, ∂ ν := ∂/∂ν, ∆ is the Laplace operator in R n , and ∆ Γ is the Beltrami -Laplace operator on Γ. This problem takes the form (1), (2), where q = 2, κ = 1, m 1 = p, m 2 = p + 2 , and r 1 = −p. Direct calculation shows that this problem is elliptic in Ω. Note that, if a(x 0 ) = 0 for some point x 0 ∈ Γ, it is impossible to exclude the unknown function v 1 from the boundary conditions and preserve the smoothness of the coefficients and right-hand side of the boundary condition obtained.
With the problem (1), (2) under investigation, we associate the linear mapping We will investigate properties of an extension (by continuity) of this mapping on appropriate pairs of Hilbert distribution spaces that form extended Sobolev scales over Ω and Γ.
To describe the range of this extension, we need the following special Green formula (formula (4.1.10), [28]): Of course, if µ = 2q (which is equivalent to m ≤ 2q − 1), the functions w 1 , . . . , w µ−2q and the relevant sums will be absent. Here, (·, ·) Ω and (·, ·) Γ stand respectively for the inner products in the Hilbert spaces L 2 (Ω) and L 2 (Γ) of functions square integrable over Ω and Γ relative to the Lebesgue measures. We also let A + denote the PDO which is formally adjoint to A relative to (·, ·) Ω . Moreover, C + j,k , R + j,k , and Q + j,k respectively denote the tangent PDOs which are formally adjoint to C j,k , R j,k , and Q j,k relative to (·, ·) Γ , the tangent PDOs R j,k and Q j,k appearing in the representation of the boundary PDOs D j−1 ν A and B j in the form We put D ν := i∂/∂ν and understand D k−1 ν as a boundary PDO on Γ; specifically, D 0 ν means the trace operator on Γ. Note that ord R j,k ≤ 2q + j − k and ord Q j,k ≤ m j − k + 1. Finally, K k := K k (x, D) is a certain boundary PDO on Γ whose order ord K k ≤ 2q − k and whose coefficients belong to C ∞ (Γ). This Green formula leads us to the following boundary-value problem in Ω: q+κ ∑ j=1 C + j,k h j = ψ µ+k on Γ, k = 1, . . . , κ.

Extended Sobolev Scale
This scale was introduced and investigated in (Section 2.4.2, [8]) and [36], first over R n and then over Euclidean domains and closed infinitely smooth manifolds. The scale consists of Hilbert generalized Sobolev spaces [44,45] whose order of regularity is a function from a certain class OR.
The space H α (R n ) is endowed with the inner product and the corresponding norm w α,R n := (w, w) 1/2 α,R n . This space is Hilbert and separable, and the set C ∞ 0 (R n ) of compactly supported test functions is dense in it. We say that α is the order of regularity of the space H α (R n ) and its versions for Ω and Γ considered below.
This space is an isotropic Hilbert case of the spaces B p,k introduced and investigated by Hörmander in (Section 2.2, [44]) and applied by him to partial differential equations (see also (Section 10.1, [51])). Namely, if p = 2 and k(ξ) = α( ξ ) for all ξ ∈ R n , then H α (R n ) = B p,k . Note that the Hörmander spaces in the Hilbert case form a subclass of the spaces introduced by Malgrange [52] and coincide with the spaces investigated by Volevich and Paneah ( § 2, [45]). If both embeddings being continuous and dense. This property is a direct consequence of the inequality (10) written for t = 1. According to (Section 2.4.2, p. 105, [8]), the class is called the extended Sobolev scale over R n . This class has remarkable interpolation properties; namely, it is obtained by means of the interpolation with function parameter between inner product Sobolev spaces, is closed with respect to the (quadratic) interpolation between Hilbert spaces, and consists of all Hilbert spaces that are interpolation ones between inner product Sobolev spaces [36]. Thus, the class (12) is the maximal extension of the Hilbert scale of Sobolev spaces with the help of the interpolation between Hilbert spaces. These properties of the extended Sobolev scale make it suitable and useful in the study of linear operators induced by elliptic PDEs and elliptic problems (see (Section 2.4.3, [8]) and [9,10,12,13,53]). The extended Sobolev scales over the domain Ω and its boundary Γ are built in a standard way on the base of (12) (see (Section 2, p. 139, [36]) and (Section 2.4.2, p. 106, [8]) respectively). Let us give the necessary definitions. Now we suppose that n ≥ 2.
As above, α ∈ OR. By definition, the linear space is Hilbert and separable with respect to this norm because it is a factor space of the Hilbert space H α (R n ) by its subspace Briefly saying, the space H α (Γ) consists of all distributions on Γ that yield elements of H α (R n−1 ) in local coordinates on Γ. Let us give a detailed definition. We arbitrarily choose a finite atlas π j : R n−1 ↔ Γ j , with j = 1, . . . , λ, from C ∞ -structure on the manifold Γ. Here, the open sets Γ j form a covering of Γ. Let functions χ j ∈ C ∞ (Γ), with j = 1, . . . , λ, satisfy the conditions χ 1 + · · · + χ λ ≡ 1 and supp χ j ⊂ Γ j . By definition, the linear space Here, (χ j h) • π j is a representation of the distribution χ j h in the local map π j . The space This space is Hilbert and separable and does not depend up to equivalence of norms on our choice of π j and χ j (Theorem 2.21, [8]). The set Thus, we have the extended Sobolev scales over Ω and Γ respectively. They contain Hilbert Sobolev scales; namely, if α(t) ≡ t s for some s ∈ R, then H α (Ω) =: H (s) (Ω) and H α (Γ) =: H (s) (Γ) are the inner product Sobolev spaces of order s. The classes (12) and (13) are partially ordered with respect to embedding of spaces. Let α, η ∈ OR and G ∈ {R n , Ω, Γ}. The function α/η is bounded in a neighbourhood of infinity if and only if H η (G) → H α (G). This embedding is dense and continuous. It is compact in the G ∈ {Ω, Γ} case if and only if α(t)/η(t) → 0 as t → ∞. This follows directly from (Theorems 2.2.2 and 2.2.3, [44]). Specifically, property (11) remains true and the relevant embeddings become compact if we replace R n with Ω or Γ.
Both the classes (13) have the same above-mentioned interpolation properties as (12). We will discuss some of them in Section 5.

The Main Results
With the problem (1), (2), we associate the following Hilbert spaces: where η ∈ OR. In these and similar designations, we use the function parameter (t) ≡ t not to write the argument t in indices. Thus, e.g., the parameter η −2q means the function to the problem (1), (2) in the case where f = 0 in Ω and each g j = 0 on Γ. Similarly, let N + stand for the linear space of all solutions to the formally adjoint problems (6)- (8) in the case where θ = 0 in Ω and all ψ k = 0 and ψ µ+k = 0 on Γ. Since both problems are elliptic in Ω, the spaces N and N + are finite-dimensional (Consequence 4.1.1, [28]).

Theorem 1.
Let η ∈ OR and σ 0 (η) > m + 1/2. Then the mapping (4) extends uniquely (by continuity) to a bounded linear operator This operator is Fredholm. Its kernel coincides with N . Its range consists of all vectors such that The index of the operator (14) is equal to dim N − dim N + and hence does not depend on η.
As to this theorem, we recall that a linear bounded operator T : X → Y between Banach spaces X and Y is called Fredholm if its kernel ker T := {x ∈ X : Tx = 0} and cokernel Y/T(X) are finite-dimensional. The Fredholm operator has the closed range T(X) (see, e.g., (Lema 19.1.1, [54])) and the finite index where T * is the adjoint of T.
Formula (16) needs commenting. Certainly, if µ = 2q, the first sum with respect to j will be absent in this formula. The first components of the forms (·, ·) Γ in (16) belong to in view of (Proposition 4, [10]) and because In addition, Thus, both the sums with respect to j are well defined in (16). (16) and does not depend on the choice of f k , which will be shown in the proof of Theorem 1.
If N = {0} and N + = {0}, then the operator (14) becomes an isomorphism between the spaces D η (Ω, Γ) and E η −2q (Ω, Γ). Generally, this operator induces an isomorphism between some of their (closed) subspaces, which have a finite codimension. It is convenient to give this isomorphism with the help of certain decompositions of the source and target spaces of (14) in direct sum of their subspaces. Let σ 0 (η) > m + 1/2; then This decomposition is well defined because it is a restriction of the relevant orthogonal decomposition of the Hilbert space L 2 (Ω) ⊕ (L 2 (Γ)) κ . Note that D η (Ω, Γ) lies in the above space due to (3). A decomposition of E η −2q (Ω, Γ) is based on the following result: whenever η ∈ OR and σ 0 (η) > m + 1/2. If µ = 2q, we may take G = N + .
Let P and P + denote respectively the projectors of the spaces D η (Ω, Γ) and E η −2q (Ω, Γ) onto the second term in the sums (17) and (18) parallel to the first. The rules that define these projectors do not depend on η.
Let us study properties of generalized solutions to the elliptic problem (1), (2) in the spaces used above. Beforehand, we will give a definition of such solutions. Put the last equality being valid due to (11). Let the right-hand sides of the problem (1), (2) satisfy the condition As usual, D (Ω) and D (Γ) denote the linear topological spaces of all distributions on Ω and Γ respectively. A vector is called a generalized solution to this problem if Λ(u, v) = ( f , g). Here, Λ means the operator (14) for a certain parameter η ∈ OR subject to σ 0 (η) > m + 1/2. This definition is reasonable because it is independent of η.
In this case, Theorem 3 deals with the global regularity of (u, v), i.e., concerns the regularity of u in Ω and v on Γ.
Here, of course, χ(u, v) := (χu, (χ Γ)v 1 , . . . , (χ Γ)v κ ) and the expression ζΛ(u, v) is similarly understood. These theorems were proved in (Sections 4 and 6, [40]) in the special case where the function η(t) varies regularly at infinity in the sense of J. Karamata and on the assumption that m ≥ 2q. If m ≤ 2q − 1 and if the function η ∈ OR satisfies the stronger condition σ 0 (η) > 2q − 1/2, Theorems 1-3 were proved in (Sections 4 and 6, [39]). (The indicated articles are published in Ukrainian.) Generally, the conclusions of these theorems are not valid for arbitrary η ∈ OR. Specifically, if s ≤ m j + 1/2 for certain j ∈ {1, . . . , q + κ} and if Q j,m j +1 = 0 in the representation of B j (x, D) in the form (5), then the mapping u → B j u, where u ∈ C ∞ (Ω), cannot be extended to a continuous linear operator from the whole Sobolev space H (s) (Ω) to D (Γ); this follows from (Chapter 1, Theorem 9.5, [43]). Hence, the bounded linear operator (14) is not well defined in the η(t) ≡ t s case under these conditions. However, if the elliptic Equation (1) is homogeneous (i.e., f = 0 in Ω), certain versions of the above theorems will hold for any η ∈ OR. We restrict ourselves to a relevant version of the key Theorem 1.
Given η ∈ OR, we put Here, Au is understood in the sense of the distribution theory. We endow the linear space H η A (Ω) with the inner product and norm in H η (Ω). The space H η A (Ω) is complete because the differential operator A is continuous on D (Ω). The set is dense in H η A (Ω) by (Theorem 7.1, [9]). Consider the linear mapping With this mapping, we associate the Hilbert spaces Let N + 0 denote the linear space of all vectors (h 1 , . . . , h q+κ ) ∈ (C ∞ (Γ)) q+κ for each of which there exist functions ω ∈ C ∞ (Ω) and w 1 , . . . , Certainly, dim N + 0 ≤ dim N + , with the strict inequality being possible (Theorem 13.6.15, [51]).
The index of the operator (22) equals dim N − dim N + 0 and hence does not depend on η.
Since the function parameter η ∈ OR is arbitrary in this theorem, components of the vector (23) may be irregular distributions on Γ. We therefore interpret the expression (g j , h j ) Γ in (24) as the value of the distribution g j ∈ D (Γ) on the test function h j ∈ C ∞ (Γ) and consider the space D (Γ) as the dual of C ∞ (Γ) with respect to the inner product in L 2 (Γ).
This theorem was given (without a complete proof) in [37,38] in the special case where the function η(t) varies regularly at infinity, paper [38] treating the m ≤ 2q − 1 case.

The Interpolation between Hilbert Spaces
As has been mentioned in Section 3, the extended Sobolev scale possesses an important interpolation property, which will play a decisive role in the proof of Theorems 1 and 5. Namely, each space H α (G), where G ∈ {R n , Ω, Γ} and α ∈ OR, can be obtained by the interpolation (with an appropriate function parameter) between inner product Sobolev spaces H (s 0 ) (G) and H (s 1 ) (G) such that s 0 < σ 0 (α) and s 1 > σ 1 (α). Therefore, we will recall the definition of the interpolation between Hilbert spaces and formulate its properties being used in our proofs.
The interpolation method we need was introduced by C. Foiaş and J.-L. Lions in (p. 278, [55]). Expounding it, we mainly follow monograph (Section 1.1, [8]), which gives its various applications to elliptic operators and elliptic boundary-value problems. It is sufficient for our purposes to restrict ourselves to separable Hilbert spaces.
Let X := [X 0 , X 1 ] be an ordered pair of separable complex Hilbert spaces X 0 and X 1 such that X 1 is a manifold in X 0 and that w X 0 ≤ c w X 1 whenever w ∈ X 1 , with the number c > 0 not depending on w. This pair is called regular. As is known, for X there exists a positive-definite self-adjoint operator J given in the Hilbert space X 0 and such that X 1 is the domain of J and that Jw X 0 = w X 1 for all w ∈ X 1 . This operator is uniquely determined by the pair X and is called the generating operator for this pair. The operator J sets an isometric isomorphism between X 1 and X 0 .
Given ψ ∈ B and applying the spectral theorem to the self-adjoint operator J, we obtain the (generally, unbounded) operator ψ(J) on X 0 . Let [X 0 , X 1 ] ψ or, briefly, X ψ denote the domain of ψ(J) endowed with the inner product (u 1 , u 2 ) X ψ := (ψ(J)u 1 , ψ(J)u 2 ) X 0 and the corresponding norm u X ψ = ψ(J)u X 0 . The space X ψ is Hilbert and separable and is continuously embedded in X 0 .
We call a function ψ ∈ B an interpolation parameter if the following condition is satisfied for all regular pairs X = [X 0 , X 1 ] and Y = [Y 0 , Y 1 ] of Hilbert spaces and for an arbitrary linear mapping T given on whole X 0 : If the restriction of T to X j is a bounded operator from X j to Y j for every j ∈ {0, 1}, then the restriction of T to X ψ is also a bounded operator from X ψ to Y ψ . We say in this case that X ψ is obtained by the interpolation, with the function parameter ψ, of the pair X (or, in other words, between X 0 and X 1 ) and that the bounded operator T : X ψ → Y ψ is the result of the interpolation applied to the operators T : X j → Y j with j ∈ {0, 1}.
A function ψ ∈ B is an interpolation parameter if and only if ψ is pseudoconcave in a neighbourhood of infinity, i.e., ψ ψ 1 there for a certain positive concave function ψ 1 (t) of t 1. (As usual, ψ ψ 1 means that the functions ψ/ψ 1 and ψ 1 /ψ are bounded on the indicated set). This fundamental fact follows from J. Peetre's [56] description of all interpolation functions of positive order. Specifically, the power function ψ(t) ≡ t s is an interpolation parameter if and only if 0 ≤ s ≤ 1.
It is useful for us to formulate the above-mentioned interpolation property of the extended scale as follows: Proposition 1. Let η ∈ OR, and suppose that real numbers s 0 and s 1 satisfy s 0 < σ 0 (η) and s 1 > σ 1 (η). Define a function ψ ∈ B by the formula Then ψ is an interpolation parameter, and up to equivalence of norms provided that G ∈ {Ω, Γ}. If G = R n , then (26) holds true with equality of norms.
Proving Theorem 5, we will use the following interpolation property of the space H η A (Ω) (Theorem 7.8(i), [9]): Proposition 2. Let η ∈ OR, s 0 , s 1 ∈ R, and ψ ∈ B satisfy the hypothesis of Proposition 1. Then up to equivalence of norms.
We also need two general interpolation properties given below.
Proposition 3. Let X = [X 0 , X 1 ] and Y = [Y 0 , Y 1 ] be regular pairs of Hilbert spaces. Suppose that a linear mapping T on X 0 satisfies the following condition: The restrictions of T to the spaces X j , with j = 0, 1, are bounded and Fredholm operators T : X j → Y j that have a common kernel and the same index. Then, for an arbitrary interpolation parameter ψ ∈ B, the bounded operator T : X ψ → Y ψ is Fredholm with the same kernel and index, and the range of the last operator equals Y ψ ∩ T(X 0 ).
with equality of norms norms whatever ψ ∈ B.

Proofs of the Main Results
Proof of Theorem 1. We first consider the Sobolev case where η(t) ≡ t s and s > m + 1/2. In this case, Theorem 1 is essentially contained in Grubb's result (Corollary 5.5, [25]). As compared with the last two sentences of Theorem 1, Grubb proved that there exists a finite dimensional subspace M of C ∞ (Ω) × (C ∞ (Γ)) µ−q+κ that plays the role N + in this theorem. If m ≥ 2q, we may take M := N + , as is shown in (Section 6, Proof of Theorem 1, [40]). Let us show that we may do so in the case where m ≤ 2q − 1.
Proof of Lemma 1. We separately consider the cases where m ≥ 2q and where m ≤ 2q − 1. If m ≥ 2q, this lemma is contained in (Lemma 1, [40]) provided that η(t) ≡ t s and s > m + 1/2. Hence, the required formula (18) follows directly from the decomposition If m ≤ 2q − 1, then (18) holds true for G := N + because the intersection of the subspaces on its right-hand side is zero space and since the finite dimension of the first subspace equals the co-dimension of the second by Theorem 1.

Proof of Theorem 3. Our reasoning is motivated by (Proof of Theorem 7, [13]). Put
We first prove that, under the hypothesis of Theorem 3, the implication holds true for every s > m − 1/2. Here, we use algebraic sums of spaces. We choose a number s > m − 1/2 arbitrarily and suppose that the premise of the implication (28) is true. Given χ ∈ Υ, we select a function ζ ∈ Υ such that ζ(·) = 1 in a neighbourhood of supp χ. By the hypothesis, Interchanging the operator of the multiplication by χ with the component-wise PDO operator Λ, we write Here, Λ is a certain component-wise PDO of the form of Λ, the orders of all components of Λ being at least in 1 less than the orders of the corresponding components of Λ. Thus, By the premise of the implication (28), we have the sum ζ(u, v) = (u * , v * ) + (u + , v + ) for certain vectors (u * , v * ) ∈ D η (Ω, Γ) and (u + , v + ) ∈ D (s) (Ω, Γ). In view of (29) we obtain Recall that χ( f , g) ∈ E η −2q (Ω, Γ) by the hypothesis of the theorem. The boundedness of the operator follows by Proposition 1 from the well-known boundedness of the operators where m − 1/2 < s 0 < σ 0 (η) and s 1 > σ 1 (η). It follows from (30)-(32) that are the unique solutions of the operator equations and Λ(u , v ) = P + ( f , g ) ∈ P + (E (s+1−2q) (Ω, Γ)), due to Theorem 2. By Theorem 1 we hence get for a certain vector (u • , v • ) ∈ N, which in view of (33) proves the required implication (28). Since (u, v) ∈ D m+1/2+ (Ω, Γ) by the hypothesis of the theorem, the premise of this implication holds true for s = m. Choose an integer p ≥ 1 such that m + p > σ 1 (η). Applying the implication (28) p times successively for s = m, s = m + 1,. . . , and s = m + p − 1, we conclude that for every χ ∈ Υ; i.e., (u, v) ∈ D η loc (Ω 0 , Γ 0 ). Proof of Theorem 4. Let c 1 and c 2 denote some positive numbers that do not depend on (u, v). According to Theorem 3, we have the inclusion χ(u, v) ∈ D η (Ω, Γ). As is known in the theory of operators (Lemma 3, [57]), it follows from the Fredholm property of the bounded operator (14) and from the compact embedding D η (Ω, Γ) → D η −1 (Ω, Γ) that Owing to (29), we get This yields the required estimate (20).
Proof of Theorem 5. We first prove this theorem in the Sobolev case where η(t) ≡ t s and s ∈ Z and then deduce it in the general case with the help of Proposition 2. To prove the theorem in the Sobolev case, we exploit some Hilbert spaces introduced by Roitberg [33,34] and make use of known results (Sections 3.4 and 4.1, [28]) on the solvability of the elliptic problem (1), (2) in these spaces. Let s ∈ Z and r ∈ N, and recall the definition of the Roitberg space H s,r (Ω). We previously introduce the Hilbert space H s,0 (Ω) used in the definition of H s,r (Ω). If s ≥ 0, then H s,0 (Ω) := H s (Ω); if s < 0, then H s,0 (Ω) denotes the dual of H −s (Ω) with respect to the inner product in L 2 (Ω). Let · s,0,Ω stand for the norm in H s,0 (Ω). By definition, the space H s,r (Ω) is the completion of C ∞ (Ω) with respect to the Hilbert norm The mapping u → (u, (D k−1 ν u) r k=1 ), where u ∈ C ∞ (Ω), extends by continuity to an isometric linear operator H (s−m j −1/2) (Γ) (35) for arbitrary s ∈ Z. Its kernel coincides with N , and its range consists of all vectors ( f , g 1 , . . . , g q+κ ) that belong to the target space in (35) and satisfy the condition Here, ( f 0 , f 1 , . . . , f µ−2q ) := I µ−2q f if m ≥ 2q, and f 0 := f if m ≤ 2q − 1 (then the first sum with respect to j is absent). The forms (·, ·) Ω and (·, ·) Γ in (36) are well defined as extensions by continuity of the inner products in L 2 (Ω) and L 2 (Γ), respectively.
Choose a number s ∈ Z arbitrarily. As we see from (35), the mapping u → Au, where u ∈ C ∞ (Ω), extends by continuity to a bounded linear operator A : H s,µ (Ω) → H s−2q,µ−2q (Ω). Using this operator, we put Let D s,µ A (Ω, Γ) and H (s) (Γ) respectively denote the source and target spaces of the operator (37). It follows from the above-mentioned properties of (35) that N is the kernel of the operator (37) and that its range consists of all vectors g := (g 1 , . . . , g q+κ ) ∈ H (s) (Γ) which satisfy (24). Hence, the range is closed in H (s) (Γ) and its codimension equals dim N + 0 . Thus, the operator (37) is Fredholm, and its index is equal to dim N − dim N + 0 . Let us show that the spaces H  (Ω, Γ)), the latter space being endowed with the norm in H (s) (Γ). Let (u (k) , v (k) ) be the preimage of P + 0 g (k) by this isomorphism. Owing to (38), we conclude that for certain (u • , v • ) ∈ N . The inclusion in (39) holds true because whenever l ∈ N. Indeed, the last property implies that (Ω, Γ) and (u k,0 , v k,0 ) ∈ N , which yields in view of (34). According to (39), we get Here, · λ,Ω is the norm in the Sobolev space H λ (Ω), and denotes equivalence of norms. The first inequality and the equivalence of norms in (40) need explanation. If s ≥ 0, this inequality becomes the equality due to the definition of H s,0 (Ω). If s < 0, then it follows from the definition that the norm of any function u ∈ C ∞ (Ω) in H s,0 (Ω) equals the norm of the extension of u by zero in H s (R n ) (see, e.g., (Chapter 1, Remark 12.4, [43])). This implies the first inequality in (40) in the s < 0 case. If s ≥ 2qp, the equivalence in (40) follows from (34). If s < 2qp, this equivalence is due to the isomorphism  Let us now deduce Theorem 5 for arbitrary η ∈ OR from the Sobolev case just treated. Choose integers s 0 and s 1 such that s 0 < σ 0 (η) and s 1 > σ 1 (η), and define the interpolation parameter ψ by (25). Applying the interpolation with the parameter ψ to the Fredholm bounded operators we obtain the Fredholm bounded operator by Proposition 3. According to Propositions 1, 2 and 4, this operator acts between the spaces D η A (Ω, Γ) and H η (Γ). It extends the mapping (21) by continuity because the set . Owing to Proposition 3, the kernel and index of this operator coincide with the common kernel N and the index dim N − dim N + 0 of the operators (41). In addition, the range of (42) equals A (Ω, Γ)) = g ∈ H η (Γ) : (24) is satisfied .
We have proved all the properties of the operator (22) stated in Theorem 5.

Applications
We will apply Theorem 3 to obtain new sufficient conditions under which components of generalized solutions to the elliptic problem (1), (2) have continuous classical derivatives of a prescribed order. To this end, we also use the following result: Proposition 5. Let d ∈ N, α ∈ OR, and 0 ≤ l ∈ Z. Suppose that U is an open nonempty subset of R d . Then This proposition follows from Hörmander's embedding theorem (Theorem 2.2.7, [44]) as is shown in (Lemma 2, [53]). The case of U = R d is possible here. As usual, C l (·) denotes the space of all l times continuously differentiable functions on a given set.
Suppose that the sets Ω 0 and Γ 0 are the same as those in Theorem 3.
(Note that, providing m ≤ 2q − 1, the condition u ∈ C 2q (Ω) ∩ C m (U ε ∪ Γ) is equivalent to u ∈ C 2q (Ω) ∩ C m (Ω).) If the solution (u, v) is classical, the left-hand sides of the problem (1), (2) are calculated with the help of classical derivatives and are continuous functions on Ω and Γ respectively.

Concluding Remarks
The results obtained in this paper form a core of a solvability theory for elliptic problems that have additional unknowns in boundary conditions and are considered in generalized Sobolev spaces. The use of OR-varying function parameters as orders of regularity of distribution spaces allows obtaining more precise results than those received in the framework of classical Sobolev spaces, whose orders of regularity are given by power functions only. This is demonstrated by applications given in Section 7. For example, analyzing Theorem 6, we see that, if the regularity order η takes the form η(t) ≡ t s for some s ∈ R, then (43) is equivalent to s > l + n/2. The latter condition cannot be weakened in the framework of Sobolev spaces. However, using generalized Sobolev spaces, we find, e.g., that the function η(t) ≡ t l+n/2 log(1 + t) satisfies (43).
The choice of the function class OR as a set of regularity orders for generalized Sobolev spaces allows using the interpolation technique in our proofs, which facilitates them essentially as compared with proofs based on the Fourier transform approach and theory of pseudodifferential operators. This class seems the broadest one in order that generalized Sobolev spaces be well defined on smooth manifolds. It contains some functions that have not a definite order at infinity (i.e., their lower and upper Matuszewska indices are different). This circumstance specifically complicates the proof of Theorem 3 as compared with the case of power function or regularly varying functions (see, e.g., (Section 5, [5])). Our proof of Theorem 5 involves special Roitberg's spaces, which allows treating rough boundary data (of arbitrarily low regularity).
It is possible to show that the Fredholm property of the operator (14) will remain valid if the boundary Γ and coefficients of PDOs involved in the elliptic problem be of some finite smoothness and if a certain condition is imposed on the upper Matuszewska index of the regularity order η (compare with (Section 4.4.5, [28]) and (Theorem 4.1.5, [34]) in the case of Sobolev spaces). This may be a subject of another article.
Our approach is applicable to elliptic problems for systems of differential equations, pseudodifferential elliptic problems, and parameter-elliptic problems. It can be extended to generalized L p -Sobolev, Besov, and Triebel-Lizorkin spaces by using various methods of interpolation with function parameter between normed spaces, as indicated in (Section 1.