Russian Journal of Mathematical Physics, Vol. 11, No. 4, 2004, pp. 474–497. Copyright \begin{matrix}c \\ ◯ \end{matrix} 2004 by MAIK “Nauka/Interperiodica” (Russia).

This paper is dedicated to L. R. Volevich on the occasion of his 70 birthday

Formally Adjoint Problems and Solvability Conditions for Elliptic Operators

A. Volpert* and V. Volpert**

*Department of Mathematics, Technion, 32000 Haifa, Israel **Laboratoire de Mathématiques Appliquées, UMR 5585 CNRS Université Lyon 1, 69622 Villeurbanne, France

Received May 15, 2004

Abstract. We formulate solvability conditions for regular elliptic problems in unbounded domains in terms of formally adjoint problems. We impose conditions at infinity which are not only sufficient for the Fredholm property of the elliptic problem in question but also necessary for the given operators and the formally adjoint operators to be normally solvable with finite-dimensional kernels.

1. INTRODUCTION

There exists a complete solvability theory of Fredholm type for elliptic problems in bounded domains (see [2] and the references therein). The most general definition of elliptic systems (mixed order systems) has been given by Douglis and Nirenberg [6]. L. R. Volevich [17, 18] gave an equivalent formulation of ellipticity, which is easily verifiable, and proved the Fredholm property of mixed order elliptic problems in bounded domains. Later on, using new principal and technical ideas, Volevich and R. Denk extended the known results for scalar equations to mixed order systems with large parameter [5] .

In this paper, we study the Fredholm property of regular scalar elliptic problems in unbounded domains and in subsequent papers we will extend these results to mixed order elliptic systems. The Fredholm theorems are formulated in terms of formally adjoint problems. In particular, this means that the cokernel of the Fredholm operator is described explicitly in terms of solutions of the homogeneous formally adjoint problem. In bounded domains, results of this kind were obtained in [11] (the L^{2} theory) and in [15] (the L^{p} theory). In unbounded domains, one must impose some conditions at infinity (Conditions NS and NS^{∗}). These conditions turn out to be not only sufficient for the Fredholm property, but also necessary for the given operators and the formally adjoint operators to be normally solvable with finite-dimensional kernels.

1.1. Formally Adjoint Problems

We consider the boundary value problem

Au = f_{0} in Ω, B_{j}u = f_{j} in Γ, j = 1, …, m.  (1.1)

Here

Au = ∑_{|α|⩽2m} a^{α}(x)D^{α}u, x ∈ Ω,

B_{j}u = ∑_{|α|⩽s_{j}} b_{j}^{α}(x)D^{α}u, x ∈ Γ, j = 1, …, m, s_{j} < 2m,

where α = (α_{1}, …, α_{n}) is a multi-index, |α| = α_{1} + ⋯ + α_{n}, D^{α} = D_{1}^{α_{1}} … D_{n}^{α_{n}}, D_{i}^{α_{i}} = ∂^{α_{i}}/∂x_{i}^{α_{i}}, Ω ⊂ R^{n} is an unbounded domain, and Γ is the boundary of Ω.

Suppose that problem (1.1) is uniformly elliptic. This means that

(i) A is a uniformly elliptic operator, (ii) A is properly elliptic, (iii) the boundary operators satisfy uniform Lopatinsky (complementing) conditions (see [12, 1]). It is also supposed that the B_{j} form a normal system, which means that 0 ⩽ s_{1} < ⋯ < s_{m} and

∑_{|α|=s_{i}} b_{j}^{α}(x)ν_{x}^{α} ≠ 0, j = 1, …, m,

for each normal vector ν_{x} on Γ, where x ∈ Γ (see [3, 11]). Such problems are referred to as regular elliptic problems.

The coefficients a^{α}(x) and b_{j}^{α}(x) in (1.1) are complex-valued functions given in Ω̄. Assume for simplicity that these functions are infinitely differentiable and all of their derivatives are bounded.

Moreover, suppose that the boundary satisfies the following Condition D (see [20]).

Condition D. For each x_{0} ∈ Γ, there exists a neighborhood U(x_{0}) such that 1. U(x_{0}) contains a sphere centered at x_{0} of radius δ, where δ is independent of x_{0},

2. There exists a homeomorphism ψ(x; x_{0}) of the neighborhood U(x_{0}) onto the unit sphere B = {y : |y| < 1} in R^{n} such that the images of Ω ∩ U(x_{0}) and Γ ∩ U(x_{0}) coincide with B_{+} = {y : y_{n} > 0, |y| < 1} and B_{0} = {y : y_{n} = 0, |y| < 1}, respectively,

3. Both the function ψ(x; x_{0}) and its inverse are infinitely differentiable, and each of the derivatives is uniformly bounded on Γ.

As is known, the following Green formula holds for the operators under consideration:

∫_{Ω} Au v̄ dx − ∫_{Ω} u \overline{A^{∗}v} dx = ∑_{j=1}^{m} ∫_{Γ} S_{j}u \overline{C_{j}v} ds − ∑_{j=1}^{m} ∫_{Γ} B_{j}u \overline{T_{j}v} ds.  (1.2)

Here u, v ∈ D(Ω̄), A^{∗} is the formally adjoint operator,

A^{∗}v = ∑_{|α|⩽2m}(−1)^{|α|}D^{α}(a^{α}(x)v),

and the boundary operators B_{j}, S_{j} and C_{j}, T_{j} form Dirichlet systems (see [11]).

The boundary value problem

A^{∗}v = f_{0} in Ω, C_{j}v = f_{j} in Γ, j = 1, …, m,  (1.3)

is called the formally adjoint problem of (1.1).

1.2. Function Spaces

Sobolev spaces W^{s,p} proved to be very convenient in the study of elliptic problems in bounded domains. However, more flexible spaces must be used to study elliptic problems in unbounded domains. We need a generalization of the space W^{s,p}. More exactly, we mean spaces which coincide with W^{s,p} in bounded domains and have a prescribed behavior at infinity in unbounded domains. It turns out that spaces of this kind can be constructed for arbitrary Banach spaces of distributions (not only Sobolev spaces) as follows.

Let us first consider functions defined on R^{n}. As usual, denote by D the space of compactly supported infinitely differentiable functions and by D′ the dual of D. Let E ⊂ D′ be a Banach space, where the inclusion is understood both in the algebraic and topological sense. Denote by E_{loc} the collection of all elements u ∈ D′ such that fu ∈ E for any f ∈ D. Let ω(x) ∈ D, 0 ⩽ ω(x) ⩽ 1, ω(x) = 1 for |x| ⩽ 1/2 and ω(x) = 0 for |x| ⩾ 1.

Definition 1.1. Let E_{q} (1 ⩽ q ⩽ ∞) be the space of all functions u ∈ E_{loc} such that

∥u∥_{E_{q}} := (∫_{R^{n}} ∥u(.)ω(. − y)∥_{E}^{q}dy)^{1/q} < ∞,  1 ⩽ q < ∞,
∥u∥_{E_{∞}} := sup_{y∈R^{n}} ∥u(.)ω(. − y)∥_{E} < ∞.

It can be proved that E_{q} is a Banach space. If Ω is a domain in R^{n}, then, by definition, E_{q}(Ω) is the space of restrictions of E_{q} to Ω with the usual norm for restrictions. One can readily see that, if Ω is a bounded domain, then

E_{q}(Ω) = E(Ω), 1 ⩽ q ⩽ ∞.

In particular, if E = W^{s,p}, then we write W_{q}^{s,p} = E_{q} (1 ⩽ q ⩽ ∞). It can be proved that

W_{p}^{s,p} = W^{s,p} (s ⩾ 0, 1 < p < ∞).

Hence, the spaces W_{q}^{s,p} generalize Sobolev spaces (related to the case q < ∞) and Stepanov spaces (related to the case q = ∞), see [8, 9].

If E = H^{s}, then we can also introduce the spaces H_{q}^{s}, 1 ⩽ q ⩽ ∞.

Suppose that

u ∈ W_{q}^{l,p}(Ω), f_{0} ∈ W_{q}^{l−2m,p}(Ω), f_{j} ∈ W_{q}^{l−s_{j}−1/p,p}(Γ), j = 1, …, m,

in problem (1.1), where l ⩾ 2m is an integer, 1 < p < ∞, and 1 ⩽ q ⩽ ∞.

1.3. Limiting Problems

To formulate the results of the work, recall the notions of limiting operators and limiting domains. Limiting operators were first investigated in [7–9] (see also [16]) for differential operators on the real axis with quasi-periodic coefficients, and then for elliptic operators in R^{n} and in domains that are cylindrical or conical at infinity [13, 14, 4].

In the general case, the limiting operators and domains were introduced in [19, 20]. Their construction can briefly be described as follows. Let x_{k} ∈ Ω be a sequence tending to infinity. Consider the shifted domains Ω_{k} corresponding to the shifted characteristic functions χ(x + x_{k}), where χ(x) is the characteristic function of the domain Ω. Consider a ball B_{r} ⊂ R^{n} of radius r centered at the origin. Suppose that, for all k, there are points of the boundaries ∂Ω_{k} placed in the interior of the ball B_{r}. If the boundaries are sufficiently smooth, then we can expect that from the sequence Ω_{k} ∩ B_{r} a subsequence that converges to some limiting domain Ω_{∗} may be chosen. After that, we take a larger ball and choose a convergent subsequence of the previous subsequence. The usual diagonal process allows us to extend the limiting domain to the entire space.

To define limiting operators, we consider the shifted coefficients a^{α}(x + x_{k}) and b_{j}^{α}(x + x_{k}) and choose subsequences that converge to some limiting functions â^{α}(x) and b̂_{j}^{α}(x) uniformly on every bounded set. The limiting operator is the operator with the limiting coefficients. Limiting operators considered in limiting domains constitute limiting problems. It is clear that the same problem can have a family of limiting problems depending on the choice of the sequence x_{k} and on the choice of both converging subsequences of domains and coefficients.

1.4. Normal Solvability

Write

L = (A, B_{1}, …, B_{m}).

Let L̂ be a corresponding limiting operator. The following condition determines the normal solvability of elliptic problems.

Condition NS. Any limiting problem

L̂u = 0, x ∈ Ω_{∗}, u ∈ W_{∞}^{l,p}(Ω_{∗}),

has the zero solution only.

This condition is necessary and sufficient for a general elliptic operator in a Hölder space to be normally solvable with a finite-dimensional kernel [19]. For scalar elliptic problems in Sobolev spaces, this fact was proved in [20]. More precisely, we prove that the elliptic operator L is normally solvable and has a finite-dimensional kernel in the space W_{∞}^{l,p} (1 < p < ∞) if and only if Condition NS is satisfied.

Similar condition can be formulated for the formally adjoint problem.

Condition NS^{∗}. Any limiting problem

L̂^{∗}v = 0, x ∈ Ω_{∗}, v ∈ W_{∞}^{l,p}(Ω_{∗}),

has the zero solution only. Here L̂^{∗} stands for the operator formally adjoint to L̂.

In Section 6, we show that Conditions NS and NS^{∗} have equivalent formulations in terms of smooth functions and hence do not depend on l and p.

1.5. Fredholm Theorem

The main result of the paper is given by the theorem below. Let us first introduce the space E.

Definition 1.2. A function f belongs to the space E if f(x) is an infinitely differentiable function defined on the entire space R^{n} and, for any multi-index α, there exists an ϵ > 0 such that

|D^{α}f(x) e^{ϵ|x|}| ⩽ C_{α} ∀x ∈ R^{n}.

Let E(Ω̄) be the restriction of E to Ω̄, and let the definition of E(Γ) be standard.

Theorem 1.3. Let Conditions NS and NS^{∗} be satisfied. Suppose that

f_{0} ∈ W_{q}^{l−2m,p}(Ω), f_{j} ∈ W_{q}^{l−s_{j}−1/p,p}(Γ), j = 1, …, m, 1 < p < ∞, 1 < q ⩽ p, l ⩾ 2m,

and l is an integer.

Then problem (1.1) has a solution u ∈ W_{q}^{l,p} if and only if

∫_{Ω} f_{0} \overline{v_{0}} dx + ∑_{j=1}^{m} ∫_{Γ} f_{j} \overline{T_{j}v_{0}} ds = 0  (1.4)

for any v_{0} in the finite-dimensional space

N^{∗} = {v_{0} : v_{0} ∈ E(Ω̄), A^{∗}v_{0} = 0, C_{1}v_{0} = 0, …, C_{m}v_{0} = 0}.  (1.5)

Since Condition NS implies that the kernel of L is finite-dimensional, it follows that if Conditions NS and NS^{∗} are satisfied, then the elliptic operator has the Fredholm property. Moreover, its cokernel belongs to the space E, and therefore does not depend on l, p, q. It is shown in Section 6 that the same is true for the kernel. Thus, the index of the operator does not depend on l, p, q.

1.6. Generalizations

In this paper, we consider the case in which the boundary operators form a normal system and Green’s formula can be obtained. However, it turns out that both the normal solvability and Fredholm property can be proved for elliptic problems in unbounded domains without these assumptions, and both the properties can be extended even to general systems of equations in the Douglis–Nirenberg sense. In this general case, one cannot use the formally adjoint problem in the above definition. For this reason, instead of the formally adjoint problem, we can use the adjoint problem in the operator sense. These results will be published elsewhere (see [21, 22]).

2. A PRIORI ESTIMATES

In this section, we derive some special a priori estimates of solutions used below. Estimates of this type were first obtained in [13, 14] for elliptic operators on the entire space R^{n}. Below we use the notation

E = W^{l,p}(Ω), F = W^{l−2m,p}(Ω) × W^{l−s_{1}−1/p,p}(Γ) × ⋯ × W^{l−s_{m}−1/p,p}(Γ).

The corresponding spaces E_{∞} and F_{∞} are defined in Section 1.2.

Theorem 2.1. Let Condition NS be satisfied. Then there exist numbers M_{0} and R_{0} such that the following estimate holds:

∥u∥_{E_{∞}} ⩽ M_{0} (∥Lu∥_{F_{∞}} + ∥u∥_{L^{p}(Ω_{R_{0}} )}) ∀u ∈ E_{∞}.  (2.1)

Here Ω_{R_{0}} = Ω ∩ {|x| ⩽ R_{0}}.

Proof. Suppose that the assertion of the theorem fails. Let {M_{k}} and {R_{k}} be given sequences, and let M_{k} → ∞ and R_{k} → ∞. Then there exists a sequence u_{k} ∈ E_{∞} such that

∥u_{k}∥_{E_{∞}} > M_{k} (∥Lu_{k}∥_{F_{∞}} + ∥u_{k}∥_{L^{p}(Ω_{R_{k}})}) .

We can suppose that

∥u_{k}∥_{E_{∞}} = 1.  (2.2)

Then

∥Lu_{k}∥_{F_{∞}} + ∥u_{k}∥_{L^{p}(Ω_{R_{k}})} < \frac{1}{M_{k}} → 0 as k → ∞.  (2.3)

Theorem 2.1 in [20] implies that

∥Lu_{k}∥_{F_{∞}} + ∥u_{k}∥_{L_{∞}^{p}(Ω)} ⩾ \frac{1}{c}.

It follows from (2.3) that ∥Lu_{k}∥_{F_{∞}} → 0. Hence

∥u_{k}∥_{L_{∞}^{p}(Ω)} > \frac{1}{2c} for k ⩾ k_{0}  (2.4)

for some k_{0}. Since

∥u_{k}∥_{L_{∞}^{p}(Ω)} = sup_{y∈Ω} ∥u_{k}∥_{L^{p}(Q_{y}∩Ω)},

where Q_{y} is the unit ball centered at y, it follows from (2.4) that there exists a y_{k} ∈ Ω such that

∥u_{k}∥_{L^{p}(Q_{y_{k}}∩Ω)} > \frac{1}{2c} .  (2.5)

By (2.3),

∥u_{k}∥_{L^{p}(Ω_{R_{k}})} → 0.

This convergence and relation (2.5) imply that |y_{k}| → ∞.

Write

Lu_{k} = f_{k}.  (2.6)

It follows from (2.3) that

∥f_{k}∥_{F_{∞}} → 0 as k → ∞.  (2.7)

We also write x = y + y_{k} and

w_{k}(y) = u_{k}(y + y_{k})  (2.8)

and rewrite (2.6) in the detailed form

∑_{|α|⩽2m} a^{α}(x)D^{α}u_{k} = f_{k}^{0}, x ∈ Ω,  (2.9)

∑_{|β|⩽s_{j}} b_{j}^{β}(x)D^{β}u_{k} = f_{k}^{j}, x ∈ ∂Ω, j = 1, …, m,  (2.10)

where f_{k} = (f_{k}^{0}, …, f_{k}^{m}). Denoting

a_{k}^{α}(y) = a^{α}(y + y_{k}), b_{kj}^{β}(y) = b_{j}^{β}(y + y_{k}), j = 1, …, m,

we see from (2.9) and (2.10) that

∑_{|α|⩽2m} a_{k}^{α}(y)D^{α}w_{k}(y) = f_{k}^{0}(y + y_{k}), x ∈ Ω_{k},  (2.11)

∑_{|β|⩽s_{j}} b_{kj}^{β}(y)D^{β}w_{k}(y) = f_{k}^{j}(y + y_{k}), x ∈ ∂Ω_{k}, j = 1, …, m,  (2.12)

where Ω_{k} is the shifted domain. It follows from (2.2) that

∥w_{k}∥_{E_{∞}(Ω_{k})} = 1.  (2.13)

Suppose that the functions w_{k} are extended to R^{n} in such a way that their W_{∞}^{l,p}(R^{n})-norms are uniformly bounded. Passing to a subsequence and retaining the same notation, we can suppose that

w_{k} → w_{0} in W^{l−ϵ,p}(R^{n}) locally (ϵ > 0),  (2.14)
w_{k} → w_{0} in W^{l,p}(R^{n}) locally weakly  (2.15)

for some w_{0} as k → ∞, and

w_{0} ∈ W_{∞}^{l,p}(R^{n}).  (2.16)

Let us prove that

L̂w_{0} = 0  (2.17)

for a limiting operator L̂. To this end, we pass to the limit in (2.11), (2.12) for some subsequence of k. We choose this subsequence in such a way that Ω_{k} converges to a limiting domain, Ω_{k} → Ω_{∗}, and keep the same notation for this subsequence.

Let us begin with equation (2.11). For any x_{0} ∈ Ω_{∗}, we take a neighborhood U in such a way that U ⊂ Ω_{k} if k is sufficiently large. For any ϕ ∈ D supported by U, it follows from (2.11) that

∫_{U} ∑_{|α|⩽2m} a_{k}^{α}(y)D^{α}w_{k}(y)ϕ(y)dy = ∫_{U} f_{k}^{0}(y + y_{k})ϕ(y)dy.  (2.18)

Passing to a subsequence if necessary, we can assume that

a_{k}^{α}(y) → â^{α}(y) in C^{l−2m}(R^{n}) locally,

where â^{α}(y) are the coefficients of the limiting operator. It follows from (2.15) that the sequence D^{α}w_{k} (|α| ⩽ 2m) converges locally weakly in W^{l−2m,p} to D^{α}w_{0} as k → ∞. Hence, we can pass to the limit in (2.18).

It follows from (2.7) that

∥f_{k}^{0}(· + y_{k})∥_{W_{∞}^{l−2m,p}(Ω_{k})} → 0 as k → ∞.

Hence, the right-hand side of (2.18) tends to zero. Passing to the limit in this equation, we obtain

∑_{|α|⩽2m} â^{α}(y)D^{α}w_{0}(y) = 0,  y ∈ Ω_{∗}.  (2.19)

Now consider (2.12). It follows from (2.14) that the sequence D^{β}w_{k} (|β| ⩽ s_{j}) tends to D^{β}w_{0} in W^{l−s_{j}−ϵ,p}(R^{n}) locally. Hence,

∑_{|β|⩽s_{j}} b_{j}_{k}^{β}(y)D^{β}w_{k}(y) → ∑_{|β|⩽s_{j}} b̂_{j}^{β}(y)D^{β}w_{0}(y), j = 1, …, m,  (2.20)

in W_{loc}^{l−s_{j}−ϵ,p}(R^{n}). Therefore, this convergence remains also valid in the space W_{loc}^{l−s_{j}−ϵ,p}(Ω_{∗}) and, consequently, in the space W_{loc}^{l−s_{j}−ϵ−1/p,p}(∂Ω_{∗}). In other words, we have proved that the convergence (2.20) is of the type W_{loc}^{l−s_{j}−ϵ−1/p,p}(∂Ω_{k} → ∂Ω_{∗}) (see [20]).

Now consider the right-hand side of (2.12). According to (2.7), we have

∥f_{k}^{j}(· + y_{k})∥_{W_{∞}^{l−s_{j} −1/p,p}(∂Ω_{k})} → 0 as k → ∞, j = 1, …, m.

We can extend f_{k}^{j}(y + y_{k}) to the entire space R^{n} in such a way that

f_{k}^{j}(· + y_{k}) → 0 in W_{∞}^{l−s_{j},p}(R^{n}).

Therefore,

f_{k}^{j}(· + y_{k}) → 0 in W_{loc}^{l−s_{j}−1/p,p}(∂Ω_{k} → ∂Ω_{∗}).

This fact, together with the convergence (2.20), implies that

∑_{|β|⩽s_{j}} b̂_{j}^{β}(y)D^{β}w_{0}(y) = 0, y ∈ ∂Ω_{∗}.  (2.21)

It follows from (2.16) that the left-hand side of this relation belongs to the space W_{∞}^{l−s_{j}−1/p,p}(∂Ω_{∗}). Hence, this formula can be regarded as a relation in the space W_{∞}^{l−s_{j}−1/p,p}(∂Ω_{∗}).

It follows from (2.19) and (2.21) that w_{0} is a solution of the limiting problem (2.17). Let us now prove that w_{0} ≠ 0. Formulas (2.5) and (2.8) imply that

∥w_{k}∥_{L^{p}(Ω_{k}∩Q_{0})} > \frac{1}{2c},  (2.22)

where Q_{0} is the unit ball centered at the origin. Let us prove that

∥w_{0}∥_{L^{p}(Ω_{∗}∩Q_{0})} ⩾ \frac{1}{2c}.  (2.23)

Indeed, relation (2.14) yields

w_{k} → w_{0} in L_{loc}^{p}(R^{n}).

RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS Vol. 11 No. 4 2004

FORMALLY ADJOINT PROBLEMS AND SOLVABILITY CONDITIONS 481

Write S_{k} = Ω_{k} ∩ Q_{0} and S_{∗} = Ω_{∗} ∩ Q_{0}. Then |∥w_{k}∥_{L^{p}(S_{k})} − ∥w_{0}∥_{L^{p}(S_{∗})}| ⩽ |∥w_{k}∥_{L^{p}(S_{k})} − ∥w_{0}∥_{L^{p}(S_{k})}| + |∥w_{0}∥_{L^{p}(S_{k})} − ∥w_{0}∥_{L^{p}(S_{∗})}| ≡ A_{k} + B_{k}. Further,

A_{k} ⩽ ∥w_{k} − w_{0}∥_{L^{p}(S_{k})} = (∫_{S_{k}} |w_{k} − w_{0}|^{p}dx)^{1/p} → 0 as k → ∞

and

B_{k} ⩽ |∥w_{0}∥_{L^{p}(S_{k})} − ∥w_{0}∥_{L^{p}(S_{∗})}| ⩽ M (∫_{S_{k}△S_{∗}} |w_{0}|^{p}dx)^{1/p} → 0 as k → ∞

because the measure of the symmetric difference S_{k}△S_{∗} converges to 0.

We have thus proved that

∥w_{k}∥_{L^{p}(Ω_{k}∩Q_{0})} → ∥w_{0}∥_{L^{p}(Ω_{∗}∩Q_{0})},

and (2.23) follows from (2.22).

Therefore, there exists a limiting problem with a nonzero solution. This contradicts Condition NS. The theorem is thus proved.

Write

ω_{μ} = e^{μ\sqrt{1+|x|^{2}}},

where μ is a real number.

Theorem 2.2. Let Condition NS be satisfied. Then there exist numbers M_{0} > 0, R_{0} > 0, and μ_{0} > 0 such that the following estimate holds for any μ, 0 < μ < μ_{0}:

∥ω_{μ}u∥_{E_{∞}} ⩽ M_{0} (∥ω_{μ}Lu∥_{F_{∞}} + ∥ω_{μ}u∥_{L^{p}(Ω_{R_{0}} )}) ∀ω_{μ}u ∈ E_{∞}.  (2.24)

Proof. According to (2.1), we have

∥ω_{μ}u∥_{E_{∞}} ⩽ M (∥L(ω_{μ}u)∥_{F_{∞}} + ∥ω_{μ}u∥_{L^{p}(Ω_{R_{0}} )}) .  (2.25)

Recall that L = (A, B_{j}). Let us first consider the operator

A(ω_{μ}u) = ∑_{|α|⩽2m} a^{α}(x)D^{α}(ω_{μ}u).

We have

A(ω_{μ}u) = ω_{μ}A(u) + Φ,  (2.26)

where

Φ = ∑_{|α|⩽2m} ∑_{\begin{matrix}β+γ=α \\ |β|>0 \end{matrix}} a^{α}(x)c_{βγ}D^{β}ω_{μ}D^{γ}u,

and c_{βγ} are some constants. Direct calculations give the following estimate:

∥Φ∥_{W_{∞}^{l−2m,p}} ⩽ M_{1}μ∥ω_{μ}u∥_{E_{∞}(Ω)}.  (2.27)

For the boundary operators, we have

B_{j}(ω_{μ}u) = ∑_{|β|⩽m_{j}} b_{j}^{β}(x)D^{β}(ω_{μ}u).

RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS Vol. 11 No. 4 2004 482 A. VOLPERT, V. VOLPERT

As above, we obtain

B_{j}(ω_{μ}u) = ω_{μ}B_{j}(u) + Ψ_{j},  (2.28)
∥Ψ_{j}∥_{W_{∞}^{l−s_{j} −1/p,p}} ⩽ M_{2}μ∥ω_{μ}u∥_{E_{∞}(Ω)}.  (2.29)

It follows from (2.26)–(2.29) that

∥L(ω_{μ}u)∥_{F_{∞}} ⩽ ∥ω_{μ}Lu∥_{F_{∞}} + Mμ∥ω_{μ}u∥_{E_{∞}}.

The assertion of the theorem now follows from this estimate and from (2.25). This completes the proof of the theorem.

Corollary 2.3. If 0 < μ < μ_{0}, u ∈ E_{∞}, and ω_{μ}Lu ∈ F_{∞}, then ω_{μ}u ∈ E_{∞}.

In particular, if u ∈ E_{∞} and Lu = 0, then ω_{μ}u ∈ E_{∞}.

We recall that the spaces W^{s,2} and H^{s} coincide. The results of this section will be used below for the spaces of Bessel potentials.

3. SPACES V^{2m}(Ω)

Denote by V^{2m}(Ω) the family of all functions u ∈ H_{∞}^{2m}(Ω) such that

∫_{Ω} |Au|^{2}dx + ∑_{j=0}^{m−1}⟨B_{j}u, B_{j}u⟩ < ∞.  (3.1)

Here ⟨ , ⟩ stands for the norm in H^{2m−s_{j}}. We suppose that Condition NS is satisfied. Let

N = {u | u ∈ H_{∞}^{2m}(Ω), Au = 0, B_{j} = 0, j = 0, …, m − 1}.

Then N is finite-dimensional, and one has u(x) → 0 exponentially as |x| → ∞, x ∈ Ω.

For u, v ∈ V^{2m}(Ω), introduce the inner product

[u, v] = ∫_{Ω} Au · \overline{Av} dx + ∑_{j=0}^{m−1}⟨B_{j}u, B_{j}v⟩ + ∫_{Ω} P u · \overline{P v} dx.  (3.2)

Here P is the projection operator in L^{2}(Ω) onto the subspace N. More precisely, let u_{1}, …, u_{k} be a basis in N, and let v_{1}, …, v_{k} ∈ D(Ω) be functions such that

∫_{Ω} u_{i}(x)\overline{v_{j}(x)} dx = δ_{ij}, i, j = 1, …, k.

Then

P u = ∑_{i=1}^{k}(u, v_{i})u_{i}.  (3.3)

Here and in what follows we use the notation

(u, v) = ∫_{Ω} u(x)\overline{v(x)} dx.  (3.4)

We also write

[u, v]_{0} = ∫_{Ω} Au · \overline{Av} dx + ∑_{j=0}^{m−1}⟨B_{j}u, B_{j}v⟩.  (3.5)

Therefore,

[u, v] = [u, v]_{0} + (P u, P v).  (3.6)

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FORMALLY ADJOINT PROBLEMS AND SOLVABILITY CONDITIONS 483

Proposition 3.1. If u ∈ V^{2m}(Ω), then

∥u∥_{H_{∞}^{2m}(Ω)} ⩽ C∥u∥_{V^{2m}(Ω)},  (3.7)

where

∥u∥_{V^{2m}(Ω)} = \sqrt{[u, u]}.

Proof. Since Condition NS is satisfied, it follows that

∥u∥_{H_{∞}^{2m}(Ω)} ⩽ C(∥Au∥_{L_{∞}^{2}(Ω)} + ∑_{j=0}^{m−1} ∥B_{j}u∥_{H_{∞}^{2m−s_{j} −1/2}(Γ)} + ( ∫_{Ω_{ρ}} |u(x)|^{2}dx)^{1/2})

for some C > 0, ρ > 0, and for any u ∈ H_{∞}^{2m}(Ω) (Theorem 2.1). Therefore,

∥u∥_{H_{∞}^{2m}(Ω)} ⩽ C(∥Au∥_{L^{2}(Ω)} + ∑_{j=0}^{m−1} ∥B_{j}u∥_{H^{2m−s_{j} −1/2}(Γ)} + ( ∫_{Ω_{ρ}} |u(x)|^{2}dx)^{1/2}).

It follows that

∥u∥_{H_{∞}^{2m}(Ω)} ⩽ C_{1}(∥u∥_{V^{2m}(Ω)} + ( ∫_{Ω_{ρ}} |u(x)|^{2}dx)^{1/2}).  (3.8)

It is sufficient to prove (3.7) for any function ∥u∥_{V^{2m}(Ω)} = 1. In this case, the estimate becomes

∥u∥_{H_{∞}^{2m}(Ω)} ⩽ C ∀u ∈ V^{2m}(Ω), ∥u∥_{V^{2m}(Ω)} = 1. (3.9)

Suppose that this estimate fails. Then there exists a sequence {u_{k}}, u_{k} ∈ V^{2m}(Ω) such that

∥u_{k}∥_{H_{∞}^{2m}(Ω)} → ∞, ∥u_{k}∥_{V^{2m}(Ω)} = 1.

Write

w_{k} = \frac{u_{k}}{∥u_{k}∥_{H_{∞}^{2m}(Ω)}} .

Then

∥w_{k}∥_{H_{∞}^{2m}(Ω)} = 1,  (3.10)
∥w_{k}∥_{V^{2m}(Ω)} → 0 as k → ∞.  (3.11)

It follows from (3.10) that there exists a subsequence of w_{k}, which we still denote by w_{k}, satisfying the condition

∫_{Ω_{ρ}} |w_{k}(x) − w_{l}(x)|^{2}dx → 0 as k, l → ∞.

The estimate (3.8) implies that

∥w_{k} − w_{l}∥_{H_{∞}^{2m}(Ω)} → 0 as k, l → ∞.

Hence, there exists a function w ∈ H_{∞}^{2m}(Ω) for which

∥w_{k} − w∥_{H_{∞}^{2m}(Ω)} → 0 as k → ∞.  (3.12)

It follows from (3.10) that

∥w∥_{H_{∞}^{2m}(Ω)} = 1.  (3.13)

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On the other hand, relation (3.11) yields

∫_{Ω} |Aw_{k}|^{2}dx → 0,  (3.14)
⟨B_{j}w_{k}, B_{j}w_{k}⟩ → 0, j = 0, …, m − 1,  (3.15)
∫_{Ω} |P w_{k}|^{2}dx → 0 as k → ∞.  (3.16)

It follows from (3.14) that

Aw = 0.  (3.17)

Indeed, by (3.14) we have

∫_{Ω} Aw_{k}\overline{ϕ(x)}dx → 0 as k → ∞  (3.18)

for any ϕ ∈ C_{0}^{∞}(Ω). It follows from (3.12) that

∥w_{k} − w∥_{H^{2m}(B∩Ω)} → 0 as k → ∞

for any ball B ⊂ R^{n}. Therefore, we can pass to the limit in (3.18):

∫_{Ω} Aw\overline{ϕ(x)}dx = 0.

Equality (3.17) follows from the last formula.

Similarly, it follows from (3.15) that

B_{j}w = 0, j = 0, …, m − 1.

This relation, together with (3.17), shows that w ∈ N. Hence,

w = P w.  (3.19)

Now (3.3) and (3.12) imply

∥P w_{k} − P w∥_{L^{2}(Ω)} ⩽ ∑_{i=1}^{k} |(w_{k} − w, v_{i})|∥u_{i}∥_{L^{2}(Ω)} → 0.

It follows from this convergence and from (3.16) that P w = 0 and it follows from (3.19) that w = 0. This contradicts (3.13). The proposition is proved.

Proposition 3.2. V^{2m}(Ω) is a Hilbert space.

Proof. We must prove that the space V^{2m}(Ω) is complete. Let

∥u_{k} − u_{l}∥_{V^{2m}(Ω)} → 0 as k, l → ∞.  (3.20)

Then, by (3.7), we have

∥u_{k} − u_{l}∥_{H_{∞}^{2m}(Ω)} → 0 as k, l → ∞.

Hence, there exists a function u ∈ H_{∞}^{2m}(Ω) such that

∥u_{k} − u∥_{H_{∞}^{2m}(Ω)} → 0 as k → ∞.  (3.21)

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FORMALLY ADJOINT PROBLEMS AND SOLVABILITY CONDITIONS 485

Write v_{k} = Au_{k}. Then it follows from (3.20) that

∥v_{k} − v_{l}∥_{L^{2}(Ω)} → 0 as k, l → ∞.

Therefore there exists a function v ∈ L^{2}(Ω) such that

∥v_{k} − v∥_{L^{2}(Ω)} → 0 as k → ∞.  (3.22)

Hence,

∥v_{k} − v∥_{L^{2}(Ω∩B)} → 0 as k → ∞

for any ball B ⊂ R^{n}, or, in other words,

∥Au_{k} − v∥_{L^{2}(Ω∩B)} → 0 as k → ∞.

On the other hand, it follows from (3.21) that

∥Au_{k} − Au∥_{L^{2}(Ω∩B)} → 0 as k → ∞.

Hence, v = Au in Ω ∩ B. Since B is arbitrary, we have

v = Au in Ω and Au ∈ L^{2}(Ω).  (3.23)

This, together with (3.22), yields

∥Au_{k} − Au∥_{L^{2}(Ω)} → 0 as k → ∞.  (3.24)

Let us now consider the boundary operators. Write B_{j}u_{k} = w_{k}. Then it follows from (3.20) that

∥w_{k} − w_{l}∥_{H^{2m−s_{j} −1/2}(Γ)} → 0 as k, l → ∞.

Hence, there exists a function w ∈ H^{2m−s_{j}−1/2}(Γ) such that

∥w_{k} − w∥_{H^{2m−s_{j} −1/2}(Γ)} → 0 as k → ∞.  (3.25)

Therefore,

∥w_{k} − w∥_{H^{2m−ms_{j} −1/2}(Γ∩B)} → 0 as k → ∞  (3.26)

for any ball B ⊂ R^{n}. Formula (3.21) yields

∥B_{j}u_{k} − B_{j}u∥_{H^{2m−s_{j} −1/2}(Γ∩B)} → 0 as k → ∞.

This fact, together with (3.26), implies the relation B_{j}u = w for the functions B_{j}u and w regarded as elements of H^{2m−s_{j}−1/2}(Γ ∩ B). Since B is an arbitrary ball, we conclude that B_{j}u = w as distributions on Γ. It follows that

B_{j}u ∈ H^{2m−s_{j}−1/2}(Γ),  (3.27)

and (3.25) yields

∥B_{j}u_{k} − B_{j}u∥_{H^{2m−s_{j} −1/2}(Γ)} → 0 as k → ∞.  (3.28)

It follows from (3.23) and (3.27) that u ∈ V^{2m}(Ω).

It remains to consider the operator P. We obviously have

∥P (u_{k} − u_{l})∥_{L^{2}(Ω)} → 0 as k, l → ∞.

Taking into account (3.3), we obtain

∥P u_{k} − P u∥_{L^{2}(Ω)} → 0 as k → ∞.  (3.29)

This convergence and the relations (3.28) and (3.24) imply

∥u_{k} − u∥_{V^{2m}(Ω)} → 0 as k → ∞.

This completes the proof of the proposition.

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Proposition 3.3. If u ∈ H^{2m}(Ω), then u ∈ V^{2m}(Ω) and

∥u∥_{V^{2m}(Ω)} ⩽ C∥u∥_{H^{2m}(Ω)}.  (3.30)

Proof. Let u ∈ H^{2m}(Ω). Then u ∈ H_{∞}^{2m}(Ω), and relation (3.1) is satisfied. Hence, u ∈ V^{2m}(Ω) and

[u, u]_{0} ⩽ C_{1}∥u∥_{H^{2m}(Ω)}^{2}.  (3.31)

It follows from (3.3) that

∥P u∥_{L^{2}(Ω)}^{2} ⩽ C_{2}∥u∥_{H^{2m}(Ω)}^{2}.  (3.32)

This estimate and the relations (3.31) and (3.6) imply inequality (3.30). The proposition is thus proved.

It follows from Propositions 3.1 and 3.3 that

H^{2m}(Ω) ⊂ V^{2m}(Ω) ⊂ H_{∞}^{2m}(Ω),

where both the inclusions are with the topology. These spaces coincide locally. The spaces are different due to different behavior of their elements at infinity.

Proposition 3.4. If f ∈ L_{1}^{2}(Ω) and v ∈ V^{2m}(Ω), then

(v, f) = ∫_{Ω} v(x)\overline{f(x)} dx

is a continuous functional on V^{2m}(Ω).

Proof. We have

|(v, f)| = | ∫_{Ω} ∑_{i=1}^{∞} ϕ_{i}(x)v(x)\overline{f(x)} dx|

(ϕ_{i} is a partition of unity, ψ_{i}(x) ∈ D, and ψ_{i}(x) = 1 for x ∈ supp ϕ_{i})

⩽ ∑_{i=1}^{∞} | ∫_{Ω} ϕ_{i}vψ_{i}\overline{f} dx| ⩽ ∑_{i=1}^{∞} ∥ϕ_{i}v∥_{L^{2}}∥ψ_{i}\overline{f}∥_{L^{2}} ⩽ C∥v∥_{L_{∞}^{2}(Ω)} ∑_{i=1}^{∞} ∥ψ_{i}f∥_{L^{2}(Ω)}

⩽ C_{1}∥v∥_{L_{∞}^{2}(Ω)}∥f∥_{L_{1}^{2}(Ω)} ⩽ C_{1}∥v∥_{H_{∞}^{2m}(Ω)}∥f∥_{L_{1}^{2}(Ω)} ⩽ C_{2}∥v∥_{V^{2m}(Ω)}∥f∥_{L_{1}^{2}(Ω)}.

The proposition is proved.

The following proposition is a generalization of Proposition 5.1 in [11].

Proposition 3.5. Let u ∈ V^{2m}(Ω), let f ∈ L_{1}^{2}(Ω) ∩ H_{loc}^{r}(Ω), where r is a positive integer, and let

[u, v]_{0} = ∫_{Ω} f\overline{v} dx ∀v ∈ V^{2m}(Ω).  (3.33)

Then u ∈ H_{loc}^{4m+r}(Ω).

Proof. Choosing v ∈ D(Ω) in (3.33), we obtain A^{∗}Au = f. Hence, u ∈ H^{4m+r}(ω) for any open bounded set ω such that ω̄ ⊂ Ω.

Since the space V^{2m}(Ω) locally coincides with the space H^{2m}(Ω), we obtain the same formula as (5.19) in [11] and use the corresponding result of [11]. The proposition is proved.

FORMALLY ADJOINT PROBLEMS AND SOLVABILITY CONDITIONS 487

4. FREDHOLM THEOREMS

Below we use Green’s formula (1.2) and Definition 1.2.

Proposition 4.1. Let Conditions NS and NS^{∗} be satisfied, and let f_{0} ∈ D(Ω̄) and f_{j} ∈ D(Γ), j = 1, …, m. Then problem (1.1) has a solution u ∈ H^{2m}(Ω) if and only if

∫_{Ω} f_{0} \overline{v_{0}} dx + ∑_{j=1}^{m} ∫_{Γ} f_{j} \overline{T_{j}v_{0}} ds = 0  (4.1)

for any v_{0} in the finite-dimensional space

N^{∗} = {v_{0} : v_{0} ∈ E(Ω̄), A^{∗}v_{0} = 0, C_{1}v_{0} = 0, …, C_{m}v_{0} = 0}.  (4.2)

Proof. Green’s formula (1.2) implies that condition (4.1) is necessary. Let us prove that this condition is sufficient.

1. We begin with the case in which

f_{0} ∈ D(Ω̄), f_{j} = 0, j = 1, …, m.  (4.3)

Introduce the space V_{∗}^{2m}(Ω) which is similar to the space V^{2m}(Ω) in Section 3. Namely, V_{∗}^{2m}(Ω) is the collection of all functions u ∈ H_{∞}^{2m}(Ω) such that

∫_{Ω} |A^{∗}u|^{2}dx + ∑_{j=1}^{m}⟨C_{j}u, C_{j}u⟩_{j} < ∞.

We also introduce the inner product

[u, v] = ∫_{Ω} A^{∗}u\overline{A^{∗}v}dx + ∑_{j=1}^{m}⟨C_{j}u, C_{j}v⟩_{j} + ∫_{Ω} P u\overline{P v}dx,  (4.4)

where P is the projection in L^{2}(Ω) onto the space N^{∗} defined by (4.2). More precisely, let u_{1}, …, u_{k} be a basis in N^{∗}, and let v_{1}, …, v_{k} ∈ D(Ω) be such that

(u_{i}, v_{j}) = δ_{ij}, i, j = 1, …, k.

Then

P u = ∑_{i=1}^{k}(u, v_{i})u_{i}.

The norm in V_{∗}^{2m}(Ω) is given by the relation

∥u∥_{V_{∗}^{2m}(Ω)} = \sqrt{[u, u]}.

Similarly to the proof in Section 3, we can show that V_{∗}^{2m}(Ω) is a Hilbert space and

∥u∥_{H_{∞}^{2m}(Ω)} ⩽ c∥u∥_{V_{∗}^{2m}(Ω)} ∀u ∈ V_{∗}^{2m}(Ω),
∥u∥_{V_{∗}^{2m}(Ω)} ⩽ c∥u∥_{H^{2m}(Ω)} ∀u ∈ H^{2m}(Ω),
H^{2m}(Ω) ⊂ V_{∗}^{2m}(Ω) ⊂ H_{∞}^{2m}(Ω).

Write

[u, v]_{0} = ∫_{Ω} A^{∗}u\overline{A^{∗}v}dx + ∑_{j=1}^{m}⟨C_{j}u, C_{j}v⟩_{j}.  (4.5)

Then, similarly to Proposition 3.5, we have the following assertion.

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Proposition 4.2. Let u ∈ V_{∗}^{2m}(Ω), let f ∈ L_{1}^{2}(Ω) ∩ H_{loc}^{r}(Ω), where r is a positive integer, and let

[u, v]_{0} = ∫_{Ω} f\overline{v} dx ∀v ∈ V_{∗}^{2m}(Ω).  (4.6)

Then u ∈ H_{loc}^{4m+r}(Ω).

Since (v, f_{0}) is a continuous linear functional on V_{∗}^{2m}(Ω) for f_{0} ∈ D(Ω̄) (see Proposition 3.4), it follows that there exists a w ∈ V_{∗}^{2m}(Ω) such that

[w, v] = ∫_{Ω} f_{0}\overline{v} dx ∀v ∈ V_{∗}^{2m}(Ω).  (4.7)

Write

w′ = w − P w.  (4.8)

Let v′ ∈ V_{∗}^{2m}(Ω) and P v′ = 0. Then, by (4.7), we have

[w, v′] = ∫_{Ω} f_{0} \overline{v′} dx.  (4.9)

However,

[w, v′] = [w′ + P w, v′] = [w′, v′] + [P w, v′].  (4.10)

Obviously, [P w, v′] = (P w, P v′) since P w ∈ N^{∗}. Hence, [P w, v′] = 0, and it follows from (4.10) that [w, v′] = [w′, v′]. Moreover, (4.9) implies

[w′, v′] = (f_{0}, v′).  (4.11)

Let v ∈ V_{∗}^{2m}(Ω). Then

[w′, v] = [w′, v′] + [w′, v′′],

where v′′ = P v, v′ = (I − P)v, and P v′ = 0. Further,

[w′, v′′] = [w′, P v] = (P w′, P v) = 0

since P w′ = 0 by virtue of (4.8). It follows that [w′, v] = [w′, v′]. Formula (4.11) yields

[w′, v] = (f_{0}, v′).  (4.12)

The assumptions (4.3) and (4.1) imply that (f_{0}, v_{0}) = 0 for any v_{0} ∈ N^{∗}. In particular, (f_{0}, v′′) = 0 since v′′ = P v ∈ N^{∗}. Hence,

(f_{0}, v) = (f_{0}, v′ + v′′) = (f_{0}, v′).

It follows from (4.12) that

[w′, v] = (f_{0}, v) ∀v ∈ V_{∗}^{2m}(Ω).  (4.13)

Formula (4.8) yields P w′ = 0. Therefore, [w′, v] = [w′, v]_{0} by virtue of (4.5). We see from (4.13) that

[w′, v]_{0} = (f_{0}, v) ∀v ∈ V_{∗}^{2m}(Ω).  (4.14)

By Proposition 4.2, we obtain

w′ = H_{loc}^{4m+r}(Ω)  (4.15)

for any integer r. Hence, w′ is a smooth function. Write

u = A^{∗}w′.  (4.16)

It follows from (4.5) and (4.14) that

∫_{Ω} u\overline{A^{∗}v} dx + ∑_{j=1}^{m}⟨C_{j}w′, C_{j}v⟩_{j} = (f_{0}, v) ∀v ∈ V_{∗}^{2m}(Ω).  (4.17)

In particular,

∫_{Ω} u\overline{A^{∗}v} dx = (f_{0}, v) ∀v ∈ D(Ω).

This gives

Au = f_{0}.  (4.18)

Formula (4.17) yields

(u, A^{∗}v) + ∑_{i=1}^{m}⟨C_{j}w′, C_{j}v⟩_{j} = (Au, v) ∀v ∈ V_{∗}^{2m}(Ω).

Let us take an element v ∈ D(Ω̄) such that

C_{j}v = 0, j = 1, …, m.  (4.19)

Then (u, A^{∗}v) = (Au, v). We use Green’s formula (1.2) for u and v ∈ D(Ω̄) satisfying (4.19). This yields

∑_{j=1}^{m} ∫_{Γ} B_{j}u \overline{T_{j}v} ds = 0,

and hence

B_{j}u = 0, j = 1, …, m  (4.20)

(see [11, Corollary 2.1]). It follows from (4.15) and (4.16) that u ∈ H_{loc}^{2m}(Ω). It remains to study the behavior of u at infinity. Since Condition NS is satisfied and ω_{μ}Lu ∈ F_{∞}, we see that μu ∈ H_{∞}^{2m} (Corollary 2.3). Therefore, u ∈ H^{2m}. Proposition 4.1 is proved in the case of (4.3).

2. Let us now consider the case

f_{0} ∈ D(Ω̄), f_{j} ∈ D(Γ), j = 1, …, m.  (4.21)

This case can be reduced to the previous one. Let u_{1} ∈ D(Ω̄) be a function satisfying the boundary conditions in (1.1),

B_{j}u_{1} = f_{j} on Γ, j = 1, …, m.

The existence of a function of this kind follows from Lemma 2.2 in [11]. Write g_{0} = f_{0} − Au_{1}. Then g_{0} ∈ D(Ω̄).

Consider the problem

Au_{0} = g_{0} in Ω, B_{j}u_{0} = 0 on Γ, j = 1, …, m.  (4.22)

For any v_{0} ∈ N^{∗}, we have

∫_{Ω} g_{0}\overline{v_{0}}dx = ∫_{Ω} f_{0}\overline{v_{0}}dx − ∫_{Ω} Au_{1}\overline{v_{0}}dx.  (4.23)

Applying Green’s formula to u_{1} and v_{0}, we see that

∫_{Ω} Au_{1}\overline{v_{0}}dx = − ∑_{j=1}^{m} ∫_{Γ} B_{j}u_{1}\overline{T_{j}v_{0}}ds = − ∑_{j=1}^{m} ∫_{Γ} f_{j}\overline{T_{j}v_{0}}ds.

It follows from (4.23) that

∫_{Ω} g_{0}\overline{v_{0}}dx = ∫_{Ω} f_{0}\overline{v_{0}}dx + ∑_{j=1}^{m} ∫_{Γ} f_{j}\overline{T_{j}v_{0}}dx.

Therefore, relation (4.1) implies

∫_{Ω} g_{0}\overline{v_{0}}dx = 0 ∀v_{0} ∈ N^{∗}.

It was proved in the previous section that there exists a solution u_{0} ∈ H^{2m}(Ω) of problem (4.22). Obviously, u = u_{0} + u_{1} is a solution of problem (1.1). Thus, Proposition 4.1 is proved in the case (4.21) as well.

Let us also introduce the following duality: for any

f = (f_{0}, f_{1}, …, f_{m}) ∈ L_{∞}^{p_{0}}(Ω̄) × L_{∞}^{p_{1}}(Γ) × ⋯ × L_{∞}^{p_{m}}(Γ)

and

v = (v_{0}, v_{1}, …, v_{m}) ∈ E(Ω̄) × E(Γ) × ⋯ × E(Γ)

we write

⟨f, v⟩ = ∫_{Ω} f_{0}\overline{v_{0}}dx + ∑_{j=1}^{m} ∫_{Γ} f_{j}\overline{v_{j}}ds.  (4.24)

Here p_{i} ⩾ 1, i = 0, 1, …, m.

Consider the subspace

N̂ = {v | v = (v_{0}, T_{1}v_{0}, …, T_{m}v_{0}), v_{0} ∈ N^{∗}}

and denote by Ẇ_{∞}^{s,p}(R^{n}), s ⩾ 0, 1 < p < ∞, the closure of D in W_{∞}^{s,p}(R^{n}). As usual, let Ẇ_{∞}^{s,p}(Ω) be the restriction of Ẇ_{∞}^{s,p}(R^{n}) to Ω. The definition of the space Ẇ_{∞}^{s,p}(Γ) is standard.

Theorem 4.3. Let Conditions NS and NS^{∗} be satisfied. Then, for any f = (f_{0}, …, f_{m}), f_{0} ∈ D(Ω̄), f_{j} ∈ D(Γ), i = 1, …, m, such that ⟨f, v⟩ = 0 for any v ∈ N̂, the problem

Lu = f  (4.25)

has a solution u ∈ E.

Proof. As was proved above, problem (4.25) has a solution u ∈ H^{2m}(Ω). It follows from a priori estimates that u ∈ H^{2m+r}(Ω) for any integer r, r ⩾ 0. Moreover, Corollary 2.3 implies that, for any l ⩾ 2m, there exists an ϵ > 0 such that ω_{ϵ}u ∈ H_{∞}^{l}(Ω), where ω_{ϵ}(x) = e^{ϵ\sqrt{1+|x|^{2}}}. According to the embedding theorems, for |α| < l − n/2, we obtain

|D^{α}(ω_{ϵ}u(x))| ⩽ c∥ω_{ϵ}u∥_{H^{l}(Ω)} ∀x ∈ Ω.

Therefore, for any α, there exists an ϵ > 0 such that

|D^{α}(ω_{ϵ}u(x))| ⩽ c_{α} ∀x ∈ Ω.  (4.26)

Denoting v = ω_{ϵ}u, we can write

|D^{α}u(x)| ⩽ |D^{α}(ω_{−ϵ}v(x))| ⩽ ω_{−ϵ}c′_{α} ∀x ∈ Ω.

This implies that u ∈ E. The theorem is proved.

FORMALLY ADJOINT PROBLEMS AND SOLVABILITY CONDITIONS 491

Let

D(Ω, Γ) = {u | u = (u_{0}, u_{1}, …, u_{m}), u_{0} ∈ D(Ω̄), u_{j} ∈ D(Γ), j = 1, …, m}.

We take a basis of N̂ : v^{1}, …, v^{l}. Let e^{1}, …, e^{l} ∈ D(Ω, Γ) be a biorthogonal system,

⟨e^{k}, v^{i}⟩ = δ_{ik}, i, k = 1, …, l.

It follows from Theorem 4.3 that the problem

Lu = f − ∑_{i=1}^{l}⟨f, v^{i}⟩e^{i}  (4.27)

has a solution u ∈ E for any f ∈ D(Ω, Γ).

Let l be an integer, l ⩾ 2m.

Theorem 4.4. Let Conditions NS and NS^{∗} be satisfied. Then, for any f = (f_{0}, f_{1}, …, f_{m}), f_{0} ∈ Ẇ_{∞}^{l−2m,p}(Ω), f_{j} ∈ Ẇ_{∞}^{l−s_{j}−1/p,p}(Γ), j = 1, …, m, such that ⟨f, v⟩ = 0 for any v ∈ N̂, the problem

Lu = f  (4.28)

has a solution u ∈ Ẇ_{∞}^{l,p}(Ω).

Proof. Let {f^{k}}, k = 1, 2, …, f^{k} ∈ D(Ω, Γ) be a sequence such that f^{k} → f in the space

F = W_{∞}^{l−2m,p}(Ω) × ∏_{j=1}^{m} W_{∞}^{l−s_{j}−1/p,p}(Γ).

Consider a solution u^{k} ∈ E of problem (4.27) for f = f^{k}. It follows from Lemma 4.5 below that

∥w^{k}∥_{W_{∞}^{l,p}(Ω)} ⩽ c ∥f^{k} − ∑_{i=1}^{l}⟨f^{k}, v^{i}⟩e^{i}∥_{F} ,

where w^{k} = (I − P)u^{k}. Hence, the convergence w^{k} → u holds in W_{∞}^{l,p}(Ω). Obviously, u is a solution of (4.28). It follows from Corollary 6.2 that w^{k} ∈ E. Therefore, u ∈ Ẇ_{∞}^{l,p}(Ω). The theorem is proved.

Lemma 4.5. Let an operator L: E → F have a closed image and a finite-dimensional kernel. Denote by P a projection operator of E onto ker L, i.e., P^{2}u = P u, P u ∈ ker L, ∀u ∈ E. Then

∥u − P u∥_{E} ⩽ c∥Lu∥_{F}

for any u ∈ E, where c is a constant independent of u.

Proof. Write E_{0} = (I − P)E, where I is the identity operator on E. Let L_{0} be the restriction of L to E_{0}. Then L_{0} is invertible. For any u ∈ E, write u_{0} = (E − P)u. Then L_{0}u_{0} = Lu_{0} = Lu. Hence, Im L_{0} = Im L. By the Banach theorem on the inverse operator, the operator L_{0}^{−1} is bounded. Therefore, u_{0} = L_{0}^{−1}Lu, ∥u_{0}∥_{E} ⩽ c∥Lu∥_{F}. The lemma is proved.

5. SPACES E_{q}

In this section, we again use the notation

E = W^{l,p}(Ω), F = W^{l−2m,p}(Ω) × W_{∞}^{l−s_{1}−1/p,p}(Γ) × ⋯ × W_{∞}^{l−s_{m}−1/p,p}(Γ).

Recall that Ė_{∞} and Ḟ_{∞} are the closures of D with respect to the norms of the spaces Ė and Ḟ, respectively. In the previous section, we proved the Fredholm property of the operator L: Ė_{∞} → Ḟ_{∞}. In this section, we use this result to prove the Fredholm property of the operator L_{q}: E_{q} → F_{q}, where L_{q} stands for the same operator which acts on the corresponding spaces.

We begin with some auxiliary results.

RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS Vol. 11 No. 4 2004

Lemma 5.1. Let an operator L from a Banach space E to another space F have a bounded inverse defined on its image R(L) ⊂ F. Suppose that the equation

L_{μ}u = f

has a solution, where

L_{μ} = L + μK

and K: E → F is a bounded operator, ∥K∥ ⩽ M. Then, for μ sufficiently small, we have

∥u∥_{E} ⩽ C∥f∥_{F} ,

where the constant C depends on μ and M but does not depend on the operator K.

Proof. Since the equation

Lu + μKu = f

is solvable, it follows that f − μKu ∈ R(L). Therefore,

u = L^{−1}(f − μKu).

The assertion of the lemma follows from the estimate

∥u∥_{E} ⩽ ∥L^{−1}∥∥f − μKu∥_{F} ⩽ ∥L^{−1}∥(∥f∥_{F} + |μ|∥K∥∥u∥_{E}).

The lemma is proved.

We generalize here the approach developed in [14] for the operators acting on H^{s}(R^{n}). As above, we use the function

ω_{δ}(x) = e^{δ\sqrt{1+|x|^{2}}}

and a system of functions e_{i} ∈ Ḟ_{∞} biorthogonal to the functions v_{j} that form a basis in N̂ (Section 4). Since the infinitely differentiable functions with bounded supports are dense in Ḟ_{∞}, we can choose functions e_{i} with bounded supports. We assume that this condition is satisfied.

Lemma 5.2. For any f ∈ Ḟ_{∞}, there exists a solution u_{j} of the equation

Lu = θ_{j}f − ∑_{i=1}^{N}⟨θ_{j}f, v^{i}⟩e^{i},  (5.1)

and the following estimate holds if δ is sufficiently small:

∥u_{j}(·)ω_{δ}(· − y_{j})∥_{E_{q}} ⩽ C∥fθ_{j}∥_{F} ,  (5.2)

where B_{j} is the unit ball centered at y_{j}, supp θ_{j} ⊂ B_{j}, and the constant C is independent of j.

Proof. Since the operator L: Ė_{∞} → Ḟ_{∞} satisfies the Fredholm property, the equation

Lu = g − ∑_{i=1}^{N}⟨g, v^{i}⟩e^{i}  (5.3)

is solvable for any g ∈ Ḟ_{∞}. Let supp g ∈ B_{j}. Consider the function

g̃(x) = (g(x) − ∑_{i=1}^{N}⟨g, v_{i}⟩e_{i}(x))ω_{δ}(x − y_{j}).

Let us show that the norm of this function in F_{∞} is independent of j. We note first of all that the function ω_{δ}(x − y_{j}) is bounded in B_{j} together with all derivatives, independently of j. Therefore,

∥g(x)ω_{δ}(x − y_{j})∥_{F_{∞}} ⩽ C∥g(x)∥_{F_{∞}}

with a positive constant C independent of j. We use here the fact that the norm of a multiplier in Ω and ∂Ω can be estimated by the norm in C^{k}. Taking (4.24) into account, we obtain

|⟨g, v^{i}⟩| = |⟨g, ψ_{j}v^{i}⟩| ⩽ C sup_{x} |ω_{−μ}(x)ψ_{j}| ∥g∥_{F_{∞}},

where ψ_{j} = ψ(x − y_{j}) is a compactly supported function equal to 1 on B_{j} and v^{i} ∈ E(Ω). We have

sup_{x} |ω_{−μ}(x)ψ_{j}| ⩽ sup_{x} |ω_{−μ}(x + y_{j})ψ(x)| ⩽ Cω_{−μ}(y_{j}),

where C does not depend on j. For δ ⩽ μ, the product ω_{−μ}(y_{j})ω_{δ}(x−y_{j}) is bounded independently of y_{j} ∈ R^{n} and of x ∈ supp e^{i}(x). Hence.

∥g̃∥_{F_{∞}} ⩽ C∥g∥_{F_{∞}},

where the constant C depends on the diameter of the supports of e_{i} and is independent of j. Since u is a solution of equation (5.3), it follows that ũ = S_{δ}u is a solution of the equation

L_{δ}ũ = g̃,  (5.4)

where L_{δ} = S_{δ}LS_{−δ} and S_{δ} is the operator of multiplication by ω_{δ}(x − y_{j}). On the other hand, L_{δ} = L + δK for some bounded operator K, ∥K∥ ⩽ C, where C depends neither on j nor on δ if δ sufficiently small. By virtue of Lemma 5.1, the solution of (5.4), which belongs to the subspace complementary to the kernel of the operator L, admits the estimate

∥ũ∥_{E_{∞}} ⩽ C_{1}∥g̃∥_{F_{∞}} ⩽ C_{2}∥g∥_{F_{∞}}

independently of j. Let δ = δ_{1} + δ_{2}, where δ_{1} and δ_{2} are positive. Then ∥u(x)ω_{δ_{1}}(x − y_{j})∥_{E_{q}} = ∥ũ(x)ω_{−δ_{2}}(x − y_{j})∥_{E_{q}} ⩽ C_{3}∥ũ(x)∥_{E_{∞}} ⩽ C_{4}∥g∥_{F_{∞}}.

Applying this estimate to equation (5.1), we obtain ∥u(x)ω_{δ_{1}}(x − y_{j})∥_{E_{q}} ⩽ C_{5}∥fθ_{j}∥_{F_{∞}} ⩽ C_{6}∥fθ_{j}∥_{F} .

The lemma is proved.

Assumption 5.3. Let u_{j} ∈ E_{q}, j = 1, 2, …, and

∑_{j=1}^{∞} ∥u_{j} ω_{δ}(x − y_{j})∥_{E_{q}}^{q} < ∞.

Then the series u = ∑_{j=1}^{∞} u_{j} is convergent, and the following estimate holds:

∥u∥_{E_{q}}^{q} ⩽ C ∑_{j=1}^{∞} ∥u_{j} ω_{δ}(x − y_{j})∥_{E_{q}}^{q}.  (5.5)

If this assumption is satisfied, then it follows from the estimate in Lemma 5.2 that ∥u∥_{E_{q}} ⩽ C∥f∥_{F_{q}}.

Therefore, for any f ∈ F_{q}(⊂ F_{∞}), there exists a solution u ∈ E_{q} of the equation

Lu = f − ∑_{i=1}^{N}⟨f, v_{i}⟩e_{i}.  (5.6)

This implies that the operator L_{q} is normally solvable and the codimension of its image is finite. Its kernel is also finite-dimensional since this is the case for the operator L_{∞}. Hence, L_{q} is a Fredholm operator.

We note that inequality (5.5) characterizes the function spaces and is not related to the operators under consideration. In the remaining part of this section, we show that this inequality is satisfied for Sobolev spaces.

Lemma 5.4 (elementary inequality; see, e.g., [10]). Let u_{i} ⩾ 0. Then

(u_{1}^{s} + u_{2}^{s} + ⋯)^{1/s} ⩽ u_{1} + u_{2} + ⋯ (s ⩾ 1),
(u_{1}^{s} + u_{2}^{s} + ⋯)^{1/s} ⩾ u_{1} + u_{2} + ⋯ (s ⩽ 1).

Lemma 5.5. Let u = ∑_{i=1}^{∞} u_{i}, let y_{i} be centers of an orthogonal lattice in R^{n}, and let p ⩾ 1. Then the following estimate holds:

∥u∥_{L^{p}(R^{n})}^{p} ⩽ C ∑_{i=1}^{∞} ∥u_{i} ω_{δ}(x − y_{i})∥_{L^{p}(R^{n})}^{p}.  (5.7)

Proof. Let k = [p] + 1, p = ks. Here k is an integer, s < 1. If p is an integer, we do not need to introduce k. All estimates below can be done directly for p. We have

∥u∥_{L^{p}(R^{n})}^{p} = ∫_{R^{n}} |u|^{ks}dx = ∫_{R^{n}} | ∑_{i=1}^{∞} u_{i}|^{ks}dx ⩽ ∫_{R^{n}} (∑_{i=1}^{∞} |u_{i}|^{s})^{k} dx

= ∫_{R^{n}} ∑_{i_{1},i_{2},...,i_{k}=1}^{∞} |u_{i_{1}}|^{s}|u_{i_{2}}|^{s} ⋯ |u_{i_{k}}|^{s}dx,  (5.8)

By virtue of the inequality between the geometrical and arithmetical mean, we have

|u_{i_{1}}|^{s}|u_{i_{2}}|^{s} ⋯ |u_{i_{k}}|^{s} ⩽ \frac{1}{k} (|u_{i_{1}}|^{ks} + ⋯ + |u_{i_{k}}|^{ks}) .

The same inequality with any positive a_{i_{j}} gives

|u_{i_{1}}|^{s}|u_{i_{2}}|^{s} ⋯ |u_{i_{k}}|^{s} ⩽ \frac{1}{k} (|u_{i_{1}}|^{ks}a_{i_{1}}^{k−1}a_{i_{2}}^{−1} ⋯ a_{i_{k}}^{−1} + ⋯ + |u_{i_{k}}|^{ks}a_{i_{1}}^{−1}a_{i_{2}}^{−1} ⋯ a_{i_{k}}^{k−1}) .  (5.9)

Set a_{i_{k}}(x) = ω_{δ_{1}}(x − y_{i_{k}}). Then

∑_{i_{k}=1}^{∞} a_{i_{k}}^{−1}(x) ⩽ C ∀x,

and, substituting (5.9) into the right-hand side of (5.8) and taking into account that there are k similar summands, we obtain

∫_{R^{n}} ∑_{i_{1},i_{2},...,i_{k}} |u_{i_{1}}|^{s}|u_{i_{2}}|^{s} ⋯ |u_{i_{k}}|^{s}dx ⩽ ∑_{i_{1},i_{2},...,i_{k}} ∫_{R^{n}} |u_{i_{1}}|^{ks}a_{i_{1}}^{k−1}a_{i_{2}}^{−1} ⋯ a_{i_{k}}^{−1}dx

⩽ C^{k−1} ∫_{R^{n}} ∑_{i_{1}} |u_{i_{1}}|^{p}ω_{δ_{1}}(x − y_{i_{1}})^{k−1}dx.

Replacing C^{k−1} by C, we obtain (5.7) for δ = δ_{1}(k − 1)/p. The lemma is proved.

We can similarly prove the lemma for the spaces W^{l,p}(R^{n}) and W^{l,p}(Ω) with an integer l ⩾ 0.

Lemma 5.6. The estimate

∥u∥_{W^{k,p}(Ω)}^{p} ⩽ C ∑_{i=1}^{∞} ∥u_{i} ω_{δ}(x − y_{i})∥_{W^{k,p}(Ω)}^{p}  (5.10)

holds for any p ⩾ 1 and any integer k ⩾ 0.

This lemma proves that Assumption 5.3 (with q = p) holds for Sobolev spaces. This enables us to prove the Fredholm theorems for elliptic operators in Sobolev spaces. Thus, we have in fact proved the theorem stated below as Theorem 5.7.

Suppose that l ⩾ 2m is an integer and that 1 < p < ∞. Then the following theorem holds.

Theorem 5.7. Let Conditions NS and NS^{∗} be satisfied. In this case, if the function f satisfies the condition

f ∈ W^{l−2m,p}(Ω) × ∏_{j=1}^{m} W^{l−s_{j}−1/p,p}(Γ),

then the problem

Lu = f

has a solution u ∈ W^{l,p}(Ω) if and only if ⟨f, v⟩ = 0 for any v ∈ N̂.

Let us now show that the Fredholm property holds not only for q = p but also for q ⩽ p. To this end, we shall verify Assumption 5.3. Let

E = L^{p}, E_{q} = (L^{p})_{q}.

In this case,

∥u∥_{E_{q}}^{q} = ∑_{j} ∥ϕ_{j}u∥_{L^{p}}^{q}.

If u = ∑_{i} u_{i}, then it follows from Lemma 5.5 that

∥ϕ_{j}u∥_{L^{p}}^{p} ⩽ C ∑_{i} ∥ϕ_{j}u_{i}ω_{δ}(x − y_{i})∥_{L^{p}}^{p}.

This estimate and the previous equality imply

∥u∥_{E_{q}}^{q} ⩽ C^{q/p} ∑_{j} ( ∑_{i} ∥ϕ_{j}u_{i}ω_{δ}(x − y_{i})∥_{L^{p}}^{p})^{q/p}.

On the other hand,

∑_{i} ∥u_{i}ω_{δ}(x − y_{i})∥_{E_{q}}^{q} = ∑_{i} ∑_{j} ∥ϕ_{j}u_{i}ω_{δ}(x − y_{i})∥_{L^{p}}^{q}.

Therefore, to verify Assumption 5.3, it is sufficient to satisfy the estimate

( ∑_{i} ∥ϕ_{j}u_{i}ω_{δ}(x − y_{i})∥_{L^{p}}^{p})^{q/p} ⩽ ∑_{i} ∥ϕ_{j}u_{i}ω_{δ}(x − y_{i})∥_{L^{p}}^{q}.

However, this estimate is satisfied if q ⩽ p (see Lemma 5.4). This completes the proof of Theorem 1.3.

6. SMOOTHNESS OF SOLUTIONS

Denote by C_{B}^{∞}(Ω) the class of infinitely differentiable functions defined on Ω and satisfying the condition

sup_{x∈Ω} |D^{α}u(x)| < ∞

for any multi-index α. Similarly, denote by C_{B}^{∞}(Γ) the class of infinitely differentiable functions with the bounded derivatives on the boundary Γ of Ω.

Consider the elliptic problem (1.1). It is not assumed to be regular.

496 A. VOLPERT, V. VOLPERT

Theorem 6.1. If u ∈ W_{q}^{l,p}(Ω) (where l ⩾ 2m, 1 < p < ∞, and 1 ⩽ q ⩽ ∞) is a solution of problem (1.1) and if f_{0} ∈ C_{B}^{∞}(Ω) and f_{j} ∈ C_{B}^{∞}(Γ) (j = 1, …, m), then u ∈ C_{B}^{∞}(Ω).

Proof. Since W_{q}^{l,p}(Ω) ⊂ W_{∞}^{l,p}(Ω), it suffices to consider the case q = ∞. The following estimate is proved in [20]:

∥u∥_{W_{∞}^{k,p}(Ω)} ⩽ c_{kp}(∥f_{0}∥_{W_{∞}^{k−2m,p}(Ω)} + ∑_{j} ∥f_{j}∥_{W_{∞}^{k−s_{j} −1/p,p}(Γ)} + ∥u∥_{L_{∞}^{p}(Ω)})  (6.1)

for any k ⩾ l (since the coefficients and the boundary of the domain are infinitely differentiable). Since f_{0} ∈ C_{B}^{∞}(Ω) and f_{j} ∈ C_{B}^{∞}(Γ), we see that

∥u∥_{W_{∞}^{k,p}(Ω)} ⩽ M_{kp} ∀k ⩾ l.

Let α be an arbitrary multi-index and k > n/p. Then by the Sobolev embedding theorem, we have

sup_{x∈Ω} |D^{α}v(x)| ⩽ c_{|α|,k,p}∥v∥_{W^{|α|+k,p}(Ω)} ∀v ∈ W^{|α|+k,p}(Ω).

It follows from Definition 1.1 that

∥u∥_{W_{∞}^{|α|+k,p}(Ω)} = sup_{y∈Ω} ∥u(·)ω(· − y)∥_{W^{|α|+k,p}(Ω)}.

Hence,

sup_{x∈Ω} |D_{x}^{α}(u(x)ω(x − y))| ⩽ C_{α}

for any y ∈ Ω, where the constant C_{α} does not depend on y. In particular, if y = x, then

sup_{x∈Ω} |D^{α}u(x)| ⩽ C_{α}.

The theorem is proved.

It follows from this theorem that Conditions NS and NS^{∗} introduced in Subsection 1.4 can be formulated in the following equivalent form.

Condition NS. Any limiting problem

L̂u = 0, x ∈ Ω_{∗}, u ∈ C_{B}^{∞}(Ω_{∗}),

has the zero solution only.

Condition NS^{∗}. Any limiting problem

L̂^{∗}v = 0, x ∈ Ω_{∗}, v ∈ C_{B}^{∞}(Ω_{∗}),

has the zero solution only.

Corollary 6.2. Let Condition NS be satisfied. Then any solution u ∈ W_{q}^{l,p}(Ω) of the problem Lu = 0 belongs to the space E(Ω̄). Here 1 < p < ∞, 1 ⩽ q ⩽ ∞, and E(Ω̄) is introduced in Subsection 1.5.

The proof follows from Corollary 2.3 if we use the estimates in the proof of Theorem 6.1.

RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS Vol. 11 No. 4 2004

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