NONLOCAL ELLIPTIC PROBLEMS IN INFINITE CYLINDER AND APPLICATIONS

. We consider a unique solvability of nonlocal elliptic problems in inﬁnite cylinder in weighted spaces and in H¨older spaces. Using these results we prove the existence and uniqueness of classical solution for the Vlasov– Poisson equations with nonlocal conditions in inﬁnite cylinder for suﬃciently small initial data.


1.
Introduction. An interest to nonlocal elliptic problems increased considerably during last years. In [5], it was studied the Laplace equation in a bounded domain with nonlocal boundary condition connecting the values of unknown function on a boundary with the values on some manifold inside domain. The solvability of general nonlocal elliptic problems of that type was formulated as an open problem [22], [17]. A solution of this problem and analysis of spectral properties and asymptotic formulas of solutions to nonlocal elliptic problems is presented in [24,25]. For other investigations of nonlocal elliptic problems, see also [10], [11], [13], and [27].
In this paper we consider solvability of nonlocal elliptic problems in infinite cylinder in weighted spaces and in Hölder spaces. An interest to this problem is motivated by its application to a problem of plasma confinement in thermonuclear reactors having the shape of long cylinder ("mirror traps"), see [18], [23]. We note that local boundary value problems for elliptic differential equations in a cylinder were studied in [15], [19], and [20]. Section 2 is devoted to statement of problem and notation. We consider a second order elliptic differential equation in the cylinder Q = G × R, where G ⊂ R n is a bounded domain with boundary ∂G ∈ C ∞ . A solution of this equation u(x, τ ) (x ∈Ḡ, τ ∈ R) satisfies a nonlocal boundary condition connected the values of solution on the boundary ∂Q = ∂G × R with its values on some cylindrical surface from Q.
In Section 3, along with nonlocal elliptic boundary value problem we consider a linear bounded operator L acting in weighted spaces with the weight e βτ , β ∈ R, and exponent p = 2. Using the Fourier transform, we obtain an operator-valued function λ →L(λ) acting in Sobolev spaces. It is shown that, for any h ∈ R, there exists λ 1 > 1 such that, for all λ ∈ {Im λ = h, | Re λ| ≥ λ 1 }, the operatorL(λ) has a bounded inverse R(λ) =L −1 (λ). Moreover, the operator-valued function λ →R(λ) is finitely meromorphic in C. This result together with a priori estimates of solutions of nonlocal elliptic boundary-value problems with a parameter allows to prove that the operator L is an isomorphism of weighted spaces with the weight e βτ and exponent p = 2 if and only if the line {Im λ = β} does not contain eigenvalues of the operator-valued function λ →L(λ).
Section 4 deals with auxiliary results devoted to the Dirichlet problem for a second order elliptic equation in the cylinder Q = G × R. There we prove a unique solvability and a priori estimates of solutions in Hölder spaces.
In Section 5, we generalize results of Section 3 to the case p ≥ 2. We prove that the operator L is an isomorphism of weighted spaces with the weight e βt and exponent p ≥ 2 if and only if the line {Im λ = β} does not contain eigenvalues of the operator-valued function λ →L(λ).
In Section 6, we show that, if the operator-valued functions λ →L 0 (λ) and λ → L(λ) have no real eigenvalues, then a nonlocal elliptic boundary-value problem in the cylinder Q = G×R has a unique solution u ∈ C 2+σ 0 (Q) for all right hand sides of equation and nonlocal boundary conditions from corresponding Hölder spaces. Here λ →L 0 (λ) is the operator-valued function with trivial nonlocal term, C 2+σ 0 (Q) is the closure of set of functions from C 2+σ (Q) with compact supports inQ, 0 < σ < 1.
Section 7 deals with applications. We consider the Vlasov-Poisson equations in infinite cylinder with nonlocal boundary condition for the electric field potential and initial conditions for density distribution functions of charged particles. Applying Theorem 6.3 on a unique solvability of nonlocal elliptic problems in a cylinder in Hölder spaces and Theorem 5.1 in [26] on solvability of abstract Vlasov equations, we prove that there is a unique classical solution of the above problem for sufficiently small initial density distribution functions. Moreover, the supports of density distribution functions belong to some interior cylinder.
Problems of existence for classical solutions and generalized solutions to the Cauchy problem and for generalized solutions to mixed boundary value problems for Vlasov equations were studied in details, see [2], [3], [7], [16] and [21]. The question of existence of classical solutions to mixed problems for Vlasov equations was studied much less. In [22], [16] it was formulated as an open problem. An interest to classical solutions of mixed boundary value problems for Vlasov-Poisson equations is associated with the design of a controlled thermonuclear fusion reactor. One of the devices for thermonuclear fusion is the mirror trap, which has the shape of a long cylinder tapered at the ends, see [18], [23].
The production of a stable high-temperature plasma in a reactor requires that the so-called plasma column be strictly inside the domain during some time interval in order to keep it away from the vacuum container wall. In most of models of thermonuclear fusion reactors an external magnetic field is used as a control ensuring the existence of plasma confinement in the reactor [18], [23]. From mathematical point of view this means that one has to ensure existence of solutions of the Vlasov-Poisson equations for which the supports of the charged-particle density distributions do not intersect the boundary. This can be achieved by the influence of the external magnetic field. However in mathematical investigations devoted to classical solutions of mixed problems for the Vlasov-Poisson equations it was studied the behavior of the trajectories of particles near the boundary with reflection boundary conditions [9], [12]. The effect of the magnetic field was not taken into account. According to [18], [23], the presence of a considerable number of particles on the boundary can result in either destruction of the reactor walls or in cooling the high-temperature plasma due to its contact with the reactor walls. As a distinct from other papers, we consider here the Vlasov-Poisson equations with external magnetic field and study solutions with supports at some distance from the boundary. We also examine a two-component plasma, since the word "plasma" is used in physics to designate a high-temperature state of an ionized gas with charge neutrality [18]. Another distinction, just as in [22], is connected with nonlocal boundary conditions for the potential of electric field.
2. Statement of problem. Some notation.

We consider the equation
with nonlocal boundary condition neighborhood Ω 0 of the boundary ∂G onto ω(Ω 0 ) so that ω(Ω 0 ) ⊂ G, χ ∈ R. If n = 1, then Q = (a, b), x ∈ R is a scalar variable, α ≥ 0 is an integer, We suppose that the operator A(x, D x , D τ ) is uniformly elliptic inQ, i.e., the following condition holds.
In some cases we also assume that the following conditions are fulfilled.
2.2. We introduce some function spaces, which will be necessary later on.
Let Ω ∈ R N be either a domain with boundary ∂Ω ∈ C ∞ , or Ω = R N . We denote by C k (Ω), k ∈ Z, k ≥ 0, the space of continuous functions onΩ, having all continuous derivatives inΩ up to the order k with the norm We denote by C k+σ (Ω), k ∈ Z, k ≥ 0, 0 < σ < 1, the Hölder space of continuous functions onΩ, having all continuous derivatives onΩ up to the order k with the finite norm where Thus we have defined the space C s (Ω) for any s ≥ 0. Similarly one can define the spaces C k (·) and C k+σ (·) on an (N − 1)-dimensional C ∞ -manifold, on a finite cylinder, or on a semi-infinite cylinder. Let C(Ω) = C 0 (Ω).
Remark 2.4. For any k and σ, the spaces C k (Ω) and C k+σ (Ω) are Banach spaces. Moreover, C k (Ω) is a separable space. However the space C k+σ (Ω) is nonseparable, and the set of infinitely differentiable functions onΩ is nondense in C k+σ (Ω) see [4].
Denote by C s 0 (Q) (C s 0 (∂Q)), s ≥ 0, the closure of the set of functions from C s (Q) (C s (∂Q)) with compact supports inQ (∂Q).
The properties of spaces C k (Ω) and C k+σ (Ω) are presented in [28]. We denote by W k p (Ω), k ∈ N, p ≥ 2, the Sobolev space of functions v ∈ L p (Ω) having all generalized derivatives ∂ α x Similarly one can define the spaces W k p (·) on a finite cylinder or on a semi-infinite cylinder. Let M ⊂Ω be an (N − 1)-dimensional C ∞ -manifold. Let W k−1/p p (M ) be the space of traces on M for functions from W k p (Ω) with the norm . For the theory of Sobolev spaces including imbedding theorems, extension theorems, and spaces of traces, see [28].
We introduce the weighted Kondrat'ev space W k p,β (Q) as the completion of the set C ∞ 0 (Q) with respect to the norm where C ∞ 0 (Q) is the set of infinitely differentiable inQ functions with compact supports inQ, 0 ≤ k ∈ Z, β ∈ R, p ≥ 2. If β = 0, then the space W k p,β (Q) coincides with the Sobolev space W k p (Q). We denote by W k−1/p p,β (∂Q), k ≥ 1, the space of traces on ∂Q for functions from W k p,β (Q) with the norm For the first time the theory of weighted spaces W k 2,β (Q) was created in [14] for investigation of elliptic equations in domains with conical or angular points. For further research on the spaces W k p,β (Q), p > 1, and bibliography, see [19]. 3. Nonlocal elliptic problems in W 2 2,β (Q). 3.1. First we consider a nonlocal elliptic problem with a parameter associated with problem (2.1), (2.2).
The left hand side of (3.10) follows from (3.16).
where N (K) and R(K) are the kernel and the range of the operator K, respectively. The index of the Fredholm operator is defined by ind Below we shall prove that the operatorL(λ) :  Definition 3.4. Let H and H be Hilbert spaces. The operator-valued function C λ → R(λ) is said to be finitely meromorphic at λ 0 ∈ C, if for some ε > 0, it can be expanded into the Laurent series converging with respect to operator norm, where R(λ), R j : H → H, j = −r, −r + 1, . . . , are linear bounded operators, and the operators R k , −r ≤ k < 0, are finite dimensional. (b) For every h ∈ R, there exists λ 1 > 1 such that for λ ∈ S h,λ1 the operatorL(λ) has a bounded inverseR(λ) =L −1 (λ) : W 0 2 (G, ∂G) → W 2 2 (G) in the norm ||| · |||.
2. Now we can prove statement (a). Let λ ∈ C be an arbitrary fixed number.
Hence from the compactness of the embedding of W 2 2 (G) into W 1 2 (G) it follows that the operator Φ 1 : N (A) → W for λ ∈ω δ,γ . Hence the setω δ,γ does not contain eigenvalues of the operator-valued function λ →L(λ).

Using
where k 2 > 0 does not depend on f . We have proved that the operator L : W 2 2,β (Q) → W 0 2,β (Q, ∂Q) is an isomorphism. 2. Now we assume that the operator-valued function λ →L(λ) has an eigenvalue λ 0 on the line {λ ∈ C : Im λ = β}. Then we prove that the operator L : W 2 2,β (Q) → W 0 2,β (Q, ∂Q) is not an isomorphism. Assume to the contrary that the operator L is an isomorphism. Hence for any f ∈ W 0 2,β (Q, ∂Q) there exists a unique solution u ∈ W 2 2,β (Q) of problem (3.1), (3.2) and inequality (3.22) holds. Let ϕ 0 (x) be an eigenfunction of the operator-valued function λ →L(λ) corresponding to the eigenvalue λ 0 . We introduce a truncation function ζ ∈ C ∞ (R) SinceL(λ 0 )ϕ 0 = 0, we have Hence the relation |e iλ0τ | = e −βτ implies that Here constants k 3 , . . . , k 6 > 0 do not depend on N . Thus for u = u N , the right hand side of (3.22) is bounded by a constant that does not depend on N .
On the other hand, using again the relation |e iλ0τ | = e −βτ , we have

ALEXANDER L. SKUBACHEVSKII
We obtained a contradiction with inequality (3.22). Thus, if the operator L : is an isomorphism, then the line Im λ = β does not contain eigenvalues of the operator-valued function λ →L(λ).
Remark 3.7. Using part 2 of the proof of Theorem 3.6, one can show that, if the line {λ ∈ C : Im λ = β} contains eigenvalues of the operator-valued function λ →L(λ), then the range of the operator L is not closed in W 0 2,β (Q, ∂Q). Further for investigation of solvability of problem (3.1), (3.2) in Hölder spaces we shall assume that the operator-valued function λ →L(λ) has no real eigenvalues. Let us demonstrate some examples, in which the realization of this assumption will be analyzed.
Example 3.8. We consider the nonlocal elliptic problem in a strip We define the operator L : The operatorL(λ) : In [25,Section 4.1] it was shown that the operator-valued function λ →L(λ) has no real eigenvalues if and only if b 1 +b 2 < 2. In particular, this is true if b 1 = b 2 = 0. Therefore the operator-valued function λ →L 0 (λ) has no real eigenvalues. Example 3.9. We consider nonlocal elliptic boundary value problem (2.1), (2.2) assuming that a 0 (x) ≥ 0 (x ∈Ḡ), a ij , a i , a 0 ∈ C ∞ (R n ) are real-valued functions. We also assume that θ < 1. We shall prove that in this case the operator-valued function λ →L(λ) has no real eigenvalues.
Assume to the contrary that there exists a λ 0 ∈ R and a complex-valued function By virtue of the theorem on interior smoothness of eigenfunctions of elliptic equations, we have u 0 ∈ C ∞ (G). On the other hand, nonlocal boundary condition (3.28) implies that u 0 | ∂G ∈ C ∞ (∂G). Therefore from the theorem on smoothness of generalized solutions of elliptic problems near a boundary and Sobolev imbedding theorem it follows that u 0 ∈ C ∞ (Ḡ). We write the function u 0 (x) in the form u 0 (x) = v 0 (x) + iw 0 (x), where v 0 (x) and w 0 (x) are real-valued functions. Since equation (3.27) has real coefficients, the functions v 0 (x) and w 0 (x) are also the solutions of this equation. Denote Taking into account the equalities A 1 v 0 (x) = 0 and A 1 w 0 (x) = 0 (x ∈ G) and collecting similar terms, we obtain Hence sup On the other hand, by virtue of inequality (3.30) and condition θ < 1, we have This inequality contradicts to nonlocal boundary condition (3.28). Thus we have proved that the operator-valued function λ →L(λ) has no real eigenvalues. Assuming that θ = 0, we obtain the Dirichlet problem for equation (2.1). Therefore the operator-valued function λ →L 0 (λ) has no real eigenvalues.
In Example 3.8 it was shown that in the case when n = 1 violation of condition θ < 1 can lead to the appearance of real eigenvalues for the operator-valued function λ →L(λ). It is easy to make sure that this phenomenon takes place also for n > 1.
We consider the auxiliary problem
Let Q N = {(x, τ ) ∈ Q : |τ | < N }. By virtue of Lemma 6.18 in [8, Chapter 6], we have u ∈ C 2+σ (Q N ) for any N > 0. Hence from Corollary 6.7 in [8,Chapter 6] it follows that for any N > 0 the following inequality is fulfilled where k 1 > 0 does not depend on N and f 0 . From (4.9) and (4.3) it follows that where k 2 > 0 does not depend on N and f 0 . From (4.10) we obtain estimate (4.8).
For a proof of theorem on the necessary and sufficient conditions of a unique solvability to problem (2.1), (2.2) in weighted space W 2 p,β (Q), we need some auxiliary results.
We introduce a partition of unity {κ k } on R subordinated to the covering of R by the intervals (k − 1, k + 1) such that |κ Here c 1 > 0 does not depend on τ and k. Proof. From a priori estimates of solutions of elliptic problems near a boundary and inside domain and estimate of norm for trace of nonlocal term it follows that Lemma 5.2. Let Condition 2.1 hold, and let the line {λ ∈ C : Im λ = β} do not contain eigenvalues of the operator-function λ →L(λ). Then for any function u ∈ W 2 2,β (Q) such that supp Lu ⊂ (m − 1, m + 1) and Lu ∈ W 0 p,β (Q, ∂Q) the following estimate takes place where c 3 , κ > 0 do not depend on k, m ∈ Z and u; p ≥ 2.
In [26], it was proved the following statement.