Principal eigenvalue of the fractional Laplacian with a large incompressible drift

We add a divergence-free drift with increasing magnitude to the fractional Laplacian on a bounded smooth domain, and discuss the behavior of the principal eigenvalue for the Dirichlet problem. The eigenvalue remains bounded if and only if the drift has non-trivial first integrals in the domain of the quadratic form of the fractional Laplacian.


Introduction
This article is motivated by the following result of Berestycki, et al. given in [4] for the Laplacian perturbed by a divergence-free drift in dimensions d ≥ 2. Let D ⊂ R d be a bounded C 2 regular open set and let b(x) = (b 1 (x), . . . , b d (x)) : R d → R d be a bounded d-dimensional vector field such that div b = 0 on D in the sense of distributions (distr.), i.e.
For A ∈ R, let (φ A , λ A ) be the principal eigen-pair corresponding to the Dirichlet problem for the operator ∆ + Ab(x) · ∇. Theorem 0.3 of [4] asserts that λ A remains bounded as A → +∞, if and only if the equation div (wb) = 0 (distr.) on D (1.2) has a solution w (called a first integral of b), such that w = 0 and w ∈ H 1 0 (D). The result can be interpreted intuitively in the following way: functions w satisfying (1.2) are constant along the flow of the vector field Ab(x) (see Section 5.2), and the existence of (non-trivial) first integrals allows for flow lines that are contained in D. On the other hand, if no such w exist, then the flow leaves D with speed proportional to A. Adding the Laplacian ∆ to b · ∇, or equivalently the Brownian motion to the flow, results in a stochastic process whose trajectories gradually depart from the integral curves of b, but the general picture is similar: if nontrivial first integrals exist, then the trajectories may remain in D with positive probability during a finite time interval, even as A → +∞. In this case we are lead to a nontrivial limiting transition mechanism between the flow lines. The result described in the foregoing enjoys many extensions and has proved quite useful in various applications describing the influence of a fluid flow on a diffusion, see for example [5,16,20,39]. In the context of a compact, connected Riemannian manifold a sufficient and necessary condition for λ A to remain bounded, as A → +∞, expressed in terms of the eigenspaces of the advection operator b(x) · ∇, has been given in [21,Theorem 1].
The purpose of the present paper is to verify that a similar property of the principal eigenvalue holds when the classical Laplacian is replaced by the fractional Laplacian ∆ α/2 with α ∈ (1, 2). We consider I α 0 defined as the set of all the nonzero first integrals in the Sobolev space H α/2 0 (D) equipped with the norm coming from the Dirichlet form E α of ∆ α/2 (see (2.16) below). The Sobolev norm condition on the first integrals reflects smoothing properties of the Green function of the fractional Laplacian, while (1.2) is related to the flow defined by b.
The main difficulty in our development stems from roughness of general elements of H α/2 0 (D) and non-locality of ∆ α/2 , which prevent us from a direct application of the differential calculus in the way it has been done in [4]. Instead, we use conditioning suggested by a paper of Bogdan and Dyda [9], approximation techniques for flows given by DiPerna and Lions in [17], and the properties of the Green function and heat kernel of gradient perturbations of ∆ α/2 obtained by Bogdan, Jakubowski in [11] and Chen, et al. in [13] for α ∈ (1, 2) and bounded C 1,1 -regular open sets D. These properties allow to define and study, via the classical Krein-Rutman theorem and compactness arguments, the principal eigen-pair (λ A , φ A ) for L A = ∆ α/2 + Ab · ∇ and α ∈ (1, 2). Our main result can be stated as follows.

3)
and the infimum is attained. Here we use the convention that inf ∅ = +∞, hence lim A→+∞ λ A = +∞ if and only if the zero function is the only first integral.

Equality (1.3) results from the following lower and upper bounds of λ
The bounds are proved in Sections 5.1 and 5.2, correspondingly. In Section 5.3 we explain that the minimum on the right hand side of (1.3) is attained, and we finish the proof of the theorem. Comparing our approach with the arguments used in the case of local operators, cf. [4,21], we note that the use of the Green function seems more robust whenever we lack sufficient differentiability of functions appearing in variational formulas. Recall that in the present case we need to deal with H α/2 0 (D), which limits the applicability of the arguments based on the usual differentiation rules of the classical calculus, e.g. the Leibnitz formula or the chain rule. We consider the use of the Green function as one of the major features of our approach. In addition, the non-locality of the quadratic forms forces a substantial modifications of several other arguments, e.g. those involving conditioning of nonlocal operators and quadratic forms in the proof of the upper bound (1.5) in Section 5.2. Finally, we stress the fact that the Dirichlet fractional Laplacian on a bounded domain D is not a fractional power of the Dirichlet Laplacian on D, e.g. the eigenfunctions of these operators have a different power-type decay at the boundary, see [31,3,35] in this connection.
As a preparation for the proof, we recall in Section 2 the estimates of [11,13] for the Green function and transition density of L A for the Dirichlet problem on D. These functions are defined using Hunt's formula (2.31), which in principle requires the drift b(x) to be defined on the entire R d . We show however, in Corollary 3.9, that they are determined by the restriction of the drift to the domain D. In Section 4 we prove that the corresponding Green's and transition operators are compact, see Lemmas 4.1 and 4.2. This result is used to define the principal eigen-pair of L A , via the Krein-Rutman theorem. In Theorem 3.6 of Section 3 we prove that the domains of ∆ α/2 and L A in L 2 (D) coincide. In Section 5 we employ the bilinear form of L A to estimate the principal eigenvalue. The technical assumption ∇b ∈ L 2d/(d+α) (D) is only needed in Sections 5.2 to characterize the first integrals of b by means of the theory of flows developed by DiPerna and Lions in [17] for Sobolev-regular vector fields.

Generalities
We start with a brief description of the setting and recapitulation of some of the results of [11,13]. Further details and references may be found in those papers (see also [8,10,26] and the references therein). In what follows, R d is the Euclidean space of dimension d ≥ 2, scalar product x · y, norm |x| and Lebesgue measure dx. All sets, measures and functions in R d considered throughout this paper will be Borel. We denote by B(x, r) = {y ∈ R d : |x − y| < r}, the ball of center x ∈ R d and radius r > 0. We will consider nonempty, bounded open set D ⊂ R d , whose boundary is of class C 1,1 . The latter means that r > 0 exists such that for every Q ∈ ∂D there are balls B(x ′ , r) ⊂ D and B(x ′′ , r) ⊂ R d \ D, which are tangent at Q (the inner and outer tangent ball, respectively). We will refer to such sets D as to C 1,1 domains, without requiring connectivity. For an alternative analytic description and localization of C 1,1 domains we refer to [10,Lemma 1]. We note that each connected component of D contains a ball of radius r, and the same is true for D c . Therefore D and D c have a finite number of components, which will play a role in a later discussion of extensions of the vector field to a neighborhood of D. The distance of a given x ∈ R d to D c will be denoted by Constants mean positive numbers, that do not depend on the considered arguments of the functions being compared. Accordingly, notation f (x) ≈ g(x) means that there is a constant C such that We will employ the function space L 2 (D), consisting of all square integrable real valued functions, with the usual scalar product Generally, given 1 ≤ p ≤ ∞, the norms in L p (D) shall be denoted by · p . Customarily, C ∞ c (D) denotes the space of smooth functions on R d with compact support in D. Also, In the last equality we have used Plancherel theorem. The Fourier transform of f is given bŷ 2.2 Isotropic α-stable Lévy process on R d The discussion in Section 2.2 and Section 2.3 is valid for any α ∈ (0, 2). Let The coefficient is chosen in such a way that We define the fractional Laplacian as the L 2 (R d )-closure of the operator Its Fourier symbol is given by ∆ α/2 φ(ξ) = −|ξ| αφ (ξ), cf (2.2). The fractional Laplacian is the generator of the semigroup of the isotropic α-stable Lévy process (Y t , P x ) on R d . Here P x and E x are the law and expectation for the process starting at x ∈ R d . These are defined on the Borel σ-algebra of the canonical cádlág path space D([0, +∞); R d ) via transition probability densities as follows. We let (Y t ) be the canonical process, i.e.

Stable process killed off D
Let τ D = inf{t > 0 : Y t / ∈ D} be the time of the first exit of the (canonical) process from D. For each (t, x) ∈ (0, +∞) × D, the measure is absolutely continuous with respect to the Lebesgue measure on D. Its density p D (t, x, y) is continuous in (t, x, y) ∈ (0, +∞) × D 2 and satisfies G. Hunt's formula (see [30], [10]), (2.7) In addition, the kernel is symmetric (see [6,10] for discussion and references): and defines a strongly continuous semigroup on L 2 (D), We shall denote P t = P R d t . The Green function of ∆ α/2 for D is defined as x, y)dt, and the respective Green operator is We have ( [11]), and The following estimate has been proved by Kulczycki [30] and Chen and Song [14] (see also [24,Theorem 21]), In particular, From (2.12) it also follows that From [12,Corollary 3.3] we have the following gradient estimate, (2.14) We define H α/2 = H α/2 (R d ) as the subspace of L 2 (R d ) made of those elements for which Here (the Dirichlet form) E α is given as follows (cf [22]): The above formula can be also used to define a bilinear form E α (·, ·) on H α/2 × H α/2 by polarization. We also have .
and g belongs to the domain of the fractional Laplacian, then Throughout the remainder of the paper we always assume that 1 < α < 2, and b : R d → R d is a bounded vector field. For t > 0 and x, y ∈ R d we let and for each n ≥ 1 It follows from [26, Theorem 2 and Example 2] that series (2.21) converges uniformly on compact subsets of (0, +∞) × (R d ) 2 . From the results of [10], we know thatp(t, x, y) is a transition probability density function, i.e. it is non-negative and R dp In addition,p is continuous on (0, +∞) × (R d ) 2 , and where c T → 1 if T → 0. In fact, this holds under much weaker, Kato-type condition on b, see [10, Theorems 1 and 2]. We denote byP x andẼ x the law and expectation on D([0, +∞); R d ) for the (canonical) Markov process starting at x and defined by the transition probability densityp, Remark 1.P x may also be defined by solving stochastic differential equation dX t = dY t +b(X t )dt. Such equations have been studied in dimension 1 in [37] under the assumptions of boundedness and continuity of the vector field; also for α = 1. We refer the reader to [34, formula (13)], for a closer description of a connection to (2.21) and (2.20).

Remark 2.
We may also define the perturbation series for −b(x). In what follows, objects pertaining to −b will be marked with the superscript hash ( # ), e.g.p # (t, x, y) = ∞ n=0 (−1) n p n (t, x, y).

Antisymmetry of the perturbation
If also g ∈ C ∞ c (D) and we substitute f + g for f in (2.24), then we obtain The last equality extends to arbitrary f, g ∈ H 1 0 (D).
[?]. Using formula (2.26) n times in space, and then integrating in time we see that p n (t, x, y) = (−1) n p n (t, y, x), (2.29) which yields the identities stated in the lemma.
Remark 3. A strengthening of Proposition 2.1 will be given in Corollary 3.11 below.

Gradient perturbations with Dirichlet conditions
We recall that D is a bounded C 1,1 domain in R d . Hunt's formula may be used to define the transition probability density of the (first) perturbed and (then) killed process [11]. Thus, for t > 0, x, y ∈ D 2 we letp We haveẼ where sup t∈[0,T ] c t < +∞ for any T > 0. By [11, formula (40)], there exist constants c, C > 0 such that We define the Green function of D for L: The main result of [11] asserts that andG D (x, y) is continuous for x = y (for estimates ofp D see [13]). We consider the integral operators In light of (2.32) and (2.36), the above operators are bounded on every , turn out to be mutually adjoint on L 2 (D), as follows from Corollary 3.11 below. From (2.22), (2.30) and (2.31) we obtain that (2. 38) It has been shown in [11] that the following crucial recursive formula holds and for all ϕ ∈ C ∞ c (D) and x ∈ D we have Later on we shall also consider the operator corresponding to the vector field Ab, where A ∈ R and we let A → ∞. Clearly, if A is fixed, then there is no loss of generality to focus on L = L 1 .

Comparison of the domains of generators
The following pointwise version of (2.39) is proved in [11,Lemma 12], We define After a series of auxiliary estimates, we will prove H to be compact on L 2 (D).
Proof. Using (2.14) we obtain that which we bound from above, thanks to (2.12), by and this yields (3.4).
Proof. We only need to examine points close to ∂D. Given such a point we consider the integral over its neighborhood O. The neighborhood can be chosen in such a way that, after a bi-Lipschitz change of variables (see [11, formula (75)], [12, formula (11)]), we can reduce our consideration to the case when

Remark 5. The result is valid for all bounded open
Lipschitz sets ( [12]) in all dimensions d ∈ N.
The above result shall be used to establish that operator K (thus also H) is L 2 bounded via Schur's test, see [23,Theorem 5.2]. Note that we have pK ≤ cq, for some constant c > 0, provided that Summarizing, we will require the following conditions: Proposition 3.5. Operator H is compact on L 2 (D).
In view of Proposition3.5 we may regard the gradient operator b · ∇ as a small perturbation of ∆ α/2 when α ∈ (1, 2). In fact b · ∇ is relatively compact with respect to ∆ α/2 with Dirichlet conditions in the sense of [27, IV.1.3]. For future reference we remark that I − H * is invertible, too.
for any bounded function g. Invoking the argument used in Proposition 3.5, we conclude that there is a number c independent of g for which ∇G D g 2 ≤ c g 2 . (3.14) By approximation, (3.13) and (3.14) extend to all g ∈ L 2 (D). Furthermore, if {g j } ⊂ C ∞ c (D) and lim j→+∞ g j = g in L 2 (D), then lim j→+∞ |G D g − G D g j | 1 = 0, cf. (2.1). Each G D g j is continuous on D and G D g j = 0 on ∂D, hence G D g j ∈ H 1 0 (D) (see [1,Theorem 5.37]). Therefore, G D g ∈ H 1 0 (D). The following result justifies our notation L = ∆ α/2 + b · ∇. Proof. Let f ∈ D(L) and h = b · ∇f . ApplyingG D to ∆ α/2 f + h and using (2.39), we obtaiñ which, thanks to (2.38), concludes the proof. We shall also observe the following localization principle for our perturbation problem.  D as operators on L 2 (D), hence also as functions defined in Section 2.6. A similar conclusion forP D t follows from the fact thatG −1 D is the generator of the semigroup. We will identify the adjoint operator of H on L 2 (D). Proof. If f, g ∈ L 2 (D), then by Corollary 3.7 and (2.25), Functions {G D g, g ∈ L 2 (D)} form a dense set in L 2 (D), which ends the proof.
When definingP D t , via (2.31), we may encounter the situation when b is given only on D. We may extend the field outside D by letting e.g. b = 0 on D c . Of course such an extension needs not satisfy divb = 0, even though the condition may hold on D. However, we still have a local analogue of Proposition 2.1. Recall that the transition probability densities and Green function corresponding to the vector field −b are marked with a hash ( # ). Proof. We shall first prove (3.17), or, equivalently, that whereG * D is adjoint toG D on L 2 (D). From(2.38) applied toG # D we have, Applying the operator G D from the left to both sides of the equality we obtaiñ By Lemma 3.10, Taking adjoints of both sides of (3.12) we also obtain, We have already noted in the proof of Theorem 3.6 that I − H * is a linear automorphism of L 2 (D). Therefore (3.20) and (3.21) give (3.18). Furthermore, let L * denote the adjoint of L on L 2 (D). We have where the last equality follows from [18, Lemma XII.1.6]. Let (P D * t ) be the semigroup adjoint to (P D t ). By [33,Corollary 4.3.7], the generator of (P D * t ) is L * . Since L # = L * , the semigroups are equal. The corresponding kernels are defined pointwise, therefore they satisfy (3.16).

Krein-Rutman eigen-pair
Proof. Let N > 0, andG y) is compact. Indeed, it has a finite Hilbert-Schmidt norm: The norm ofG D −G (N ) D on L 2 (D) may be directly estimated as follows, see, e.g., Theorem 3, p. 176 of [32]. We let N → ∞. The functionsG D (x, ·) (andG D (·, y)) are uniformly integrable on D by (2.35). SinceG D is approximated in the norm topology by compact operators, it is compact.
In the special case when b(x) ≡ 0, (P D t ) equals (P D t ), a symmetric contraction semigroup on L 2 (D), whence the Green operator G D is symmetric, compact and positive definite. The spectral theorem yields the following. Corollary 4.3. G β D is symmetric and compact for every β > 0. By (2.36) and (2.12),G D is also irreducible. Krein-Rutman theorem (see [29]) implies that there exists a unique nonnegative φ ∈ L 2 (D) and a number λ > 0 such that φ 2 = 1 and We shall call (λ, φ) the principal eigenpair corresponding to L. From (2.38) we have φ ∈ D(L) and   To estimateG (n) D , we use basic properties of the Bessel potentials, which can be found in [2, Ch.II. §4]. Recall that for α > 0 the Bessel potential kernel G α is the unique, extended-continuous, probability density function on R d , whose Fourier transform iŝ Thus, G α * G β = G α+β for all α, β > 0, see (4.7) of ibid. If α < d, then by [2, (4.2)], G α (x) is locally comparable with |x| α−d . By (2.35) there is a constant c > 0 such that [2, (4.2)] again. Considering such n we conclude that φ is bounded. The boundary decay of φ follows from (4.2), (2.36) and (2.13). The continuity of φ is a consequence of the continuity ofG D (x, y) for y = x, and the uniform integrability of the kernel, which stems from (2.35).

Proof of Theorem 1.1
and w is not equal to 0 a.e. Recall that for A ∈ R, the operator L A = ∆ α/2 + Ab · ∇ is considered with the Dirichlet exterior condition on D, i.e. it acts on G D (L 2 (D)), see Theorem 3.6. The Green operator and Krein-Rutman eigen-pair of L A shall be denoted byG A and (λ A , φ A ), respectively. We also recall that φ A ∈ D(L A ) and The proof of (1.3) shall be obtained by demonstration of lower and upper bounds for λ A (as A → ∞).

Proof of the lower bound (1.4)
Proof. Let f ∈ D(L A ). According to Proposition 3.8, f belongs to D(∆ α/2 ) and H 1 0 (D). In addition, L A f = ∆ α/2 f + Ab · ∇f . Taking the scalar product of both sides of the equality against f and using (2.25) we get the result, because the second term vanishes.
According to Proposition 5.1, Suppose A n → +∞, as n → ∞, but λ An stay bounded. By (5.3) and Corollary 4.3, the sequence Suppose that w is a weak limit of φ An in H α/2 0 (D), thus a strong limit in L 2 (D). We have w 2 = 1, and for ψ ∈ C ∞ c (D), Dividing both sides by A n and passing to the limit, we obtain that D wb · ∇ψdx = 0, thus w ∈ I α . Fatou's lemma and (5.3) yield lim inf n→+∞ λ An ≥ E α (w, w), therefore (1.4) follows.

The proof of the upper bound (1.5)
The proof of (1.5) uses "conditioning" of truncations of w 2 by the principal eigenfunction inspired by [4] and [9] (see (5.21) below). Here w is a first integral in H α/2 0 (D). An important part of the procedure is to prove that the truncation of w 2 is also a first integral in H α/2 0 (D). This is true if w ∈ H 1 0 (D): suppose that f : R → R and f ′ are bounded and f (0) = 0. Let div b = 0 and that div(wb) = b · ∇w = 0 a.e. Then, a.e. we have thus f (w) is also a first integral of b. However, the a.e. differentiability of w ∈ H α/2 0 (D) is not guaranteed for α < 2. In fact, for w ∈ H

Flows corresponding to Sobolev regular drifts
Unless stated otherwise, in this section we consider general b : R d → R d such that b ∈ W 1,1 loc (R d ) and div b = 0 a.e. on R d . According to [17,Theorem III.1], there exists a unique a.e. defined jointly Borelian family of mappings X(·, x) : R → R d (flow generated by b) with the following properties: first, for a.e.
x ∈ R d , the function R ∋ t → b(X(t, x)) is continuous and so that, in particular, X(0, x) = x and X(t, X(s, x)) = X(t + s, x), s, t ∈ R, (5.6) second, for all t ∈ R and Borel measurable sets A ⊂ R d , We let p ∈ [1, +∞], u 0 ∈ L p (R d ), and define Note that, (t, u 0 (·)) → u(t, ·), t ∈ R, defines a group of isometries on L p (R d ). If for a.e. x ∈ R d and all t ∈ R, because div b = 0 implies b · ∇v = div(vb). Then for all t ∈ R, Let g ∈ C ∞ c (B(0, 1)), g ≥ 0 and R d g dx = 1 (a mollifier). Define The function is an approximate solution of the transport equation 13) in the following sense: if  ). A word of explanation may be helpful. Suppose that b ∈ W 1,q loc (R d ), div b = 0 a.e., and w is an L p -integrable first integral, i.e. b · ∇w = 0, as distributions on space-time R × R d , see [17, (13)]. By [17,Corollary II.1], there is a unique solution to the transport equation (5.13) with the initial condition u(0, ·) = w. The equation is understood in the sense of distributions on space-time, too. By [17,Theorem III.1], the solution has the form (t, x) → w(X(t, x)). However, since w is a first integral, the mapping (t, x) → w(x) defines another solution. Thus, by uniqueness, w(X(t, x)) = w(x) a.e. In our case this argument needs to be slightly modified since the first integral is defined only on D and not on the entire R d . In particular, the identity w(X(t, x)) = w(x) is bound to hold only for small times t.
As an immediate consequence of Lemma 5.2 we obtain the following. . We apply f to (5.16), and use Lemma 5.2. In particular, we may consider truncations of w at the level N > 0, Corollary 5.4. If N > 0 and w ∈ I α 0 , then w N ∈ I α 0 .

The upper bound when the drift is defined on entire R d
We shall first prove the upper bound (1.5) under the assumptions that b is bounded and of zero divergence on the whole of R d , and b ∈ W 1,2d/(d+α) loc (R d ).

5.2.3
The upper bound when the drift is defined only on D Suppose that b ∈ W 1,q (D), with q = 2d/(d + α), and div b(x) ≡ 0 on D. Here, as usual, D is a bounded domain with the C 1,1 class boundary. By the discussion in Section 2.1, R d \D has finitely many, say, N +1 connected components. Denote them by O 0 , . . . , O N , and assume that ∞ belongs to the compactification of O 0 . We start with the following extension result.
b has compact support and is smooth outside of D δ := x ∈ R d : dist(x, D) < δ .
Proof. Suppose that w ∈ I α 0 . Sinceb = b on D, we can repeat the proof of the respective part of Lemma 5.2. Namely, keeping the notation from that lemma, for every u 0 ∈ C ∞ c (D) we have κ > 0 such that D w(x)u 0 (X(−t, x))dx = D w(x)u 0 (x)dx, |t| < κ. (5.25) To complete the proof of (5.16) it suffices to conclude that κ > 0 can be so adjusted that D w(x)u 0 (X(−t, x))dx = D u 0 (x)w(X(t, x))dx, |t| < κ. (5.26) This part cannot be guaranteed directly from the definition of the flow, as the extended fieldb needs not be divergence-free. Equality (5.26) holds however, when X(t) is replaced by X ε (t) for a sufficiently small ε > 0. Indeed, by the Liouville theorem, the Jacobian J X ε (t, x) of X ε (t, x) satisfies d dt J X ε (t, x) = div b (ε) (X ε (t, x))J X ε (t, x), J X ε (0, x) = 1, t ∈ R, x ∈ R d .
Since div b (ǫ) = 0 in an open neighborhood ofD we conclude that J X ε (t, x) ≡ 1 on D for (sufficiently small) |t| < κ. Since w and u 0 are supported in D, D u 0 (x)w(X ε (t, x))dx = D w(x)u 0 (X ε (−t, x))dx, |t| < κ. Example 1. We consider the principal eigen-pair, say (λ 0 , φ 0 ), of ∆ α/2 for the unit ball in R d . By rotation invariance of ∆ α/2 and uniqueness, φ 0 is a smooth radial function in the ball. For i, j = 1, . . . , d we take radially symmetric functions h ij (|x| 2 ) ∈ C ∞ c (R d ), such that h ij = −h ji . Let b i = d j=1 ∂ j h ji , i = 1, . . . , d. The vector field b = (b 1 , . . . , b d ) is of zero divergence and tangent to the spheres |x| = r for all r > 0. Indeed, since h As a result b(x) · ∇w(x) = 0 for any C 1 smooth radially symmetric function w(·). Thus, we conclude that φ 0 is the principal eigenfunction of L A for every A. We have λ A ≡ λ 0 , and E α (w, w) attains its infimum, λ 0 , at w = φ 0 , see Section 5.1. In passing we note that the considered limiting eigenproblems are essentially different for different values of α, in accordance with the fact that the "escape rate" x → D c ν(y − x)dy of the isotropic α-stable Lévy processes from D depends on α. We refer the interested reader to [19], [31], [3] for more information on the eigenproblem of ∆ α/2 , see also [15].

Existence of a minimizer
To complete the proof of Theorem 1.1 we only need to explain the attainability of infimum appearing on the right hand side of (1.3). This is done in the following.
Proof. If e * < ∞ is the infimum, then we can choose functions w n in the set given in the statement of the lemma, such that E α (w n , w n ) → e * and, by choosing a subsequence, that w n weakly converge to w * ∈ H α/2 0 (D). Since {w n } is precompact in L 2 (D), we may further assume that w n → w in L 2 (D) and a.e. This implies that w * = 1, w * is a first integral and, by (2.16) and Fatou's lemma, that E α (w * , w * ) ≤ e * . From the definition of e * , we conclude that equality actually occurs.