Fractional Laplacians on ellipsoids

We show explicit formulas for the evaluation of (possibly higher-order) fractional Laplacians of some functions supported on ellipsoids. In particular, we derive the explicit expression of the torsion function and give examples of $s$-harmonic functions. As an application, we infer that the weak maximum principle fails in eccentric ellipsoids for $s\in(1,\sqrt{3}+3/2)$ in any dimension $n\geq 2$. We build a counterexample in terms of the torsion function times a polynomial of degree 2. Using point inversion transformations, it follows that a variety of bounded and unbounded domains do not satisfy positivity preserving properties and we give some examples.


Introduction
The fractional Laplacian (−∆) s , s > 0, is a pseudodifferential operator with Fourier symbol | · | 2s which can be evaluated pointwisely via a hypersingular integral (see (2.1) below). This operator has many applications in mathematical modeling and the set of solutions of boundary value problems involving the fractional Laplacian has a rich and complex mathematical structure, see [6,9,18].
One of the main obstacles in the study of this operator is the difficulty of evaluating explicitly (−∆) s , even on simple functions, see for example [1,3,15,16] and the references therein for some of the few exceptions that are available in the literature. For the same reason, explicit solutions of boundary value problems are rare.
In this paper, we show some explicit formulas for the evaluation of the fractional Laplacian of polynomial-like functions supported in ellipsoids. Our first result concerns the explicit expression of the torsion function of an ellipsoid. Let and H s (R n ) denotes the usual fractional Sobolev space of order s > 0 (see, for example, [4], for standard existence and uniqueness results in this setting). If s = m ∈ N, then H s 0 (Ω) is the usual Sobolev space H m 0 (Ω). Theorem 1.1. Let n ≥ 2, s > 0, A ∈ R n×n be a symmetric positive definite matrix, and let E := {x ∈ R n : Ax · x < 1}.
Then, there is κ = κ(n, s, A) > 0 such that u s : R n → R given by u s (x) := (1 − Ax · x) s + solves pointwisely (−∆) s u s = κ in E, (1.1) and u s is the unique (weak) solution of (1.1) in H s 0 (E). Here f + denotes the positive part of f . The explicit value of κ(n, s, a) can be computed in terms of hypergeometric functions 2 F 1 (see (2.8), (2.4), and (3.11)). In particular, for (two-dimensional) ellipses with axes of length 1 ; 1; 1 − a 1 a 2 for a 1 , a 2 > 0, see Remark 3.4. The name torsion function comes from elasticity theory, where u 1 denotes the Prandtl torsion stress function describing the deformation of an elastic body subject to surface forces. The function u 1 also has applications in fluid mechanics (modelling the pressure gradient of a flow in a viscous fluid), see [25] and the references therein. A solution of (1.1) in general domains for any s > 0 is usually also called torsion function, and its explicit expression is often useful for checking inequalities and to formulate or disproof general conjectures (see, for example, [24,25,29]). Theorem 1.1 relies on the following more general result.
Theorem 1.2. Under the assumptions of Theorem 1.1, let j ∈ Z, j ≥ − s − 1, and u s+ j (x) = (1 − Ax · x) s+ j + , x ∈ R n . Then u s+ j solves pointwisely where f j is the polynomial of degree (2 j) + given by
Using this approach, we can also calculate the evaluation of (−∆) s of functions such as x i u s and x 2 i u s (1 .3) for i = 1, . . . , n, see Lemmas 3.6 and 3.7. With a similar strategy one may compute the fractional Laplacian (−∆) s of x k i u s for any k ∈ N (although the length of the expressions increases considerably with k). These formulas are of independent interest since, as mentioned earlier, there are very few examples of explicit computations regarding fractional Laplacians. However, one of our main motivations in studying these expressions is related to the problem of the positivity preserving property (p.p.p., from now on) for higher-order elliptic operators, which we describe next.
We say that the operator (−∆) s satisfies a p.p.p. (in Ω) if u ≥ 0 a.e. in Ω, whenever u ∈ H s 0 (Ω) and (−∆) s u ≥ 0 pointwisely in Ω. (1.4) Property (1.4) is sometimes called weak maximum principle and it holds for general domains if s ∈ (0, 1]. The p.p.p. is one of the cornerstones in the analysis of linear and nonlinear second-order elliptic problems, and it is involved in results regarding existence of solutions, uniqueness, regularity, symmetry, monotonicity, geometry of level sets, etc. Whenever s > 1, the verification of (1.4) is a delicate issue; it can be shown that (1.4) holds for any s > 0 whenever Ω is a ball [2,13] or a halfspace [1]; however, (1.4) does not hold in general. For s > 1, the validity of (1.4) depends strongly on the geometry of Ω, but hitherto there is no way of knowing which domains satisfy (1.4) and which ones do not. The classification of domains satisfying (1.4) is a long-standing open problem in the theory of higher-order elliptic equations, see [19,Section 1.2].
One way of approaching this problem is to find first some examples of domains where (1.4) does not hold, and to try to identify a common nature. In particular, the ellipse is known to be incompatible with the p.p.p. whenever it is eccentric enough. This striking example shows that convexity, smoothness, and symmetry are not properties that guarantee the validity of (1.4). Next we include a list of references concerned with ellipses and the absence of a p.p.p.: i) The first available result dates back to [17] for the bilaplacian s = 2 in dimension n = 2, where it is shown that an ellipse with axes ratio 5/3 does not satisfy (1.4). Later, in [23], it is mentioned a ratio of about 1.17 is enough.
ii) In [27] a machinery is designed to extend the two-dimensional examples to higher dimensions. We remark that this approach strongly relies on a separation of variables that is not available for the fractional Laplacian (2.1).
iii) For s = n = 2, [32] builds an explicit and elementary example: an ellipse with axes ratio equal to 5; the explicit sign-changing solution is a polynomial of degree 7.
iv) A thorough analysis for s = n = 2 is performed in [28], finding a counterexample in terms of a polynomial of degree 6 in an ellipse with axes ratio equal to √ 19 ≈ 4.359. The authors also show that it is not possible to construct a counterexample in an ellipse with polynomials of degree less than 6; moreover, it is also shown that counterexamples with degree 6 polynomials are only possible if the axes ratio is larger than ≈ 4.352 (this threshold also appears in our analysis, see Section 4.1).
v) The first example for s = 3 and n = 2 was given in [33]: in this case, the ellipse has an axes ratio equal to 12 and the explicit sign-changing solution is a polynomial of degree 8.
vi) Finally, [34] suggests that, for s = 4 and the same ellipse as in [33], it is possible to find an explicit nodal solution which is a polynomial of degree 12.
Other domains where a general p.p.p. fails are some domains with corners [10] (in particular squares), cones [26], domains with holes [20], elongated rectangles [14], some large cylindrical domains [22], and some limaçons and cardioids [12]. For a survey on this subject for the bilaplacian in the context of the "Boggio-Hadamard conjecture", we refer to [19,Section 1.2] and the references therein.
All the techniques mentioned above are either incompatible or very hard to extend to the fractional setting s ∈ (0, ∞)\N, this case requires new ideas. Nevertheless, we believe that the study of p.p.p. in the fractional regime is relevant, since it offers a novel perspective on the subject using the continuity of the solution mapping, see [24].
For fractional powers there is only one known counterexample to (1.4), given in [4] (see also [2, Theorem 1.11]), where it is shown that, for s ∈ (k, k + 1) with k a positive odd integer, two disjoint balls and dumbbell shaped domains do not satisfy p.p.p.
In the following, we show that, using our explicit computations in ellipsoids, we can construct counterexamples to (1.4) in any dimension n ≥ 2 and for s ∈ (1, √ 3 + 3/2), where √ 3 + 3/2 ≈ 3.232. We follow the ideas from the above mentioned paper [32], where a counterexample in ellipses is built in terms of an explicit polynomial. For n ≥ 2, let (1.5) For functions in H s 0 (E a ) and s > 1, the fractional Laplacian can be evaluated via the hypersingular integral (2.1), but it can also be evaluated as a composition of operators (see [5,Corollary 1.4]), namely, We emphasize that the order of the differential operators cannot be interchanged freely in the context of boundary value problems. For more details, see [5,30]. Theorem 1.3. Let n ≥ 2 and s ∈ (1, 2). There are a 0 = a 0 (s, n) > 1 and ε 0 = ε 0 (s, n) ∈ (0, 1) such that, for every a > a 0 and ε ∈ (0, ε 0 ), the function U ε : R n → R given by belongs to H s 0 (E a ), it changes sign in E a , and (−∆) s U ε > 0 in E a .
For larger values of s one can still construct a counterexample, but the shape of U ε is slightly more involved.
Theorem 1.4. Let n ≥ 2 and s ∈ (1, √ 3 + 3/2). There are constants a 0 > 1, ε 0 ∈ (0, 1), γ ≥ 0, and δ ≥ 0, depending only on s and n, such that, for every a > a 0 and ε ∈ (0, ε 0 ), the function U ε : R n → R given by belongs to H s 0 (E a ), U ε changes sign in E a , and (−∆) s U ε > 0 in E a . We emphasize that Theorem 1.4 is the first counterexample to (1.4) in the range s ∈ (2, 3). In contrast to the results in [32] and [33] which rely on explicit computations of polynomials that can be verified quickly with a computer, the fractional case is much more complex, even with the explicit form of the fractional Laplacian (−∆) s U ε , since these formulas are given in terms of hypergeometric functions which are in general difficult to manipulate. To overcome this difficulty, we use an asymptotic analysis as the length of one of the axis in the ellipsoid goes to zero; it turns out that a suitable normalization of the hypergeometric functions simplifies in the limit and its asymptotic behavior can be determined with precision (see Lemma A.1). This is enough to guarantee the positivity of (−∆) s U ε for thin enough ellipsoids.
As to the upper bound √ 3 + 3/2 for s in Theorem 1.4, it is a technical limitation of our asymptotic approach involving polynomials of the form (1.6). Surprisingly, for some (relatively) small values of a one can obtain counterexamples for slightly larger s (up to around 3.8), and we explore this fact in Section 4.1, where we do a computer-assisted analysis in two dimensions. We also remark that, as expected, a 0 ↑ ∞ as s ↓ 1, as can be seen in Figure 2.
We believe that counterexamples for any s > 3 can be found in suitable ellipses, but this requires a more involved analysis with polynomials p of degree strictly higher than two, and we do not pursue this here. See the discussion in Section 4.1 and see [34] for a counterexample to the p.p.p. for s = 4 in terms of a polynomial of degree 12.
Via a point inversion transformation, one can use Theorem 1.4 to show that a wide variety of shapes do not satisfy (1.4) either. To be more precise, in [1] (see also [13]) the following result is shown. (1.7) To understand the geometrical meaning of the point inversion transformation σ , see Figure 4. We have the following consequences of Theorems 3.2, 1.4, and Proposition 1.5. Let n ≥ 1, c > 0, ν ∈ R n \∂ E a , and (1.8) Corollary 1.6. Let n ≥ 1, c > 0, a ∈ R n with a i > 0, and ν ∈ R n \∂ E a . Then −ν ∈ Ω(a, c, ν) and, for s > 0, the function is a pointwise solution of (−∆) s w s (x) = k |x + ν| n+2s in Ω(a, c, ν), w s = 0 in R n \ Ω(a, c, ν) (1.10) for some constant k = k(n, s, c, a) > 0.
To see some of the different (bounded and unbounded) domains represented by Ω(a, c, ν) for n = 2 and n = 3, see Appendix B.
The paper is organized as follows. In Section 2 we introduce some of the most relevant notation and important definitions. In Section 3 we show Theorems 1.1 and 1.2 and deduce the explicit formulas regarding functions of the type (1.3) in ellipsoids. Section 4 is devoted to the construction of counterexamples, and contains the proofs of Theorems 1.3 and 1.4, as well as those of Corollaries 1.6 and (1.7).

Notation and definitions 2.1 The higher-order fractional Laplacian
Any positive power s > 0 of the (minus) Laplacian, i.e. (−∆) s , has the same Fourier symbol (see [31,Chapter 5] or [5, Theorem 1.8]) as the following hypersingular integral, is a finite difference of order 2m, and c n,m,s is the positive constant given by c n,m,s := (2.2)

Ellipsoids
Let n ≥ 1, a ∈ R n , a i > 0, and A = diag(a k ) n k=1 a diagonal matrix. Then, for x, y ∈ R n , x , y a := Ax · y and |x| a := x , x a define an equivalent scalar product and norm in R n (note that the converse is also true for any symmetric positive definite matrix A, after a suitable rotation of the axes). Let E a ⊂ R n denote the open unitary ball with respect to the a-norm, i.e., In Section 4 we use a to denote a positive real number, in this case we use the convention given in (1.5).
For β > −1, let the function u β : R n → R be given by We also let where dθ denotes the surface measure of ∂ E a , and These integrals appear frequently in our explicit evaluations. In the particular case a 1 = . . . = a n−1 = 1, the integrals J 0 and J (k) i can be computed explicitly as well as their asymptotic profile as a n ↑ ∞, see Lemma A.1.

Special functions
We use the gamma, beta, and hypergeometric functions in our analysis, see [7, Chapter 6 and Chapter 15] for general properties of these functions. We collect here the definitions and some integral representations.

(Gamma function)
For z > 0 we denote by the beta function. Note that in this case where (q) k is the Pochhammer symbol given by (q) 0 = 1 and (q) Moreover, if c > b > 0, then by using the meromorphic extension of the hypergeometric function we have for z < 1 where δ i, j is the Kronecker delta. In particular, Proof. Let us first notice that, for any β > 0 and x ∈ E a , Identity (3.5) directly gives (3.1). Iterating the same idea, from (3.6) one deduces where c n,m,s is given in (2.2).
Proof. We consider spherical coordinates with respect to the a-norm by writing any z ∈ R n as z = tθ with t > 0 and θ ∈ ∂ E a . This transformation has the Jacobian t n−1 /|Aθ |, since, by the coarea formula (notice that ∇|x| a = Ax/|x| a ), We recall notation (2.3) and write We now focus on the inner integral: recall that Apply the change of variables dτ which amounts to (after a translation in the τ variable) Therefore, by (3.8) and (3.9), . 1 In the notations of [16, Corollary 4], we fix V (x) ≡ 1, l = 0, δ = n = 1, σ = β and ρ = s.
In the next corollaries we collect some consequences of Theorem 3.2. For this let k n,s := 2 2s−1 Γ(n/2 + s) Moreover, for any ∈ N such that s − > −1, it also holds with J 0 as in (2.4).
Remark 3.4 (Torsion function in an ellipse). The torsion function given in Theorem 1.1 can be extended to any ellipse and the constant can be computed explicitly. Let α 1 , α 2 > 0 and E = {x ∈ R 2 : α 1 x 2 1 +α 2 x 2 2 < 1}. For a = α 2 α 1 let τ be given by (3.13). Finally, let τ : E → R be given by since, by Lemma A.1, The case β = s + j with j ∈ N in Theorem 3.2 is particularly useful, and therefore we state it as a corollary.
Corollary 3.5. Let j ∈ N. Then, for x ∈ E a , In the particular cases j = 1, 2, Table 1 hold. Proof. Identity (3.16) simply follows by considering β = s + j in (3.7). In order to deduce the particular cases listed in Table 1, we need to remark that, as one of the arguments in the hypergeometric function is a negative integer, then the hypergeometric function reduces to a polynomial, see (2.7). Such polynomials for j = 1, 2 can be found in Table 2. The calculation of k n,s follows as in the proof of Corollary 3.3.
Proof of Theorem 1.2. As mentioned above, this Theorem follows immediately from Corollary 3.5, since for j ∈ N 0 and v, w ≥ 0 we have Then, for any x ∈ E a , Proof. From Lemma 3.1 and Corollary 3.5 it follows that Then for any x ∈ E a we have Proof. Using (3.3) and (3.4) of Lemma 3.1, we have Using the identities in Table 1, we have In order to compute (3.19), we consider the following differential identities

20)
∂ k x, θ 2 a = 2a k x, θ a θ k , In view of (3.20)-(3.21), equation (3.19) can be rewritten Proof. Using Lemma 3.1, By Table 1 and by suitably adjusting (3.20)-(3.21) to the current situation, we deduce Remark 3.9. Consider a 1 = . . . = a n−1 = 1 and a n = a. In this particular case one has Note also that, in this case, For the sake of clarity we summarize the above in the following table for the particular case n = 2. Table 3: The particular case n = 2, a 1 = 1, a 2 = a, where we simply write J i for J (1) i .

Counterexample to positivity preserving properties in ellipsoids
In the following, we give a counterexample to the positivity preserving property (see (1.4)) of (−∆) s , s > 1, in an ellipsoid E a , where we choose a 1 = . . . = a n−1 = 1, and a n = a > 1 sufficiently large. To this end, we consider where p is a polynomial of degree two such that p − ε is sign-changing for every ε > 0. Note that once we have shown that there is a constant k > 0 such that it follows, by linearity, that for a suitable ε > 0 small the function U ε := (p − ε)u s has a nonnegative fractional Laplacian while the function itself is sign-changing in E a . We begin with a heuristic explanation of the strategy. We choose p(x) = p 2 (x 1 ) + γ p 1 (x 1 ) − δ q(x) for constants γ, δ ≥ 0 to be fixed later and where From Lemmas 3.6, 3.7, and 3.8 it follows that (−∆) s (p 2 u s ) = P 2 (x 1 ) + R 2 (x 2 , . . . , x n ) for some degree 2 polynomials P 2 and R 2 , (−∆) s (p 1 u s ) = P 1 (x 1 ), for some degree 1 polynomial P 1 , for some degree 2 polynomials Q and R 0 .
To achieve (4.2) we then need, in particular, that δ satisfies The choice of γ is far more delicate, but from a geometric point of view it can be made intuitively optimal: indeed, in the worst case scenario, the polynomial P 2 (x 1 ) has two real roots P 2,− < P 2,+ < 1, while P 1 (x 1 ) always has one P 1,+ . In this case, it holds that P 2,+ and P 1,+ are both of the order But then, if we aim at having P 2 (x 1 ) + γP 1 (x 1 ) > 0 in (−1, 1), it is enough to verify (see Figure 1) and consequently choose γ = − P 2 (P 2,+ ) P 1 . (4.5) (noticing that the derivative of P 1 is a negative constant): with this choice of γ, we will have P 2 (x 1 ) ≥ −γP 1 (x 1 ) in (−1, 1) by convexity. By taking δ > 0 such that (4.3) is satisfied, and replacing P 2 (x 1 ) with P 2 (x 1 ) + δ Q(x 1 ), the range of possible choices of s so that (4.4) is satisfied can even be enlarged.
The conditions that need to be verified in this argument and their compatibility (on top of an asymptotic analysis as a ↑ ∞) is basically the technical reason why the strategy stops working at finite s: nevertheless we expect that increasing the degrees of the involved polynomials could give some more flexibility in the computations, resulting in a wider range for s. Figure 1: A choice of γ that implies P 2,δ + γP 1 > 0. Theorem 1.4 follows directly from the next result.
Then, for every s ∈ (1, √ 3 + 3/2), there are γ, δ ≥ 0, and a 0 > 1 such that the following holds: for every a ≥ a 0 there is K > 0 such that In particular, for every a ≥ a 0 there is ε > 0 such that the function U ε = (p − ε)u s ∈ H s 0 (E a ) satisfies Proof of Theorem 4.1. In the following, we perform an asymptotic analysis letting a ↑ ∞. To this end, let us first recall (2.4) and (2.5). By Lemma A.1, we have (4.7) Moreover, lim = 0 for all s > 1. Let  We denote by so that, for x ∈ E a , we have We first note that the discriminant of P 2,δ is given by 2 . If s ∈ (1, 3/2] and δ = 0, then so that there is a 0 > 0 such that P 2,0 is positive for all a ≥ a 0 . On the other hand, if s ∈ (3/2, 2) and δ = 0, then, using (4.7) and the notation f ∼ g (for a ↑ ∞) whenever lim a↑∞ f (a) g(a) = 1, The claim in the case s ∈ (1, 2) hence follows by choosing δ = γ = 0, noting that for δ = 0 we have 1 by (A.2) in Lemma A.1 and the definition of the hypergeometric function. In the following we assume s ≥ 2. Moreover, we assume that δ is such that this is asymptotically satisfied as a ↑ ∞ if sδ < 1.
For the positivity of Q δ we first consider identities from which we deduce that Q δ ≥ 0 for a sufficiently large if In view of the last inequality, we choose ; (4.11) remark how this choice for δ also fulfills (4.10) for a large. Note that, in view of (4.10), the largest root of P 2,δ is given by provided 2 B 2 ≥ AC. We remark that 3 as a ↑ ∞.
The root of P 1 is given by As explained above, with γ as in (4.5) we have P 2,δ + γP 1 > 0 in [−1, 1], if (and only if) we can find δ such that P 2,+ < P 1,+ , (4.14) where the strict inequality is needed due to the asymptotic analysis. This inequality is moreover equivalent to Asymptotically, this is satisfied if and only if which is equivalent to i.e., As the condition aδ < 1/(2s − 1) is already implied by (4.11), we are left to verify what values of s allow for a non-empty range of δ as resulting from (4.11) and (4.15): these are those values that satisfy which in particular holds for s ∈ [2, √ 3 + 3/2).
Proof of Theorem 1.3. This follows directly from the first part of the proof of Theorem 4.1.

A computer-assisted analysis in two dimensions
Theorem 4.1 shows that the fractional Laplacian (−∆) s does not satisfy a positivity preserving property in the ellipse E a for a large enough. Its proof uses an asymptotic analysis as a ↑ ∞ and constructs an explicit counterexample for any a sufficiently large (a > a 0 for some a 0 > 1) and for s ∈ (1, s 0 ) with s 0 := √ 3 + 3/2 ≈ 3.232. In this section we fix n = 2 and address the following questions: i) How large is a 0 ?
ii) What can be said for s ≥ s 0 ?
The answer to these questions depends on the explicit calculations developed in Section 3, which involve several hypergeometric functions. These functions can be expressed as a series (2.6) or as an integral (2.8). However, direct calculations using these representations are usually hard to perform; nevertheless, computers are very efficient and precise manipulating and approximating the values of hypergeometric functions, and we use this to answer questions i) and ii). x ∈ R 2 (4. 16) then the value of (−∆) s (pu s ) in E a can be computed explicitly in terms of hypergeometric functions, see Table 3.
In particular, where A, B, and C are given in (4.8). In Figure 2 we  In particular, Figure 2 shows that (4.17) holds for all s ∈ (1, 2) and a > a 0 for some a 0 > 0, as stated in Theorem 4.1, however a 0 ↑ ∞ as s ↓ 1, whereas for s = 3/2 we have a 0 < 115. Note that, if s ↑ 2, then we also have that a 0 ↑ ∞ whenever p has the simple form (4.16); but, by using a more general polynomial p as in (4.6) for suitable δ and γ, one can obtain a counterexample for s larger.

Extended range for counterexamples
If s ≥ s 0 , then the asymptotic analysis in the proof of Theorem 4.1 cannot be successfully implemented. However, one can show that a counterexample can be obtained for some s ≥ s 0 if a is not very large.
To be more precise, let γ be as in (4.5) and let .
Then we can compute numerically that h(11, s) > 0 for s ∈ [3, 3.8456), see Figure 3. Observe also that h(20, 3.8) < 0; in particular, this implies that large values of a are not always optimal to construct a counterexample.
To argue the optimality and the consistency of our approach, we remark that the root of the mapping a → h(a, 2) can be computed numerically, and it is given by b 0 ≈ 18.94281916344395 (see Figure 3), which is the same threshold found in [28,Theorem 5.2], obtained with different arguments than ours in the study of the bilaplacian in two-dimensional ellipses.
Proof of Corollary 1.7. We use the notation from the proof of Corollary 1.6. Assume that Ω is bounded or that n > 4s and let U ε be given by Theorem 1.4. Then, a direct calculation shows that W := K s U ε ∈ L 2 (R n ). Moreover, W is sign changing and, by Proposition 1.5 and Plancherel's Theorem, where W is the Fourier transform of W and P is a polynomial of degree two given by Lemmas 3.6 and 3.7. In particular W ∈ H s 0 (R n ). Arguing as in Corollary 1.6, we obtain that (−∆) s W > 0 pointwisely in Ω.

A Asymptotic behavior of J (k) i
Recall that µ is defined in (2.3) with a diagonal matrix A with entries a 1 = . . . = a n−1 = 1 and a n = a.