A fractional Hadamard formula and applications

We consider the domain dependence of the best constant in the subcritical fractional Sobolev constant, $$ \lambda_{s,p}(\Omega):=\inf \left\{ [u]_{H^s(\mathbb{R}^N)}^2,\,\, u\in C^\infty_c(\Omega),\,\, \|u\|_{L^p(\Omega)}=1 \right\}, $$ where $s\in (0,1)$, $\Omega$ is bounded of class $C^{1,1}$ and $p\in [1, \frac{2N}{N-2s})$ if $2s<N$, $p\in [1, \infty)$ if $2s\geq N=1$. Explicitly, we derive formula for the one-sided shape derivative of the mapping $\Omega\mapsto \lambda_{s,p}(\Omega)$ under domain perturbations. In the case where $ \lambda_{s,p}(\Omega)$ admits a unique positive minimizer (e.g. $p=1$ or $p=2$), our result implies a nonlocal version of the classical variational Hadamard formula for the first eigenvalue of the Dirichlet Laplacian on $\Omega$. Thanks to the formula for our one-sided shape derivative, we characterize smooth local minimizers of $\lambda_{s,p}(\Omega)$ under volume-preserving deformations, and we find that they are balls if $p\in \{1\}\cup [2,\infty)$. Finally, we consider the maximization problem for $\lambda_{s,p}(\Omega)$ among annular-shaped domains of fixed volume of the type $B\setminus \overline B'$, where $B$ is a fixed ball and $B'$ is ball whose position is varied within $B$. We prove that, for $p\in \{1,2\}$, the value $\lambda_{s,p}(B\setminus \overline B')$ is maximal when the two balls are concentric.


Introduction
Let s ∈ (0, 1) and Ω ⊂ R N be a bounded open set. The present paper is devoted to the study of best constants λ s,p (Ω) in the family of subcritical Sobolev inequalities As a consequence of the subcriticality assumption on p and the boundedness of Ω, the space H s 0 (Ω) compactly embeds into L p (Ω). Therefore a direct minimization argument shows that λ s,p (Ω) admits a nonnegative minimizer u ∈ H s 0 (Ω) with u L p (Ω) = 1. Moreover, every such minimizer solves, in the weak sense, the semilinear problem (−∆) s u = λ s,p (Ω)u p−1 in Ω, u = 0 in R N \ Ω. (1.4) where (−∆) s stands for the fractional Laplacian. It therefore follows from regularity theory and the strong maximum principle for (−∆) s that u is strictly positive in Ω, see Lemma 2.3 below. We recall that, for functions ϕ ∈ C 1,1 c (R N ), the fractional Laplacian is given by 2ϕ(x) − ϕ(x + y) − ϕ(x − y) |y| N +2s dy. 1 Of particular interest are the cases p = 1 and p = 2 which correspond to the fractional torsion problem (−∆) s u = λ s,1 (Ω) in Ω, u = 0 in R N \ Ω, (1.5) and the eigenvalue problem (−∆) s u = λ s,2 (Ω)u in Ω, u = 0 in R N \ Ω, (1.6) associated with the first Dirichlet eigenvalue of the fractional Laplacian, respectively. In these cases, the minimization problem for λ s,p (Ω) in (1.3) possesses a unique positive minimizer. Indeed, it is a well-known consequence of the fractional maximum principle that (1.5) admits a unique solution, and that (1.6) has a unique positive eigenfunction with u L 2 (Ω) = 1. Incidentally, the uniqueness of positive minimizers extends to the full range 1 ≤ p ≤ 2, as we shall show in Lemma A.1 in the appendix of this paper. Our first goal in this paper is to analyze the dependence of the best constants on the underlying domain Ω. For this we shall derive a formula for a one-sided shape derivative of the map Ω → λ s,p (Ω). We assume from now on that Ω ⊂ R N is a bounded open set of class C 1,1 , and we consider a family of deformations {Φ ε } ε∈(−1,1) with the following properties: Φ ε ∈ C 1,1 (R N ; R N ) for ε ∈ (−1, 1), Φ 0 = id R N , and the map (−1, 1) → C 0,1 (R N , R N ), ε → Φ ε is of class C 2 . (1.7) We note that (1.7) implies that Φ ε : R N → R N is a global diffeomorphism if |ε| is small enough, see e.g. [7,Chapter 4.1]. To clarify, we stress that we only need the C 2 -dependence of Φ ε on ε with respect to Lipschitz-norms, while Φ ε is assumed to be a C 1,1 -function for ε ∈ (−1, 1) to guarantee C 1,1 -regularity of the perturbed domains Φ ε (Ω). From the variational characterization of λ s,p (Ω) it is not difficult to see that the map ε → λ s,p (Φ ε (Ω)) is continuous. However, since λ s,p (Ω) may not have a unique positive minimizer, we cannot expect this map to be differentiable. We therefore rely on determining the right derivative of ε → λ s,p (Φ ε (Ω)) from which we derive differentiability whenever λ s,p (Ω) admits a unique positive minimizer, thereby extending the classical Hadamard shape derivative formula for the first Dirichlet eigenvalue of the Laplacian −∆.
Throughout this paper, we consider a fixed function δ ∈ C 1,1 (R N ) which coincides with the signed distance function dist(·, R N \ Ω) − dist(·, Ω) in a neighborhood of the boundary ∂Ω. We note here that, since we assume that Ω is of class C 1,1 , the signed distance function is also of class C 1,1 in a neighborhood of ∂Ω but not globally on R N . We also suppose that δ is chosen with the property that δ is positive in Ω and negative in R N \ Ω, as it is the case for the signed distance function.
Our first main result is the following.
Here the function u/δ s is defined on ∂Ω as a limit. Namely, for x 0 ∈ ∂Ω, the limit u δ s (x 0 ) = lim x→x 0 x∈Ω u δ s (x) (1.9) exists, as the function u/δ s extends to a function in C α (Ω) for some α > 0, see [20]. In addition, the function δ 1−s ∇u also admits a Hölder continuous extension on Ω satisfying δ 1−s ∇u·ν = su/δ s on ∂Ω, see [8]. As a consequence, the expression u/δ s , restricted on ∂Ω, plays the role of an inner fractional normal derivative. Note that, for s = 1, the limit on the RHS of (1.9) coincides with the classical inner normal derivative of u at x 0 . We observe that the constant Γ(1 + s) 2 appears also in the fractional Pohozaev identity, see e.g. [21]. This is, to some extend, not surprising at least in the classical case since Pohozaev's identity can be obtained using techniques of domain variation, see e.g. [24].
We also remark that one-sided derivatives naturally arise in the analysis of parameter-dependent minimization problems, see e.g. [7,Section 10.2.3] for an abstract result in this direction. Related to this, they also appear in the analysis of the domain dependence of eigenvalue problems with possible degeneracy, see e.g. [11] and the references therein.
A natural consequence of Theorem 1.1 is that the map ε → θ(ε) = λ s,p (Φ ε (Ω)) is differentiable at ε = 0 whenever λ s,p (Ω) admits a unique positive minimizer. Indeed, applying Theorem 1.1 to the map ε → θ(ε) := λ s,p (Φ −ε (Ω)) yields where H is given as in Theorem 1.1. As a consequence, we obtain the following result. Corollary 1.2. Let λ s,p (Ω) be given by (1.3) and consider a family of deformations Φ ε satisfying (1.7). Suppose that λ s,p (Ω) admits a unique positive minimizer u ∈ H s 0 (Ω). Then the map ε → θ(ε) = λ s,p (Φ ε (Ω)) is differentiable at ε = 0. Moreover As mentioned earlier, λ s,p (Ω) admits a unique positive minimizer u ∈ H s 0 (Ω) for 1 ≤ p ≤ 2, see Lemma A.1 in the appendix. Therefore Corollary 1.2 extends, in particular, the classical Hadamard formula, for the first Dirichlet eigenvalue λ 1,2 (Ω) of −∆, to the fractional setting. We recall, see e.g. [15], that the classical Hadamard formula is given by An analogue of Corollary 1.2 for the case of the local r-Laplace operator was obtained in [2,12]. We also point out that, prior to this paper, a Hadamard formula in the fractional setting of the type (1.10) was obtained in [6] for the special case p = 1, s = 1 2 , N = 2 and Ω of class C ∞ . We are not aware of any other previous work related to Theorem 1.1 or 1.2 in the fractional setting.
If an open subset Ω of R N of class C 3 is a volume constrained local minimum for Ω → λ s,p (Ω), then Ω is a ball. Corollary 1.3 is a consequence of Theorem 1.1, from which we derive that if Ω is a constrained local minimum then any element u ∈ H satisfies the overdetermined condition u/δ s ≡ constant on ∂Ω. Therefore by the rigidity result in [9] we find that Ω must be a ball. We point out that we are not able to include the case p ∈ (1, 2) in Corollary 1.3, since the rigidity result in [9] is based on the moving plane method and therefore requires the nonlinearity in (1.4) to be Lipschitz. The case p ∈ (1, 2) therefore remains an open problem in Corollary 1.3. We note that the authors in [6] considered the shape minimization problem for λ s,p (Ω) in the case p = 1, s = 1 2 , N = 2 among domains Ω of class C ∞ of fixed volume. They showed in [6] that such minimizers are discs.
Next we consider the optimization problem of Ω → λ s,p (Ω) for p ∈ {1, 2} and Ω a punctured ball, with the hole having the shape of ball. We show that, as the hole moves in Ω then λ s,p (Ω) is maximal when the two balls are concentric. In the local case s = 1 and N = 2, this is a classical result by Hersch [16]. For subsequent generalizations in the case of the local problem, see [5,14,18]. Theorem 1.4. Let p ∈ {1, 2}, B 1 (0) be the unit centered ball and τ ∈ (0, 1). Define Then the map A → R, a → λ s,p (B 1 (0) \ B τ (a)) takes its maximum at a = 0.
The proof of Theorem 1.4 is inspired by the argument given in [14,18] for the local case s = 1. It uses the fractional Hadamard formula in Corollary 1.2 and maximum principles for anti-symmetric functions. Our proof also shows that the map a → λ s,p (B 1 (0) \ B τ (a)) takes its minimum when the boundary of the ball B τ (a) touches the one of B 1 (0), see Section 5 below.
The proof of Theorem 1.1 is based on the use of test functions in the variational characterization of λ s,p (Ω) and λ s,p (Φ ε (Ω)). The general strategy is inspired by the direct approach in [11], which is related to a Neumann eigenvalue problem on manifolds. In the case of λ s,p (Φ ε (Ω)), it is important to make a change of variables so that λ s,p (Φ ε (Ω)) is determined by minimizing an ε-dependent family of seminorms among functions u ∈ H s 0 (Ω), see Section 2 below. An obvious choice of test functions are minimizers u and v ε for λ s,p (Ω) and λ s,p (Φ ε (Ω)), respectively. However, due to the fact that u is only of class C s up to the boundary, we cannot obtain a boundary integral term directly from the divergence theorem. In particular, the integration by parts formula given in [21,Theorem 1.9] does not apply to general vector fields X which appear in (1.8). Hence, we need to replace u with ζ k u, where ζ k is a cut-off function vanishing in a 1 k -neighborhood of ∂Ω. This leads to upper and lower estimates of λ s,p (Φ ε (Ω)) up to order o(ε), where the first order term is given by an integral involving (−∆) s (ζ k u) and ∇(ζ k u). We refer the reader to Section 4 below for more precise information. A highly nontrivial task is now to pass to the limit as k → ∞ in order to get boundary integrals involving ψ := u/δ s . This is the most difficult part of the paper. We refer to Proposition 2.4 and Section 6 below for more details.
The paper is organized as follows. In Section 2, we provide preliminary results on convergence properties of integral functional, inner approximations of functions in H s 0 (Ω) and on properties of minimizers of (1.3). In Section 3, we introduce notation related to domain deformations and related quantities. In Section 4 we establish a preliminary variant of Theorem 1.1, which is given in Proposition 4.1. In this variant, the constant Γ(1 + s) 2 in (1.8) is replaced by an implicitly given value which still depends on cut-off data. The proofs of the main results, as stated in this introduction, are then completed in Section 5. Finally, Section 6 is devoted to the proof of the main technical ingredient of the paper, which is given by Proposition 2.4.
Acknowledgements: This work is supported by DAAD and BMBF (Germany) within the project 57385104. The authors would like to thank Sven Jarohs for helpful discussions. M.M Fall's work is supported by the Alexander von Humboldt foundation.

Notations and preliminary results
Throughout this section, we fix a bounded open set Ω ⊂ R N . As noted in the introduction, we define the space H s 0 (Ω) as completion of C ∞ c (Ω) with respect to the norm [ · ] s given in (1.2). Then H s 0 (Ω) is a Hilbert space with scalar product where c N,s is given in (1.2). It is well known and easy to see that H s 0 (Ω) coincides with the closure of C ∞ c (Ω) in the standard fractional Sobolev space H s (R N ). Moreover, if Ω has a continuous boundary, then H s 0 (Ω) admits the highly useful characterization Throughout the remainder of this paper, we fix ρ ∈ C ∞ c (−2, 2) with 0 ≤ ρ ≤ 1, ρ ≡ 1 on (−1, 1), and we define ζ ∈ C ∞ (R), Moreover, for k ∈ N, we define the functions We note that the function ρ k is supported in the 2 k -neighborhood of the boundary, while the function ζ k vanishes in the 1 k -neighborhood of the boundary.
Let Ω ⊂ R N be a bounded Lipschitz domain and let u ∈ H s 0 (Ω). Moreover, for k ∈ N, let u k := uζ k ∈ H s 0 (Ω) denote inner approximations of u. Then we have u k → u in H s 0 (Ω). Proof. In the following, the letter C > 0 stands for various constants independent of k. Since ρ k = 1 − ζ k , it suffices to show that Now, since Ω has a Lipschitz boundary, using R N \Ω |x − y| −N −2s dy ∼ δ −2s (x) see e.g [3], we get and therefore Combining (2.5), (2.6) and (2.7), we obtain (2.4), as required.
From now on, we fix a bounded C 1,1 -domain Ω ⊂ R N . We also let C s 0 (Ω) = w ∈ C s (Ω) : w = 0 in R N \ Ω , and we recall the following regularity and positivity properties of nonnegative minimizers for λ s,p (Ω) as defined in (1.3).
Proof. By standard arguments in the calculus of variations, u is a weak solution of (1.4). By [22,Proposition 1.3] we have that u ∈ L ∞ (Ω), and therefore the RHS of (1.4) is a function in L ∞ (Ω). Thus the regularity up to the boundary u ∈ C s 0 (Ω) is proved in [20], where also the C α -bound (2.8) for the function ψ = u δ s is established for some α > 0. Moreover, (2.9) is proved in [8]. It also follows from (1.4), the strong maximum principle and the Hopf lemma for the fractional Laplacian that ψ is a strictly positive function on Ω. In particular, u > 0 in Ω, Therefore u ∈ C ∞ loc (Ω) follows by interior regularity theory (see e.g. [19]) and the fact that the function The computation of one-sided shape derivatives as given in Theorem 1.1 will be carried out in Section 4, and it requires the following key technical proposition. Since its proof is long and quite involved, we postpone the proof to Section 6 below.
, and assume that ψ := u δ s extends to a function on Ω satisfying (2.8) and (2.9). Moreover, put and ζ given in (2.2), and where we use the notation The minus sign in the definition of the constant κ s in (2.10) might appear a bit strange at first glance. We shall see later that, defined in this way, κ s has a positive value. A priori it is not clear that the value of κ s does not depend on the particular choice of the function ζ. This follows a posteriori once we have established in Proposition 4.1 below that this constant appears in Theorem 1.1. This will then allow us to show that κ s = Γ(1+s) 2 2 by applying the resulting shape derivative formula to a one-parameter family of concentric balls, see Section 5 below. A more direct, but somewhat lengthy computation of κ s is possible via the logarithmic Laplacian, which has been introduced in [4].

Domain perturbation and the associated variational problem
Here and in the following, we define Ω ε := Φ ε (Ω). In order to study the dependence of λ s,p (Ω ε ) on ε, it is convenient to pull back the problem on the fixed domain Ω via a change of variables. For this we let Jac Φε denote the Jacobian determinant of the map Φ ε ∈ C 1,1 (R N ), and we define the kernels Then (1.7) gives rise to the well known expansions and therefore divX is a.e. defined on R N . From (1.7), we also get Moreover by (3.2) and the fact that ∂ ε Φ ε , X ∈ C 0,1 (R N ), we have that and In particular, it follows from (3.3) and (3.5) that there exist ε 0 , C > 0 with the property that 1 For v ∈ H s 0 (Ω) and ε ∈ (−ε 0 , ε 0 ), we now define Then, by (1.3), (1.7) and a change of variables, we have the following variational characterization for λ s,p (Ω ε ): As mentioned earlier, we prefer to use (3.8) from now on where the underlying domain is fixed and the integral terms depend on ε instead. It follows from (3.
The second limit follows from Lemma 2.1, (3.5) and (3.9) by noting that µ ∈ L ∞ (R N × R N ) for the function

One-sided Shape derivative computations
We keep using the notation of the previous sections, and we recall in particular the variational characterization of λ ε s,p = λ s,p (Ω ε ) given in (3.8). The aim of this section is to prove the following result.
The proof of Proposition 4.1 requires several preliminary results. We start with a formula for the derivative of the function given by (3.7).
Proof. By (3.5), (3.11) and Fubini's theorem, we have Applying, for fixed y ∈ R N and µ > 0, the divergence theorem in the domain {x ∈ R N : |x−y| > µ} and using that ∇ x |x − y| −N −2s = −(N + 2s) Since U ∈ C 1,1 c (Ω), we have that c N,s 2 lim Moreover, since U is compactly supported, we may fix R > 0 large enough such that ( |x − y| = µ} for 0 < µ < 1 and using that U, X ∈ C 0,1 (R N ), we thus deduce that We cannot apply Lemma 4.2 directly to minimizers u ∈ H s 0 (Ω) of λ s,p (Ω) since these are not contained in C 1,1 c (Ω). The aim is therefore to apply Lemma 4.2 to U k := uζ k ∈ C 1,1 c (Ω) with ζ k given in (2.3), and to use Proposition 2.4. This leads to the following derivative formula which plays a key role in the proof of Proposition 4.1.
Proof. By Lemma 2.3 and since Ω is of class Applying Lemma 4.2 to U k , we find that By the standard product rule for the fractional Laplacian, we have (−∆) s U k = u(−∆) s ζ k + ζ k (−∆) s u − I(u, ζ k ) with I(u, ζ k ) given by (2.11). We thus obtain where we used that (−∆) s u = λ s,p (Ω)u p−1 in Ω. Consequently, Proposition 2.4 yields that Moreover, integrating by parts, we obtain, for k ∈ N, Since u p ∈ C s 0 (Ω) by Lemma 2.3, it is easy to see from the definition of ζ k that the last two terms in (4.8) tend to zero as k → ∞, whereas Plugging this into (4.7), we obtain (4.5), as required.
Lemma 4.5. We have Proof. Let (ε n ) n be a sequence of positive numbers converging to zero and with the property that Since v εn remains bounded in H s 0 (Ω) by (3.12), we may pass to a sub-sequence with the property that v εn ⇀ u in H s 0 (Ω) for some u ∈ H s 0 (Ω). Moreover, v εn → u in L p (Ω) as n → ∞ since the embedding H s 0 (Ω) → L p (Ω) is compact. In the following, to keep the notation simple, we write ε in place of ε n . By (3.10), (3.11) and (4.11), we have and therefore where the last inequality follows from Lemma 4.4. In view of (3.2) and the strong convergence v ε → u in L p (Ω), we see that as ε → 0, and hence u L p (Ω) = 1. Combining this with (4.13), we see that u ∈ H is a minimizer for λ 0 s,p , and that equality must hold in all inequalities of (4.13). From this we deduce that v ε → u strongly in H s 0 (Ω). (4.15) Now (4.12) and the variational characterization of λ 0 s,p imply that whereas by (4.14) we have and therefore Plugging this into (4.16), we get the inequality Since, moreover, V ′ vε (0) → V ′ u (0) as ε → 0 by Lemma 3.1 and (4.15), it follows that and therefore by Lemma 4.3. We thus conclude that Taking the infinimum over u ∈ H in the RHS of this inequality and using (4.10), we get the result.

Proof of the main results
In this section we complete the proofs of the main results stated in the introduction.
Proof of Theorem 1.1 (completed). In view of Proposition 4.1, the proof of Theorem 1.1 is complete once we show that 2κ s = Γ(1 + s) 2 , (5.1) where Γ is the usual Gamma function. In view of (2.10), the constant κ s does not depend on N , p and Ω, we consider the case N = p = 1 and the family of diffeomorphisms Φ ε on R N given by Φ ε (x) = (1 + ε)x, ε ∈ (−1, 1), so that X := ∂ ε ε=0 Φ ε is simply given by X(x) = x.
We thus conclude that Thus, by Proposition 4.1, we get the result as stated in the theorem.
Moreover, by definition, w ≡ u ≥ 0 in H \Θ, and w ≡ u > 0 in the subset [r H (B 1 )∩ H]\Θ which has positive measure since t > 0. Using that w is anti-symmetric with respect to H and the fact that λ s,p (Θ) > c p (which follows since Θ is a proper subdomain of Ω(t)), we can apply the weak maximum principle for antisymmetric functions (see [9,Proposition 3.1] or [17,Proposition 3.5]) to deduce that w ≥ 0 in Θ. Moreover, since w ≡ 0 in R N , it follows from the strong maximum principle for antisymmetric functions given in [17,Proposition 3.6] that w > 0 in Θ. Now by the fractional Hopf lemma for antisymmetric functions (see [9,Proposition 3.3]) we conclude that From this and (5.11) we get (5.8), since ν 1 > 0 on ∂B t ∩ Θ.
To conclude, we observe that the function t → λ s,p (t) = λ s,p (Ω(t)) is even, thanks to the invariance of the problem under rotations. Therefore the function θ attains its maximum uniquely at t = 0.

Proof of Proposition 2.4
The aim of this section is to prove Proposition 2.4. For the readers convenience, we repeat the statement here. Proposition 6.1. Let X ∈ C 0 (Ω, R N ), let u ∈ C s 0 (Ω) ∩ C 1 (Ω), and assume that ψ := u δ s extends to a function on Ω satisfying (2.8) and (2.9). Moreover, put U k := uζ k ∈ C 1,1 and ζ given in (2.2), and where we use the notation for u ∈ C s c (R N ), v ∈ C 0,1 (R N ) and x ∈ R N .
Next we consider the functions G 1 k defined in (6.13), and we first state the following estimate. Proposition 6.3. There exists ε ′ > 0 with the property that for k ∈ N, 0 ≤ r < kε ′ (6.16) with a constant C > 0. Moreover, Before giving the somewhat lengthy proof of this proposition, we infer the following corollary related to the functions G 1 k . Corollary 6.4. There exists ε ′ > 0 with the property that with a constant C > 0. Moreover, Proof. Since u = ψδ s we have u(Ψ(σ, r k )) = k −s ψ(σ + r k ν(σ))r s for k ∈ N, 0 ≤ r < kε, and lim k→∞ k s u(Ψ(σ, r k )) = ψ(σ)r s for σ ∈ ∂Ω, r > 0.
Since moreover ψ L ∞ (Ωε) < ∞, the claim now follows from Proposition 6.3 by recalling the definition in G 1 k in (6.13). We now turn to the proof of Proposition 6.3, and we need some preliminary considerations. Since ∂Ω is of class C 1,1 by assumption, there exists an open ball B ⊂ R N −1 centered at the origin and, for every σ ∈ ∂Ω, a parametrization f σ : B → ∂Ω of class C 1,1 with the property that f σ (0) = σ and df σ (0) : R N −1 → R N is a linear isometry. For z ∈ B we then have and therefore where we used in (6.21) that df σ (0)z belongs to the tangent space T σ ∂Ω = {ν(σ)} ⊥ . Here and in the following, the term O(τ ) stands for a function depending on τ and possibly other quantities but satisfying |O(τ )| ≤ Cτ with a constant C > 0.
We now have all the tools to study the quantity A k (σ, r) in (6.24).
Proof of Proposition 6.3. The proof is completed by combining (6.24) with Lemmas 6.5 and 6.7.
It finally remains to estimate the function G 2 k in (6.12).