A Hopf lemma for the regional fractional Laplacian

We provide a Hopf boundary lemma for the regional fractional Laplacian (-Δ)Ωs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\Delta )^s_{\Omega }$$\end{document}, with Ω⊂RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset \mathbb {R}^N$$\end{document} a bounded open set. More precisely, given u a pointwise or weak super-solution of the equation (-Δ)Ωsu=c(x)u\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\Delta )^s_{\Omega }u=c(x)u$$\end{document} in Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega$$\end{document}, we show that the ratio u(x)/(dist(x,∂Ω))2s-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u(x)/(\text {dist}(x,\partial \Omega ))^{2s-1}$$\end{document} is strictly positive as x approaches the boundary ∂Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega$$\end{document} of Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega$$\end{document}. We also prove a strong maximum principle for distributional super-solutions.


Introduction and main results
Let s ∈ (1/2, 1) and let Ω ⊂ R N (N ≥ 2) be a bounded domain with C 1,1 boundary.The regional fractional Laplacian (−∆) s Ω of a function u : Ω → R is defined as provided that the limit exists.We recall that "P.V." stands for the Cauchy principal value and that the normalization constant c N,s is explicitly given by c N,s := 2 ) π N/2 Γ(2 − s) .
The study of the regional fractional Laplacian has received some growing attention in recent years.However, in contrast to that of the 1 fractional Laplacian (−∆) s u(x) = c N,s P.V.
the theory of elliptic problems driven by the regional fractional Laplacian is less developed in spite of some known results.We are concerned here in particular with the Hopf boundary lemma, which is a powerful tool for the study of qualitative properties of solutions like, for example, their monotonicity and symmetry, also via moving plane arguments.
In [13], the authors obtained a Hopf lemma for pointwise super-solutions for an elliptic equation involving the fractional Laplacian (−∆) s under the assumption that an interior ball condition holds.For the Hopf boundary lemma for weak super-solutions related to the fractional p-Laplacian, we refer to [9] and references therein.Other references on the Hopf boundary lemma for fractional Laplacian can be found in [1,5,7,12,16,17].However, to the best of our knowledge, an analogue result for the regional fractional Laplacian has not been investigated before.Let us mention here that while the Hopf lemma is usually used to run a moving plane method in the case of the fractional Laplacian, as recalled above, this does not seem to be the case for the regional fractional Laplacian.
Date: May 18, 2022. 1 Sometimes it is also called restricted fractional Laplacian.
The moving plane method for (−∆) s Ω remains indeed a challenging question: the main difficulty relies on the fact that the operator depends on the domain and therefore, upon scaling the domain, the operator changes as well.We expect a symmetry breaking in the case of the regional fractional Laplacian defined on bounded domains.
Here, we investigate the validity of a suitable Hopf-type lemma for super-solutions of the equation We analyse this both for the case of pointwise and weak super-solutions.Moreover, we also study a strong maximum principle for distributional super-solutions to (1.3).So, before stating our main results, let us recall the following definitions (notations are defined in Section 2).
Definition 1.1.We say that a function u : In this case, we briefly write Remark 1.4.Sub-solutions can be defined in similar ways as in Definitions 1.1, 1.2, and 1.3.Also, in the case of Definition 1.2, by density the test function ϕ can be chosen in H s 0 (Ω) + if c is somewhat well-behaved (see Lemma 4.1 below for more details).
Let us first comment on the proof of Theorem 1.5.Starting with a strong maximum principle, we obtain the strict positivity of non-trivial super-solutions of (1.3): this is where the lower semicontinuity of u is needed.In a next step, we construct a barrier from below for u in terms of the torsion function u tor , i.e., the solution to the boundary value problem This function is known to satisfy, on smooth domains, the double-sided estimate for some C > 1, see [3,6] which are based on some estimates in [2,8,14].Intuitively, (1.9) gives that the boundary behaviour of super-solutions described by (1.4) and (1.7) is optimal.We notice that, in contrast to what happens for the fractional Laplacian, there are no explicit examples of torsion functions for the regional fractional Laplacian, even in the case when Ω is a ball.In [10], a numerical analysis is performed in the one-dimensional case Ω = (−1, 1).We mention that the existence and uniqueness of pointwise and weak solutions to the Dirichlet problem (1.8) with general bounded right-hand side was obtained in [6].We notice also that the Hölder regularity up to the boundary of any weak solution of (1.8) was recently proved in [11], while regularity up to the boundary of pointwise solution of (1.8) was obtained earlier in [6].We also mention that the boundary regularity of the ratio u tor /δ 2s−1 Ω has been established in [11] in the case when Ω is of class C 1,β for some β > 0. Thus, it makes sense to evaluate u tor /δ 2s−1 Ω pointwisely on Ω .
The proof of Theorem 1.6 follows the same line of thought as the one of Theorem 1.5, although with some more technical difficulties due to the weak character of super-solutions involved.For example, when c ∈ L q (Ω) the strong maximum principle involved in our strategy takes the following form.

Proposition 1.7 (Strong maximum principle for distributional super-solutions). Let
The paper is organized as follows.In Section 2, we present some notations and definitions.Section 3 is devoted to the proof of Theorem 1.5, whereas in Section 4 we prove Theorem 1.6.Finally, in Section 5 we prove Proposition 1.7.

Preliminaries
We collect in this section some notations and useful tools.For s ∈ (0, 1), H s (Ω) denotes the space of functions u ∈ L 2 (Ω) such that It is a Hilbert space endowed with the norm We denote by (which is equivalent to the usual one in H s (Ω) thanks to a Poincaré-type inequality) and it can be characterized as follows , we consider the symmetric, continuous, and coercive bilinear form The first Dirichlet eigenvalue of (−∆) s Ω in Ω can be defined by . ( It holds λ 1 (Ω) > 0, with the corresponding eigenfunction unique and strictly positive in Ω.Given x ∈ Ω and r > 0, we denote by B r (x) the open ball centred at x with radius r.We denote by u + := max{u, 0} and u − := max{−u, 0} the positive and negative part of u respectively.We also recall that, if u ∈ H s (Ω), then u + , u − ∈ H s (Ω) as well: this follows from a simple calculation, indeed |x − y| N +2s dx dy ≤ 0.

Proof of the Hopf lemma: the case of pointwise super-solutions
The aim of this section is to prove Theorem 1.5.Before doing this, we need one key result: we state and prove a strong maximum principle for pointwise super-solutions of (1.3).Proof.Before going into the proof, we start by proving that the function u is nonnegative in Ω as long as the hypotheses of assertion (i) are satisfied.Let us assume that c ≤ 0 in Ω, u ≥ 0 on ∂Ω, and that u does not vanish identically on Ω.Then we claim that u ≥ 0 in Ω.
(3.1) Assume to the contrary that (3.1) does not hold, that is, u is negative somewhere in Ω.Then, using that Ω is compact together with the hypotheses of lower semicontinuity of u, a negative minimum of the function u must be achieved in Ω.In other words, there exists x 0 ∈ Ω such that But, since by assumption c(x 0 ) ≤ 0, we have that c(x 0 )u(x 0 ) ≥ 0. Therefore So we can now suppose u ≥ 0 in Ω. Suppose that u ≡ 0 in Ω and let us prove that First of all, we recall that by the lower semicontinuity of u, there exist |x − y| N +2s dy ≤ c N,s P.V.
Having the above strong maximum principle, we can now give the proof of Theorem 1.5 by following some ideas in [13].
Proof of Theorem 1.5.From Proposition 3.1 it follows that u(x) > 0 for all x ∈ Ω (3.4) provided that u does not vanish identically in Ω.In other words, if u does not vanish identically in Ω, then for every compact subset K ⊂ Ω we have inf y∈K u(y) > 0. (3.5) Now suppose that u does not vanish identically in Ω and let us prove (1.4).To this end, it suffices to construct a barrier for u in terms of the solution problem (1.8).Let u tor denote the pointwise solution of (1.8).
Next, for n ∈ N, we set Then, by definition and (1.9), by the boundedness of Ω it follows that We wish now to show that there exists some n 0 ∈ N such that u ≥ v n in Ω, for any n ≥ n 0 . (3.8) In order to prove (3.8), we argue by contradiction: suppose that for every n ∈ N the function w n defined by is positive somewhere in Ω.Then, using that w n = v n − u = −u ≤ 0 on ∂Ω and the compactness of Ω, a positive maximum of the upper semicontinuous function w n (since u is lower semicontinuous by assumption) must be achieved at some x n ∈ Ω, that is, there exists x n ∈ Ω such that This implies together with (3.4) that 0 < u(x n ) < v n (x n ).From this and thanks to (3.7), we find that lim Recalling (3.5), we deduce from (3.10) that x n → ∂Ω as n → ∞.Taking this into account, one deduces that for any compact set K ⊂ Ω there exists h > 0 such that |x n − y| ≥ h > 0 for any y ∈ K and n sufficiently large.As a direct consequence, there exist two positive constants γ 1 , γ 2 > 0, independent of n such that for n sufficiently large (depending on K).
Thus we have |x n − y| N +2s dy.(3.12) We now aim at estimating the integrals on the right-hand side of the above inequality.Concerning the first integral, we notice that by (3.5), there exists a positive constant γ 3 > 0 such that u(y) ≥ γ 3 for y ∈ K.As a consequence of this and by using (3.10) and (3.11), it follows that lim sup Regarding the second integral in (3.12), we first recall that since x n is the maximum of w n in Ω, then by (3.9) Using this, the second integral in (3.12) can be estimated as follows: Moreover, a simple calculation yields Now, from (3.6) and (1.8), it follows that Finally, (3.17) and (3.13) into (3.12),lead to a contradiction with (3.18).Therefore, the inequality (3.8) follows for some n ∈ N large enough.

Proof of the Hopf lemma: the case of weak super-solutions
In this section, we aim at proving Theorem 1.6.Here, the function u tor defined via (1.8) above is understood to be a weak solution.Recall the double-sided estimate (1.9).We first state and prove a technical lemma and a strong maximum principle for weak super-solutions of (1.3).
(Ω) a sequence of nonnegative functions converging to v in the H s (Ω)-norm.By Definition 1.2 we have for any n ∈ N.
On the left-hand side we have the convergence E(u, ψ n ) → E(u, v) as n → ∞ by construction; so, let us deal with right-hand side.By the Sobolev embedding we have . So, we have the convergence which is what we show next.This indeed follows from the Hölder inequality: (ii) If u ≥ 0 in Ω, then either u vanishes identically in Ω or u > 0 in Ω.
Proof.We first recall the following elementary inequality: Assume then c ≤ 0 in Ω and u ≥ 0 on ∂Ω.Then u − = 0 on ∂Ω.Moreover, by standard arguments, we also know u − ∈ H s (Ω).Therefore u − ∈ H s 0 (Ω) + .Hence, by testing (4.1) on u − (which is allowed by Lemma 4.1), we have from inequality (4.2) that where λ 1 (Ω) has been defined in (2.1).Since λ 1 (Ω) > 0, then from the nonpositivity of c it follows So we can at this point assume that u ≥ 0 in Ω.Note that the fact that u is a weak super-solution implies in particular that u is also a distributional super-solution.Indeed, for any Using this remark, we can use Proposition 1.7.
Remark 4.3.It is possible to drop the assumption u ∈ L ∞ loc (Ω) in Proposition 4.2 by paying the price of assuming c ∈ L ∞ (Ω).In this case, the first part of the proof still holds, while, instead of using Proposition 1.7, the second part simply follows from [15,Theorem 1.2].
We now prove Theorem 1.6.For the sake of clarity, we split its proof into two different arguments.
Proof of Theorem 1.6 under assumption (1.5).Suppose that u does not vanish identically in Ω and let us prove (1.7).In other words, we want to prove that there exists a positive constant 3) From Proposition 4.2 and Remark 4.3 it follows that u > 0 in Ω.This means that for any K ⊂⊂ Ω there exists ε > 0 such that it holds Now, let w n := v n − u where v n is the function defined in (3.6).Then, thanks to (3.7) and (4.4), we can assume without any ambiguity that Now, since w + n ∈ H s 0 (Ω) + (because w + n ≥ 0 in Ω, w + n ∈ H s (Ω) since w n is, and w + n = 0 on ∂Ω since v n = 0 on ∂Ω and u ≥ 0 on ∂Ω), one can use it as a test function in Definition 1.2 (by Lemma 4.1) in order to have we have On the other hand, Since the first term on the right-hand side of the above equality is nonpositive and Now, Recall that, by definition (see also in Ω, so, upon plugging this into (4.9),we obtain for some C 0 > 0 and n sufficiently large.Plugging (4.7) into (4.6), using (4.8) and this last obtained inequality, we get For n sufficiently large, we deduce from (4.10) that Therefore, (4.3) follows.
Proof of Theorem 1.6 under assumption (1.6).The very first part of the proof follows the argument given above.We start here from (4.6).We know from Proposition 4.2 that u > 0 in Ω and so w n < v n , from which it follows By the fractional Sobolev inequality we have that u ∈ L p (Ω) for any 1 ≤ p ≤ 2 * s = 2N/(N − 2s).As the conjugate exponent of 2N/(N − 2s) is 2N/(N + 2s) which is smaller than N/(2s), we have by an application of the Hölder's inequality that By repeating the calculations in the preceding argument we then get the analog of (4.10) which reads in this case This last inequality, for n sufficiently large, gives Therefore, (4.3) follows also in this case.

Proof of the strong maximum principle for distributional super-solutions
This last section is devoted to the proof of Proposition 1.7.In the following, we assume that u : Ω → R is a distributional super-solution (in the sense of Definition 1.3) of (1.3) and that c satisfies the assumptions in (1.10).
5.1.Regional v. restricted fractional Laplacian.Note that where we recall (1.2), so that Definition 1.3 is equivalent to (if we extend u = 0 in R N \ Ω) Ω) for ε small independently of ψ and we can say As u * η ε ∈ C ∞ (Ω ′ ), the above inequality also holds in a pointwise sense.We can then exploit a Green representation on u * η ε (see [4]) to deduce that for any x ∈ Ω ′′ ⊂⊂ Ω ′ and 0 Here we have used the kernels G and P which are respectively the Green function and the Poisson kernel of the fractional Laplacian (−∆) s on the unitary ball B 1 , which are explicitly known, see [4]: From now on, we assume that u ≥ 0 in Ω.We want to send ε → 0 in (5.3) and deduce a representation for u.For the Poisson integral we use the nonnegativity of u and the Fatou's Lemma to say For the Green integral we use that where, moreover, by the Hölder inequality where the second inequality holds for β close to N 2s in view of (5.2).Therefore, using the weak topology in Lebesgue spaces, G(0, y) c + κ Ω (x + ry) u(x + ry) dy.(5.5) In the following we are going to need the following fact for p > 1 and K ⊂⊂ R N measurable.
(5.6) 5.4.The strong maximum principle.Having the above ingredients, in this subsection, we are ready to give the proof of Proposition 1.7.
In the notations of the previous subsection, and without loss of generality, we assume that there exist (x j ) j∈N ⊂ Ω ′′ and (r j ) j∈N ⊂ (0, ∞), r j → 0 as j → ∞, such that lim j→∞ 1 (2r j ) N B 2r j (x j ) u = 0.
Without loss of generality, we can assume that (r j ) j∈N is decreasing.Extract a subsequence (ρ j ) j∈N ⊂ (r j ) j∈N in such a way that 2 1 ρ N j Bρ j (x j ) u ≤ r 2s j j and ρ j ≤ r j for any j ∈ N. (5.7) In order to ease notation, relabel c Ω = c + κ Ω .We apply representation (5.4) with r = r j and we then integrate it over B ρ j (x j ), obtaining Here we briefly comment on inequality (5.7).As we know by assumption that 1 (2r j ) N B 2r j (x j ) u → 0 as j → ∞, one has also 2 N (2r j ) N B 2r j (x j ) u → 0 as j → ∞.Now, using that Br j (xj) ⊂ B2r j (xj) and that u is nonnegative, one can write 0 ≤ 1 r N j Br j (x j ) u ≤ 2 N (2rj ) N B 2r j (x j ) u −→ 0 as j → ∞.

Proposition 3 . 1 (
Strong maximum principle for pointwise super-solutions).Let Ω ⊂ R N be a bounded open set.Let c ∈ L ∞ (Ω) and u : Ω → R be a lower semicontinuous function super-solution (in the sense of Definition 1.1) of (1.3).(i)If c ≤ 0 in Ω and u ≥ 0 on ∂Ω, then either u vanishes identically in Ω, or u > 0 in Ω.(ii)If u ≥ 0 in Ω, then either u vanishes identically in Ω, or u > 0 in Ω.

Lemma 4 . 1 .
Let Ω ⊂ R N be an open bounded set and c ∈ L N 2s (Ω).Then u is a weak super-solution (in the sense of Definition 1.2) of (1.3) if and only if