A Hopf lemma and regularity for fractional $p-$Laplacians

In this paper, we study qualitative properties of the fractional $p$-Laplacian. Specifically, we establish a Hopf type lemma for positive weak super-solutions of the fractional $p-$Laplacian equation with Dirichlet condition. Moreover, an optimal condition is obtained to ensure $(-\triangle)_p^s u\in C^1(\mathbb{R}^n)$ for smooth functions $u$.


Introduction and main results
The fractional p−Laplacian is defined by the singular integral (−△) s p u(x) := C n,s,p P.V. |u(x) − u(y)| p−2 (u(x) − u(y)) |x − y| n+sp dy, where C n,s,p is a positive constant depending only on n, s, and p, s ∈ (0, 1), and p > 1. Denote If u ∈ C 1,1 loc ∩ L sp (R n ), then (1.1) is well defined. Clearly, when p = 2, (1.1) becomes the fractional Laplacian which arises in many fields such as phase transitions, flame propagation, stratified materials and others (see [1,6,27]). In particular, the fractional Laplacian can be understood as the infinitesimal generator of a stable Levy process (see [28]). The fractional * The first author is partially supported by the Simons Foundation Collaboration Grant for Mathematicians 245486. The second author is partially supported by NSFC-11571233. The third author is supported by the Fundamental Research Funds for the Central Universities (lzujbky-2017-it53). E-mail: wchen@yu.edu (W. Chen), cli@colorado.edu (C. Li), qishj15@lzu.edu.cn (S. Qi).
p−Laplacian also has many applications, for instance, it is used to study the non-local "Tugof-War" game (see [2,3,22]). The interest on these nonlocal operators continues to grow in recent years. We refer to [24] for the recent progress on these nonlocal operators.
Due to the non-locality of these kinds of operators, many traditional methods in studying the local differential operators no longer work. To overcome this difficulty, Cafarelli and Silvestre [16] introduced the extension method which turns nonlocal problems involving the fractional Laplacian (p = 2) into local ones in higher dimensions, then the classical theories for local elliptic partial differential equations can be applied. We refer to [5,15] and references therein for broad applications of this method.
Another useful method to study the fractional Laplacian is the integral equations method, which turns a given fractional Laplacian equation into its equivalent integral equation, and then various properties of the original equation can be obtained by investigating the integral equation, see [7,14,29] and references therein.
However, so far as we know, there has neither been any extension method nor the integral equations method that work for the fractional p−Laplacian equation when p = 2. The nonlinearity, the singularity (1 < p < 2) and degeneracy (p > 2) of the operator (−△) s p render many powerful methods to study the fractional Laplacian (p = 2) no longer effective.
Recently, Chen et al. have developed a direct method of moving planes to investigate the nonlocal problems, which can be used to study not only the fractional Laplacian but also the fully nonlinear nonlocal operator where α > 0, G : R → R is a locally Lipschitz continuous function. The fractional p-Laplacian is a special case in which G(t) = |t| p−2 t and α = sp. This direct method has been successfully applied to obtain symmetry, monotonicity, nonexistence and other qualitative properties of solutions for various nonlocal problems, see e.g., [8,10,11,12,13].
In the present paper, we will continue to study qualitative properties for fractional p-Laplacian. We will establish a Hopf type lemma in general domains for super solutions to fractional p-Laplacian equations with a Dirichlet condition; and for any given smooth function u, we will obtain an optimal condition for (−∆) s p u to be continuously differentiable. It is well-known that the Hopf lemma is a very powerful tool in the study of various differential equations. For example, it has been successfully used in the "second" step of the moving planes method.
In the case of fractional Laplacian (p = 2), Fall and Jarohs [19, Proposition 3.3 ] proved a Hopf lemma for the entire antisymmetric supersolution of the problem Greco and Servadei [20] obtained a Hopf type lemma to (1.2) under the assumptions that c(x) ≤ 0 and Ω ⊂ R n is a bounded domain. Chen and Li [9] established a Hopf lemma for anti-symmetric function on a half space through a rather delicate analysis. More recently, Jin and Li [23] extended the results in [9] to the fractional p−Laplacian with p > 3 for positive anti-symmetric functions on the boundary of a half space. In this paper, we shall establish a Hopf type lemma for the positive weak supersolution of (1.3) on the boundary of more general domains. Before stating our main results, we first introduce some definitions on fractional Sobolev spaces, and one can see [18,21] for more details. For any domain Ω ⊂ R n with smooth boundary, define If Ω ⊂ R n is unbounded, set Furthermore, if u is both a supersolution and a subsolution of (1.3), then we say it is a solution to (1.3).
, we say that u ∈ W s,p (Ω) is a weak supersolution of (1.3), if there hold (u + ǫ) − ∈ W s,p 0 (Ω) for any ǫ > 0, and C n,s,p for any φ ∈ W s,p 0 (Ω) with φ ≥ 0 in Ω. The weak subsolution can be defined similarly. Moreover, if u is both a weak supersolution and a weak subsolution of (1.3), then we say it is a weak solution to (1.3).

One of our main results is
Let Ω ⊂ R n be a domain with C 1,1 boundary. If it is bounded, we assume u ∈ W s,p (Ω) ∩ C(Ω); if it is unbounded, we assume u ∈ W s,p loc (Ω) ∩ C(Ω).
in the weak sense, then The other main result is concerning the regularity of (−∆) s p u. The regularity of solutions of the fractional p−Laplacian equations has attracted considerable attention in recent years, and it has been well understood for the fractional Laplacian equations (p = 2). Specifically, the Schauder interior estimate of the solution is similar to that of the Poisson equation (associated with the regular Laplacian), which states roughly that if f ∈ C γ (Ω) and u ∈ C 1,1 then the regularity of the solution u can be raised by the order of 2s in any proper subset of Ω, the same order as the operator (−∆) s . By introducing the proper weighted Hölder norms as in the case of Poisson equations, one shall be able to control a weighted C 2s+γ norm of u in Ω in terms of another weighted C γ norm of f in Ω. However, when considering the regularity of the solution up to the boundary, the situation in the fractional order equation is quite different from that in the integer order equation (when s = 1, the Poisson equation). In fact, Ros-Oton and Serra [25] proved that if u ∈ C 1,1 loc (Ω) ∩ L 2s (R n ) is a solution of (1.6) with f ∈ L ∞ (Ω), then u is C s up to the boundary; and this is optimal in general. Later, Chen et al. [14] proved the similar results by a simpler method.
For the fractional p−Laplacian, the study of the regularity becomes quite complicated. So far there are very few results. Di Castro and Kuusi [17] showed that if u ∈ W s,p (Ω) satisfies (−△) s p u = 0 in Ω, then u is locally γ-Hölder continuous for small γ. Brasco et al. [4] established a higher Hölder regularity for the fractional p−Laplacian equation in the superquadratic case (p > 2). Indeed, the authors have verified that if u ∈ W s,p then u ∈ C δ loc (Ω) for every 0 < δ < Θ(n, s, p, q) with Θ(n, s, p, q) = min Iannizzotto et al. [21] proved that the solutions of (1.7) with f ∈ L ∞ (Ω) belong to C α (Ω) for some α ∈ (0, s]. Concerning the regularities of (−△) s p u for a given smooth function u, there are more substantial technical difficulties than the local case.
While for the fractional p−Laplacian, the singularity (0 < p < 2) and degeneracy (p > 2) of operator (−△) s p make it more complex. For example, even for the local operator △ p and the sufficient smooth function u( In this paper, we shall consider the differentiability of (−△) s p u for p > 2 and establish an optimal condition such that (−△) s p u ∈ C 1 (R n ). Specifically, we prove that The condition p > 3 2−s is optimal as shown in the following The rest of the paper is organized as follows. Section 2 is devoted to establishing the Hopf type lemma for the positive solution of (1.5). In section 3, we first prove the differentiability of (−△) s p u under the condition p > 3 2−s . Then we show that this condition is optimal by giving a counterexample when p ≤ 3 2−s . In the Appendix, we state some results in [21] used in the present paper for convenience.

Hopf type lemma
In this section, we prove the Hopf type lemma for the positive weak solution of (1.5) by constructing a suitable subsolution.
Proof of Theorem 1.1. For any given x 0 ∈ ∂Ω, it follows from the C 1,1 property of the boundary of Ω that there exist x 1 ∈ Ω on the normal line to ∂Ω at x 0 and a positive constant Without loss of the generality, we suppose that x 0 is the origin, α = 1 and x 1 = e n with e n = (0, · · · , 1) the last vector of the canonical basis of R n . Let r ∈ 0, 1 3 √ 5 be a constant, O denote the origin and Next we show the following two claims.
Now we show that h is strictly monotone. Direct calculation implies that (2.6) Noting that if |X ′ | = 0 then h ′ = 1. Furthermore, if |X ′ | = 0, the last term in (2.6) is positive and the second term can be rewritten as It then follows from |X ′ | ≤ 3r and (2.1) that |J 2 | ≤ 3. Moreover, thanks to (2.3), we can verify that J 1 ≤ 1 that is, h is strictly increasing in {X n ∈ R|(X ′ , X n ) ∈ B 3r (O)}. This together with (2.4) shows that Ψ ∈ C 1,1 (R n , R n ) is a diffeomorphism of R n . Claim 2. There holds Indeed, it suffices to show that for any Then The claim is then verified. Now, we define ρ : R n → R as Then there holds ρ(Ψ(X)) = (X n ) + for any X ∈ B 2r (O). (2.9) Indeed, It follows from (2.1) and (2.4) that By a direct calculation, we see that Ψ(X) ∈ ∂B 1−Xn (e n ) for any X ∈ B 2r (O), which together with (2.8) leads to (2.9). Next we show by a similar calculation as in [21] that there exists a positive constant C 1 such that Indeed, thanks to lemma A.3, we only need to show that there exists f ∈ L ∞ (B 1 (e n )∩B r (0)) such that where G(t) = |t| p−2 t for any t ∈ R. Making a change of variables X = Ψ −1 (x), then for any where the second equality follows from (2.7) and (2.9). Noting that Ψ is a C 1,1 diffeomorphism of R n and Ψ = I in B c 3r (O), Lemma A.2 then yields that there exists (2.12) Thanks to (2.7), there exists a positive constant C such that for any x ∈ B 1 (e n ) ∩ B r (0) and Hence, we have where the notation C Ψ above may denote different positive constants. This together with (2.11) and (2.12) shows that lim ǫ→0 B c ǫ (X) G (ρ s (Ψ(X)) − ρ s (Ψ(Y ))) |Ψ(X) − Ψ(Y )| n+ps J Ψ (Y )dY = f 1 (Ψ(X)) + J 2 (X) in L 1 (Ψ −1 (B 1 (e n ) ∩ B r (0))), with f 1 • Ψ and J 2 belong to L ∞ (Ψ −1 (B 1 (e n ) ∩ B r (0))) . It then follows that Consequently, (2.10) follows. Now let D ⊂⊂ B c 1 (e n ) ∩ Ω be a bounded smooth domain, and β > 0 be a positive constant to be determined below. Set where ρ is defined by (2.8), and χ D is the characteristic function of D, namely, It follows from D ⊂⊂ B c 1 (e n ) ∩ Ω that there is a positive constant C D such that |x − y| ≥ C D for any x ∈ B 1 (e n ), y ∈ D. (2.14) For any x ∈ B 1 (e n ) ∩ B r (0), direct calculation (we omit the term 'C n,s,p lim ǫ→0 ' in the following calculation for convenience) shows that (2.16) and the last inequality holds due to (2.10). Let then it follows from the monotonicity of G that , where M 1 and M 2 are some positive constants. In view of (2.8) and (2.13), there holds The comparison principle then yields that By the definition of ρ, we have ρ(te n ) = d(te n ) for any t ∈ (0, 1), and The proof is complete.

Regularity
This section is devoted to the study of regularity of (−△) s p u. We first prove the differentiability of (−△) s p u under the assumptions of Theorem 1.2, then we show that the condition p > 3 2−s is optimal by giving a counterexample when p ≤ 3 2−s . Proof of theorem 1.2. For any x ∈ R n , by making change of variables, (−△) s p u(x) can be rewritten as where G(t) = |t| p−2 t for any t ∈ R. Note that (3.1) By a direct calculation, we obtain and where v(x) := ∂u ∂x i (x). Now, we verify that |I 2 | < ∞. Indeed, It follows from u ∈ L sp (R n ), |∇u| ∈ L sp (R n ) and the Hölder inequality that |I 2 | < ∞.
For the term I 1 , using the Taylor expansion formula, there hold where the notation O(|y| 2 ) denotes that there exist some positive constant C such that |O(|y| 2 )| ≤ C|y| 2 . Consequently, there is a positive constant C such that Now, we consider the terms J 1 and J 3 .
Cases 2. ∇u(x) · y = 0. Then we rewrite J 1 and J 3 respectively as It follows that To summary, we conclude that there exists a positive constant C independent of y such that The assumption p > 3 2−s further implies By exchanging the order of integration and differentiation, we derive that (−△) s p u is differentiable in R n , and then we conclude (−△) s p u ∈ C(R n ) by exchanging the order of integration and limit. The proof is complete. Theorem 1.2 verifies that in the case p > 2, if one assumes in addition that p > 3 2−s , then (−△) s p u ∈ C 1 (R n ) for any u satisfying u ∈ C 3 loc (R n ) ∩ L sp (R n ) and |∇u| ∈ L sp (R n ). It seems from the proof of Theorem 1.2 that p > 3 2−s is a technical assumption. While, the counterexample in Theorem 1.3 shows that this condition is optimal to ensure (−△) s p u ∈ C(R n ) for any u satisfying u ∈ C 3 loc (R n ) ∩ L sp (R n ) and |∇u| ∈ L sp (R n ). Proof of Theorem 1.3. By virtue of the definition, we have For the convenience of writing, we set It follows from a straightforward calculation that then we can rewrite (3.6) as Note that for any 0 < (3.7) For I 3 , in view of y > 5 2 and 0 < x < 1 8 , there hold |x − y| > 2 and |x + y| > 2, which along with the properties of η implies that f (x, y) + f (x, −y) = 4x 2p−3 .

Hence, we have
where C 1 is a positive constant independent of x.
The proof is complete.
Another key ingredient is the following "change of variables" lemma.
Lemma A.2. Let Ψ be a C 1,1 diffeomorphism of R n such that Ψ = I in B c r (0), r > 0. Then the function v(x) = (Ψ −1 (x) · e n ) s + belongs to W s,p loc (R n ) and is a weak solution of where C (||DΨ|| ∞ , ||DΨ −1 || ∞ , r) is a positive constant. Moreover, The following lemma implies that the point-wise solution is also a weak solution.
Lemma A.3. Let u ∈ W s,p loc (Ω) and D denote the diagonal of R n × R n . For any ǫ > 0, assume A ǫ ⊂ R n × R n is a neighborhood of D and satisfies (i) (x, y) ∈ A ǫ for all (y, x) ∈ A ǫ , (ii) sup x∈Aǫ dist(x, D) → 0 as ǫ → 0.
If g ǫ → f in L 1 loc (Ω), then u is a weak solution of (−△) s p u = f in Ω.