Hölder estimates for viscosity solutions of equations of fractional p-Laplace type

We prove Hölder estimates for viscosity solutions of a class of possibly degenerate and singular equations modelled by the fractional p-Laplace equation PV∫Rn|u(x)-u(x+y)|p-2(u(x)-u(x+y))|y|n+spdy=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$PV {\int_{\mathbb{R}^{n}}\frac{|u(x)-u(x + y)|^{p-2}(u(x) - u(x + y))}{|y|^{n+sp}}\,dy = 0},$$\end{document}where s∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${s \in (0,1)}$$\end{document} and p>2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p > 2}$$\end{document} or 1/(1-s)<p<2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${1/(1-s) < p < 2}$$\end{document}. Our results also apply for inhomogeneous equations with more general kernels, when p and s are allowed to vary with x, without any regularity assumption on p and s. This complements and extends some of the recently obtained Hölder estimates for weak solutions.


Introduction
We study the local Hölder regularity for viscosity solutions of possibly degenerate and singular non-local equations of the form PV R n |u(x) − u(x + y)| p−2 (u(x) − u(x + y))K(x, y) dy = f (x), where f is bounded and K(x, y) essentially behaves like |y| −n−sp .Here PV stands for the principal value.
This type of equations is one possible non-local counterpart of equations of p-Laplace type and arises for instance as the Euler-Lagrange equation of functionals in fractional Sobolev spaces.Solutions can also be constructed directly via Perron's method (cf.[IN10]).In the case K(y) = |y| −n−sp , when properly rescaled, solutions converge to solutions of the p-Laplace equation ∆ p u = div(|∇u| p−2 ∇u) = 0 as the parameter s tends to 1, see [IN10].
Our first and main result is that bounded viscosity solutions (see Section 2) of the homogeneous equation are locally Hölder continuous, see the theorem below.Throughout the paper we denote by B r , the ball of radius r centered at the origin.
Theorem 1. Assume K satisfies K(x, y) = K(x, −y) and there exist Λ ≥ λ > 0, M > 0 and γ > 0 such that where s ∈ (0, 1) and p ∈ (1, ∞).In the case p < 2 we require additionally p > 1/(1 − s).Let u ∈ L ∞ (R n ) be a viscosity solution of Then u is Hölder continuous in B 1 and in particular there exist α and C depending on λ, Λ, M, p, s and γ such that In particular, Theorem 1 applies for the fractional p-Laplace equation PV |y| n+sp dy = 0.
Let f ∈ C(B 2 ) ∩ L ∞ (B 2 ) and let u ∈ L ∞ (R n ) be a viscosity solution of Then u is Hölder continuous in B 1 and in particular there exist α and C depending on λ, Λ, M, p 0 , p 1 , s 0 , s 1 , γ and τ such that . Remark 1.It might seem odd that the two conditions on K in our main theorems are supposed to be satisfied in overlapping regions, B 2 and B 1 4 .This is only for notational convenience.It would be sufficient to have the first condition satisfied in B ρ for some ρ > 0 and the second one satisfied outside B R for some large R as long as we ask K to be bounded in B R \ B ρ .

Known results
Equations similar to the ones in Theorem 1 were, to the author's knowledge, introduced in [IN10], where existence and uniqueness is established.It is also shown that the solutions converge to solutions of the p-Laplace equation, as s → 1.Similar equations were also studied in [CLM12], where the focus lies in the asymptotic behaviour as p → ∞.Related equations have also been suggested to be used in image processing and machine learning, see [EDL12] and [EDLO14].
Recently, in [DCKP13b] and [DCKP13], Hölder estimates and a Harnack inequality were obtained for weak solutions of a very general class of equations of this type.The difference between these results and the ones in the present paper can be seen as the difference between equations in divergence form and those in non-divergence form in the non-local setting.In other words, their results are more in the flavour of Di Giorgi-Nash-Moser (cf.[DG57], [Nas58] and [Mos61]) while the results in this paper are more in the flavour of Krylov-Safonov (cf.[KS79]).
In the case p = 2, corresponding to equations of the form PV a similar development has already taken place.In [Sil06], a surprisingly simple proof of Hölder estimates for viscosity solutions were given for a very general class of equations corresponding to equations of non-divergence form.An adaptation of the method used therein is used in the present paper.In [Kas09], Hölder estimates were obtained for weak solutions for a class of equations corresponding to equations of divergence form, including equations of the form (1.1).
Related is also [BCF12b] and [BCF12], where another type of degenerate (or singular) non-local equation is studied.Hölder estimates and some higher regularity theory are established.It is also proved that these equations approach the p-Laplace equation in the local limit.

Comments on the equation
Let us very briefly point out the difference between the class of equations considered in [DCKP13b] and [DCKP13], and the class of equations considered here (see also [Sil06] for a similar discussion).There, weak solutions are considered, in the sense that , where G(x, y) behaves like |x−y| −n−sp .These solutions arise for instance as minimizers of functionals of the form In the most favorable of situations, we are allowed to change the order of integration and write (1.2) as . Moreover, we are not always allowed to perform the transformations above.Hence, the two types of equations overlap but neither is contained in the other.In other words, the results in [DCKP13b] and [DCKP13] do not always apply to the equations considered in this paper, and vice versa, the results in this paper do not always apply to the equations studied therein.
Another important remark is that the estimates obtained in this paper are not uniform as s → 1, i.e., in the limit in which the equation becomes local.This is also the case in [Sil06].For fully nonlinear equations of fractional Laplace type, uniform estimates as s → 1 have been obtained (see for instance [CS09]), but they are more involved, and they follow the same strategy as the estimates for fully nonlinear (local) equations.
In our case, if φ ∈ C 2 0 and p > 2, then as s → 1.If we instead have a kernel of the form , as s → 1, where the matrix (a ij )(∇φ) is positive definite and can be given explicitly as integrals over the sphere in terms of G.This type of degenerate (or singular) equations of non-divergence form, remained fairly unstudied until quite recently.Starting with [BD04], these equations have attracted an increasing amount of attention.See also [Imb11] and [IS13] where C α and C 1,α -estimates are established, respectively.

Viscosity solutions
In this section, we introduce the notion of viscosity solutions (as in [CS09]) and prove that viscosity solutions can be treated almost as classical solutions.
Definition 1.Let D be an open set and let L be as defined in Theorem 1 or Theorem if the following holds: whenever x 0 ∈ D and φ ∈ C 2 (B r (x 0 )) for some r > 0 are such that A supersolution is defined similarly and a solution is a function which is both a sub-and a supersolution.
The following result verifies that whenever we can touch a subsolution from above with a C 2 function, we can treat the subsolution as classical subsolution.The proof is almost identical to the one of Theorem 2.2 in [CS09].
Proposition 1. Assume the hypotheses of Theorem 1 or Theorem 2. Suppose Lu ≤ C in B 1 in the viscosity sense and that for some r > 0. Then Lu is defined pointwise at x 0 and Lu (x 0 ) ≤ C.
Proof.Since the result is only concerned with the behavior at one fixed point x 0 , we see that there is no difference between assuming the hypotheses of Theorem 1 or Theorem 2. Hence, we give the proof under the hypotheses of Theorem 1.For 0 < s ≤ r, let By simply interchanging y → −y we have since one can easily see that the integral is well defined since φ s is C 2 near x 0 .Moreover, Since |δ(φ r , x 0 , y)K(x, y)| is integrable, so is δ − (u, x 0 , y)K(x, y).In addition, by (2.1) Since δ + (φ s , x 0 , y) ր δ + (u, x 0 , y), the monotone convergence theorem implies and by (2.2) for any 0 < s < r.We conclude that δ + (u, x 0 , y)K(x 0 , y) is integrable.By (2.2) and the bounded convergence theorem, we can pass to the limit in the right hand side of (2.3) and obtain This implies that Lu (x 0 ) exists in the pointwise sense and Lu (x 0 ) ≤ C.

Hölder regularity for constant exponents
In this section we give the proof of our main theorem for the case of constant s and p.This is based on Lemma 4, sometimes referred to as the oscillation lemma.Throughout this section, L denotes an operator of the form in Theorem 1, i.e., Let us also, by abuse of notation, introduce the function The exact form of β is not important, we could have chosen any radial function which is C 2 and zero outside B 1 and non-increasing along rays from the origin.
We start with a couple of auxiliary inequalities.Here a, b ∈ R.
Proof.We have Proof.We split the proof into two cases.
Case 1: |a| ≤ 2|b|.Then Proof.The inequality is trivial for p = 2 so we assume p > 2. Since a+b ≥ 0, |a| p−2 a + |b| p−2 b ≥ 0. Without loss of generality we can assume a > 0 and define t = b/a.The statement of the lemma is then equivalent to This is trivially true for t = −1.Hence we are lead to study the function We find that f has critical points at t = 1 and t = 0.In addition, We conclude that f (t) ≤ 2 p−2 for all t ≥ −1, and the result follows.
Below we prove that a kernel K behaving like y −n−sp satisfies certain inequalities that might look strange at a first glance, but they are exactly the ones that will appear in the proof of our key lemma later.
Proof.The proof is split into two different cases.
Case 1: p > 2 The first term in the left hand side of (3.1) reads Since β is uniformly bounded by a constant C, we can, using the upper bound on K outside B 1/4 , obtain which is finite and converges to zero as k → 0.
For I 2 we proceed as follows Introducing the notation I 2 can be written as p(1 − s) , (3.4) where C only depends on the C 2 -norm of β, which is fixed.Clearly the left hand side of (3.4) goes to zero as k → 0.
For the rest of the terms in the left hand side we observe first that if η < γ/(p − 1) then from the upper bound on K outside B 1/4 which is uniformly bounded and tends to zero as η → 0, by the dominated convergence theorem.
In addition, since β is uniformly bounded by some constant C > 0 we have which is finite and converges to zero as k → 0, where we again have used the upper bound on K outside B 1/4 .
Thus, if we choose η and k small enough (depending on Λ, M, p, s and γ) we can make all the terms in the left hand side as small as desired.
Now we turn our attention to the right hand side.We have, due to the lower bound on K in B 2 Then it is clear that we can choose η and k, depending only on λ, Λ, M, p, s, γ and δ, so that the left hand side is larger than the right hand side.
Case 2: 1/(1 − s) < p < 2 The only difference from the case p > 2 is the first term in the left hand side.We need to show that for k small enough, the term is small.We split the integral into two parts, one in B 1 and one in Hence, where we have used the upper bound on K in B 2 .For the part outside B 1 we have from the upper bound on K outside B 1/4 .By choosing k small (depending on Λ, M, p, s, γ) we can make both of these terms as small as desired.Hence, the result follows as in the case p > 2.
Remark 2. We remark that in the proof above, nothing would change if the exponents would depend on x, since x is a fixed point.This is important later when we redo the proof for the case of variable exponents.
The lemma below is the core of this paper.The proof is an adaptation of the proof of Lemma 4.1 in [Sil06].
Proof.We argue by contradiction.Let where k is as in Proposition 2. If there is Moreover, for any y ∈ B 1 \ B 3/4 there holds Hence, the maximum of u + kβ in B 1 is attained inside B 3/4 and it is strictly larger than 1.Suppose that the maximum is attained at the point x.
The rest of the proof is devoted to estimating L(u + kβ) (x) from above and from below in order to obtain a contradiction with Proposition 2. At this point, we remark that −kβ + (u + kβ)(x) touches u from above at x. Hence, by Proposition 1, Lu (x) ≤ 0 in the pointwise sense.
We first estimate L(u + kβ) (x) from below.We split the integrals into two parts and write where there is no need for the principal value in the second integral, since x ∈ B 3/4 .Using that u(x) + kβ(x) > 1 is the maximum of u + kβ in B 1 we see that the integrand in I r is non-negative and we have the estimate where Since β ≤ 1 and k ≤ 1/2 we conclude Now we estimate I 2 from below.Using that u(x) + kβ(x) > 1 and u(z Adding the two estimates together we can summarize The next step is to estimate L(u + kβ) (x) from above.This part of the proof is split into two cases: p ≥ 2 and p < 2.
Case 1: p ≥ 2 Again we split the integral defining L(u + kβ) (x) into two parts where again, there is no need for the principal value in the second integral.We first treat I 1 by noting that when x + y ∈ B 1 , we know recalling that u + kβ attains its maximum (in B 1 ) at x.
From Lemma 3 Hence, Now we turn our attention to I 2 .We note that when x + y ∈ B 1 , we cannot apply Lemma 3 directly, but we still have from the hypothesis In other words, By adding the term 2(|2(x + y)| η − 1) > 0 to the the expression, we increase the integrand, and we also make the integrand non-negative so that we can, once more, apply Lemma 3. It follows that Adding the estimates for I 1 and I 2 together we arrive at from which it follows that Finally, we arrive at a contradiction by observing that (3.9) combined with either (3.10) or (3.11) results in a contradiction with (3.1) or (3.2) in Proposition 2.
Once the lemma above is established, the proof of the Hölder regularity is standard.We follow the lines of the proof of Theorem 5.1 in [Sil06].
Proof of Theorem 1.We first rescale u by the factor Then the new u satisfies We will now show that for j = 0, 1, . . .
where α is chosen so that where θ is from Lemma 4 and η is from Proposition 2, with δ = |B 1 |/2.This will imply the desired result with In what follows we will find constants a j and b j so that (3.12) We construct these by induction.For j ≤ 0, (3.12) holds true with b j = inf R n u and a j = b j + 1.
Assume (3.12) holds for all j ≤ k.We need to construct a k+1 and b k+1 .Put where which satisfies the same assumptions as K itself.We also remark for |y| > 1 such that 2 ℓ ≤ |y| ≤ 2 ℓ+1 we have where we have used that (3.12) holds for j ≤ k.Suppose now that |{v ≤ 0} ∩ B 1 | ≥ |B 1 |/2 (if not we would apply the same procedure to −v).Then v satisfies all the assumptions of Lemma 4, with δ = |B 1 |/2 and we obtain where θ = θ(λ, Λ, M, p, s, γ), since δ is fixed.Scaling back to u this yields by our choice of α.Hence, if we let b k+1 = b k and a k+1 = b k + 2 −α(k+1) we obtain (3.12) for the step j = k + 1 and the induction is complete.

Variable exponents
In this section we show that our results also apply to the case when both p and s vary with x.In particular we prove Theorem 2. Throughout this section L denotes the operator We follow the same strategy as in the case of constant exponents and prove slightly modified versions of Proposition 2 and Lemma 4. The proof of Hölder continuity is then similar.Proposition 3. Assume K satisfies K(x, y) = K(x, −y) and there exist Λ ≥ λ > 0, M > 0 and γ > 0 such that where 0 < s 0 < s(x) < s 1 < 1 and 1 < p 0 < p(x) < p 1 < ∞.In the case p(x) < 2 we require additionally that there is τ > 0 such that Then for any δ > 0 there are 1/2 ≥ k > 0 and η > 0 such that for p ∈ (2, ∞) for any x ∈ B 3/4 .Here k and η depend on λ, Λ, M, p 0 , p 1 , s 0 , s 1 , γ, τ and δ.
Proof.We point out the differences to the proof of Proposition 3 and briefly explain how they can be dealt with.
Case 1: p(x) ≥ 2 By the exact same computation as in (3.3), (3.4), (3.5) and (3.6) in the proof of Proposition 2 (since the computation is made for a fixed x), we can conclude that the left hand side is bounded by plus terms involving the quantities Due to the assumptions on p and s, the terms are all uniformly bounded.Thus, if we choose η and k small enough (depending on Λ, M, p 0 , p 1 , s 0 , s 1 and γ) we can make all the terms in the left hand side as small as desired.
For the right hand side, we again have Then it is clear that we can choose η and k, depending only on λ, Λ, M, p 0 , p 1 , s 0 , s 1 , γ and δ, so that the left hand side is larger than the right hand side.
Case 2: 1/(1 − s(x)) < p(x) < 2 We can again estimate the left hand side by as in (3.7) and (3.8).By choosing k small (depending on Λ, M, p 0 , p 1 , τ , γ) we can make both these terms as small as desired.The result follows also in this case.
Proof.The first part of the proof is exactly the same as the one of Proposition 2. Then it comes to estimating L(u + kβ) (x) from below.Since x is a fixed point throughout all the calculations, we obtain as in (3.9)The combination of (4.3) with either (4.4) or (4.5) is a contradiction to (4.1) or (4.2).
Proof of Theorem 2. The proof is very similar to the proof of Theorem 1.We first rescale u by the factor , where ε is chosen as in Lemma 5 with δ = |B 1 |/2.Then one readily verifies that Lu = f in B 2 , f L ∞ (B 2 ) ≤ ε 2 p 1 −1 , osc R n u ≤ 1.Next we proceed as before: we find a j and b j such that Recalling our rescaling factor in the beginning and rescaling back to our original u yields osc Br(x 0 ) u r α , which is the desired result.
y) dy, by Lemma 1.Since β is C 2 , |F | ≤ C|y| and |G| ≤ C|y| 2 .Invoking the upper bound on K in B 2 yields the estimate