A note on global regularity for the weak solutions of fractional Laplacian equations
Abstract.
We consider a boundary value problem driven by the fractional Laplacian operator with a bounded reaction term. By means of barrier arguments, we prove Hölder regularity up to the boundary for the weak solutions, both in the singular () and the degenerate () case.
Key words and phrases:
Fractional Laplacian, fractional Sobolev spaces, global Hölder regularity.2010 Mathematics Subject Classification:
35D10, 35R11, 47G201. Introduction and main result
In this short note we summarize some new results, whose full proofs are displayed in the forthcoming paper [6]. Our aim is to study regularity of the minimizers of the functional
over the functions such that a.e. in . Here and in the sequel, () is a bounded domain with a boundary , and are real numbers, and . The notion of minimizer of the above functional corresponds to that of a weak solution of the Dirichlettype boundary value problem
(1.1) 
where is the fractional Laplacian operator, defined pointwisely for sufficiently smooth ’s by
(1.2) 
The operator in problem (1.1) is both nonlocal and nonlinear. It embraces, as a special case, the wellknown fractional Laplacian (), as well as nonlinear singular () and degenerate () cases. Interior regularity results are not new for problems of the type (1.1): see for instance [1], [3], [7], and [8]. Most results, providing Hölder regularity even for more general operators, are based on Caccioppolitype or logarithmic estimates, nonlocal Harnack inequalities, and (in the case of [8]) a Krylovtype approach. Interior and boundary regularity results involving fully nonlinear, uniformly elliptic nonlocal operators, are obtained in [2] and [12] respectively. Boundary regularity for degenerate or singular problems such as (1.1), on the other hand, is still a terra incognita. In the linear case with , the global behavior of solutions is well understood. In particular, we focus on the results of [10]: regularity is obtained for the weak solutions, and is proved to be optimal by means of explicit examples, while higher regularity, namely for any , is achieved in the interior. Furthermore, a detailed analysis of the boundary behavior of the solution reveals that is Hölder continuous as well in , where
The study of boundary regularity is particularly important in view of applications to problems with a nonlinear reaction , as it allows to prove fractional versions of the Pohozaev identity [11] and of the BrezisNirenberg characterization of local minimizers in critical point theory [5, 4]. Our longterm aim is to extend to the nonlinear case these latter results, which is also the reason why we focus on weak solutions rather than other types of generalized (e.g. viscosity) solutions. A first, but important, step towards such aim consists in proving global Hölder regularity for problem (1.1). Our main result is the following:
Theorem 1.1.
There exist and , depending only on , , , with also depending on , such that, for all weak solution of problem (1.1), and
Our method differs from those of the aforementioned papers by the fact that we do not use ’hard’ regularity theory, but we prefer to employ rather elementary methods based on barriers, a comparison principle from [9], and a special ’nonlocal lemma’ describing how changes in the presence of a perturbation of supported away from . We shall divide our study in two main steps:

[leftmargin=0.7cm]

Interior regularity: we prove a weak Harnack inequality for positive solutions, then we localize it and develop a strong induction argument to achieve local bounds () with a multiplicative constant which may blow up approaching ;

Boundary regularity: we find an explicit solution for on the halfspace, then, by means of a variable change, we produce an upper barrier near and by comparison we estimate by a multiple of near . Note that, due to the nonlinear nature of the problem, we cannot use fractional Kelvin transform.
Step (b) allows us to stabilize the constant of step (a) as we approach the boundary, thus yielding the conclusion. In view of possible future developments, we remark here that the nonoptimal Hölder exponent is the outcome of interior regularity rather than of the boundary behavior. In fact, it is reasonable to expect interior regularity, which would ensure global regularity at once.
2. Preliminary results and notation
Let be open, not necessarily bounded. First, for all measurable we define the Gagliardo seminorm
then we introduce some Sobolevtype function spaces:
If is unbounded, then the space contains the functions such that for all . The nonlocal tail for a measurable outside a ball is
For all bounded and all , by a weak solution of in we will mean a function such that, for all ,
(2.1) 
(we denote for all ). We remark that the lefthand side of (2.1) is finite and continuous with respect to , since . If is unbounded, is a weak solution if it is so in any . Corresponding notions of weak super and subsolution can be given. Though we are mainly concerned with weak solutions, we also define a notion of strong solution: if , is said to solve strongly in if
(2.2) 
Now we introduce two major tools for our results. The first lemma enlightens a consequence of the nonlocal character of :
Lemma 2.1.
(Nonlocal lemma) Let be a weak (resp. strong) solution of in , with , and satisfy
Set for a.e. Lebesgue point for
Then, and weakly (resp. strongly) in .
Another important tool is the following, whose proof follows almost immediately from [9]:
Theorem 2.2.
(Comparison principle) Let be bounded, and satisfy a.e. in and
for all , in . Then, a.e. in .
Remark 2.3.
The pointwise definition of , even if is smooth, is a delicate issue in the singular case. Roughly speaking, if , whenever the limiting procedure in (2.2) is well defined, so that formula (1.2) makes sense. If , on the other hand, such a representation is possible only for , and in fact explicit examples can be detected, of very smooth functions such that the integral in (1.2) does not converge at a given point. This is a well known drawback of the viscosity solution approach for singular nonlinear equations.
3. Interior regularity
We will address the interior regularity problem with a simple proof peculiar to nonlocal problems. It can be seen as an analogue for ”divergence form” nonlocal equation of the elementary proof of [13], which is restricted to nonlocal operators in ”nondivergence” form. We begin with a weak Harnack inequality for globally nonnegative supersolutions (all balls are intended as centered at , except when otherwise specified).
Theorem 3.1.
(Weak Harnack inequality) Let satisfy
for some . Then,
(3.1) 
with , , depending only on .
Sketch of the proof. For simplicity we consider only the case , and by scaling we can also assume . We produce a lower barrier for , as follows. Pick a cutoff taking values in , such that in , in , and is bounded in in a weak sense. We choose (to be determined later) and set
Applying Lemma 2.1 and the elementary inequality
we get weakly in
By the homogeneity properties of we thus have
weakly in . Therefore either (3.1) is trivial for large due to in , or for suitably small we have
hence by Theorem 2.2 in , and in particular in , which gives (3.1). ∎
Remark 3.2.
It is worth noting that, despite the proof being quite elementary, the constant in the previous weak Harnack inequality degenerates as . This is due to the fact that, in the previous proof,
as , however regular is. This gives and as a consequence, all the following Hölder estimates blow up as . More involved proofs (see e.g. [3]), closely related to the classical regularity approach for local nonlinear variational equations, can however give Hölder estimates which are stable when .
Now we can prove a local Hölder estimate for bounded weak solutions on a ball:
Theorem 3.3.
(Local Hölder regularity) Let satisfy weakly in , . Then
with and depending on , where is the seminorm.
Sketch of the proof. First we need to localize Theorem 3.1, that is, to prove a weak Harnack inequality for supersolutions which are nonnegative in a ball only. If satisfies
then we may apply Lemma 2.1 to (its positive part), producing a tail term depending on (the negative part). Using Theorem 3.1, we see that for all there exists (depending also on ) such that
(3.2) 
again with , depending only on . We then use a strong induction argument to produce two sequences , , with nondecreasing and nonincreasing, such that for all
with depending only on and depending on . This is done by applying (3.2) to the functions , in , where they are both nonnegative, in the inductive step. Then, we obtain the following oscillation estimate for all :
(3.3) 
with depending on . A standard argument then provides the claimed estimate. ∎
4. Boundary regularity and conclusion
In this final section we turn back to weak solutions of (1.1). First, by applying Theorem 2.2 to and a multiple of the weak solution of
we prove the following:
Theorem 4.1.
(A priori bound) Let satisfy weakly in . Then
with depending on , and .
Now we produce a local upper barrier. We set , .
Lemma 4.2.
There exists , , , such that
Sketch of the proof. We divide our argument in four steps:

[leftmargin=0.7cm]

We find an explicit solution on a halfspace: namely, belongs in and satisfies both strongly and weakly in .

For all big enough we find a diffeomorphism such that outside a ball, mapping to and the plane to locally around , and satisfying further .

We prove stability of under changes of variable of the type above: setting , we see that weakly in , for some with .

We truncate at a convenient heigth , i.e., we set , and we apply Lemma 2.1 to obtain weakly in ().
By scaling and translating , we find as required. ∎The barrier is used to prove an estimate of weak solutions, near , by means of a multiple of :
Theorem 4.3.
Let satisfy weakly in . Then for a.e.
with depending on and .
Sketch of the proof. We may reduce to the case . By Theorem 4.1, the desired estimate is easily obtained away from . Due to the reguarity of , we can find such that in the set
decreases linearly on segments normal to . Fix and denote its unique metric projection on . Let be as in Lemma 4.2. By scaling and translating , we construct such that on the line segment , and moreover
with a small . By Theorem 2.2 we see that . An analogous argument applies to , yielding the conclusion. ∎
Remark 4.4.
As a byproduct, by arguments analogous to those displayed in steps (a)–(c) above, we prove that, for convenient , we have both weakly and strongly in . In fact, it can be proved that for some , which is an interesting information, as it shows that the boundary behavior of weak solutions in the general case is similar to that in the linear case (see [10, Lemma 3.9]).
We are now ready to conclude: Sketch of the proof of Theorem 1.1. Set , so weakly in . By Theorem 4.1, we only need to prove our estimate on the Hölder seminorm. Recalling Theorem 3.3, by a covering argument we find (depending only on ) and, for all , a constant (depending also on ) such that and . Let be as in the proof of Theorem 4.3. For all let . Theorem 3.3 produces the following estimate:
(4.1) 
The first term in the righthand side of (4.1) is bounded due to , . For the second term we invoke Theorem 4.3 and the inequalities for to obtain
Finally, the tail term is bounded by means of Theorem 4.3 again, together with Hölder continuity of , thus we have from (4.1)
with depending on and . Patching together the above estimates, we reach the conclusion. ∎
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