A remark on nonlocal Neumann conditions for the fractional Laplacian

We show how nonlocal boundary conditions of Robin type can be encoded in the pointwise expression of the fractional operator. Notably, the fractional Laplacian of functions satisfying homogeneous nonlocal Neumann conditions can be expressed as a regional operator with a kernel having logarithmic behaviour at the boundary.


Introduction
The purpose of this short note is to put in evidence a special feature of the fractional Laplacian when coupled with nonlocal Robin boundary conditions.
By fractional Laplacian we mean the nonlocal operator of order 2s ∈ (0, 2) where c n,s is a positive normalizing constant. We refer to Valdinoci and the author [1] for an introduction to the basic traits of this operator, with particular emphasis on the differences with the classical elliptic theory for the Laplacian, and other possible notions of fractional Laplacian. We underline how the definition in (1) makes sense 1 only for functions defined in all of R n : this yields an associated boundary value problem on some domain Ω ⊂ R n which looks like In this problem, the function g : R n \ Ω → R accounts as a boundary condition of Dirichlet sort. A Neumann boundary condition has been proposed by Dipierro, Ros-Oton, and Valdinoci [4], by means of some nonlocal normal derivative (for which we keep the original notation from the authors) in its normalized version, see [4, equation (1.5)]. With (2), it is possible to study the elliptic boundary value problem 2010 Mathematics Subject Classification: 35S15, 35J99, 47G20. Keywords: fractional Laplacian, nonlocal normal derivative, nonlocal Neumann conditions, regional Laplacian. 1 More precisely, the concerned function u must enjoy some regularity and some growth control at infinity, in order to have the integral finite: we skip here on these important details and we refer the interested reader to [1]. 1 or also the parabolic one We show how the fractional Laplacian of a function satisfying homogeneous Neumann conditions N s u = 0 in R n \ Ω can be reformulated as a regional type operator, i.e., under the form for some suitable measurable kernel K(x, y) : Ω × Ω → R. See equations (9) and (10) below for the detailed expression of the kernel.
The binary operations ∧ and ∨ will respectively denote the "min" and "max" operations between real numbers: In our computations, we will also make use of the particular choice K(x, y) = |x − y| −n−2s in (4), yielding the usually called regional fractional Laplacian Finally, for a measurable β : R n → [0, 1], let x, y ∈ Ω.
We simply write k(x, y) when β = 0 in R n \ Ω.

Main result.
The precise statements go as follows.
A particular form of the mixed boundary condition in (7) (namely, when β is a characteristic function) has been used by Leonori, Medina, Peral, Primo, and Soria [5] to study a principal eigenvalue problem.
In the particular case when β = 1 in R n \ Ω in the above theorem, we have where χ Ω denotes the characteristic function of Ω. Conversely, when β = 0 in R n , we entail the following.
is positive, symmetric, continuous, and for any fixed Moreover, it also holds This last corollary is saying that, when homogeneous nonlocal Neumann conditions are assumed, the fractional Laplacian amounts to be a perturbation of the regional operator (−△) s Ω defined in (5): this nice property becomes particularly interesting when thinking of its stochastic repercussions.
1.3. Stochastic heuristics. The coupling of (−△) s with N s u = 0 has indeed a precise interpretation from the stochastic perspective. We quote from [4]:

"The probabilistic interpretation of the Neumann problem (1.4) [(3) in this note] may be summarized as follows:
(1) u(x, t) is the probability distribution of the position of a particle moving randomly inside Ω. (2) When the particle exits Ω, it immediately comes back into Ω.
(3) The way in which it comes back inside Ω is the following: If the particle has gone to x ∈ R n \ Ω, it may come back to any point y ∈ Ω, the probability density of jumping from x to y being proportional to |x − y| −n−2s . These three properties lead to the equation (1.4) [(3) in this note], being u 0 the initial probability distribution of the position of the particle." We refer to [4, Section 2.1] for further details. The described process might look quite similar to a censored process, as introduced by Bogdan, Burdzy, and Chen [2], or at least to a stable-like process (following the wording of Chen and Kumagai [3]). The class of stable-like processes is the one induced by infinitesimal generators of the type (see [3, equation where the kernel j is supposed to be positive, symmetric, and bounded between two constants In view of Corollary 1.2, and in particular of estimate (11), the process built in [4] does not fall into the stable-like class because of its singular boundary behaviour, although such singularity is rather weak.

Estimates
As a standing hypothesis, we consider here Ω ⊂ R n open, bounded, and with C 1,1 boundary satisfying the interior and the exterior sphere condition.

Lemma 2.1.
There exists c > 0 such that, for any z ∈ R n \ Ω, Proof. As Ω is bounded, there exists R > 0 such that Therefore, using polar coordinates, Also from which we conclude that there exists c > 0 such that To get also the inverse inequality, we split the analysis for d(z) small and large. Indeed for d(z) large one has, by Fatou's Lemma, for d(z) sufficiently large.
Let us turn now to the case d(z) small. Since ∂Ω is compact and smooth, d is a continuous function. For any z ∈ R n \ Ω there exists π(z) ∈ ∂Ω such that d(z) = |π(z) − z| and, by the interior sphere condition, there are w 0 ∈ Ω and r > 0 such that B r (w 0 ) ⊂ Ω and ∂B r (w 0 )∩∂Ω = {π(z)}. So Up to a rotation and a translation, we can suppose z = 0 and w 0 = r + d(z) e 1 . Then As d(z) ↓ 0 we have the convergence for d(z) sufficiently small, from which we conclude the validity of (13).
The following Lemma computes an upper bound on k β .
Then, for any x ∈ Ω, there exists C = C(x) > 0 such that has been taken care of with Lemma 2.1. In order to prove (14), we plug (13) into (6), so that we are left with estimating |x − z| n+2s |y − z| n+2s dz in which we split the integration as follows The second addend is obviously uniformly bounded (above and below away from 0) in x, y ∈ Ω. Therefore, we must concentrate on the first one: since x is fixed inside Ω, we can drop term |x − z| −n−2s . Then, since d(z) < |y − z|, where diam(Ω) = sup{|x − y| : x, y ∈ Ω}.
Finally, this Lemma shows how the upper bound in (14) is optimal. Lemma 2.3. For a measurable β : R n \ Ω → [0, 1] and x, y ∈ Ω let k β be defined as in (6). Suppose that, for some ε > 0, N ε := {β < 1 − ε} is nonempty. Then, for any x ∈ Ω, there exists C = C(x, ε) > 0 such that Proof. To deduce such lower bound, remark that up to a smooth change of variable, the behaviour of the concerned integral (6) defining k β is the same as |d N (y)e 1 + z| n+2s dz which is easily computable as for some c > 0.

Proof of the main results
Proof of Theorem 1.1. In the following, consider a function u : R n → R satisfying (7), which, by (2), can be rewritten for a.e. x ∈ R n \ Ω.