Guy Barles

Christine A Georgelin

Espen R Jakobsen

ON NEUMANN AND OBLIQUE DERIVATIVES BOUNDARY CONDITIONS FOR NONLOCAL ELLIPTIC EQUATIONS

Keywords: Mathematics Subject Classification. 35R09 (45K05), 35B51, 35D40 Nonlocal Elliptic equation, Neumann-type boundary conditions, general nonlocal operators, reflection, viscosity solutions, Lévy process

Inspired by the penalization of the domain approach of Lions & Sznitman, we give a sense to Neumann and oblique derivatives boundary value problems for nonlocal, possibly degenerate elliptic equations. Two different cases are considered: (i) homogeneous Neumann boundary conditions in convex, possibly non-smooth and unbounded domains, and (ii) general oblique derivatives boundary conditions in smooth, bounded, and possibly non-convex domains. In each case we give apropriate definitions of viscosity solutions and prove uniqueness of solutions of the corresponding boundary value problems. We prove that these boundary value problems arise in the penalization of the domain limit from whole space problems and obtain as a corollary the existence of solutions of these problems.

Introduction

Inspired by the penalization of the domain approach of Lions & Sznitman [START_REF] Lions | Stochastic Differential Equations with reflectiong Boundary conditions Com[END_REF] (see also [START_REF] Lions | Sznitman Construction de processus de diffusion réfléchis par pénalisation du domaine[END_REF][START_REF] Menaldi | Robin Reflected Diffusion Processes with Jumps[END_REF]), we give a sense to Neumann and oblique derivatives boundary value problems for nonlocal degenerate elliptic partial integro-differential equations (PIDEs in short). Because of the nonlocal nature of our PIDEs posed in a domain Ω, the boundary conditions have then to be imposed not only at the boundary ∂Ω, but possibly in all of the complement Ω c . At least boundary conditions must be imposed in the union of the supports of the jump measures (see below).

To be more specific, we consider PIDEs of the form

F (x, u, Du, D 2 u, I[u](x)) = 0 in Ω, (1.1)
with extended Neumann/oblique derivatives boundary conditions

Du(x) • γ(x) = g(x) in Ω c . (1.2)
Here Ω is a domain in R N and F is a real-valued, continuous function defined on

R N × R × R N × S N × R
, where S N is the space of N × N symmetric matrices. We assume that F is degenerate elliptic which, in this nonlocal setting, means that for any x ∈ R N , u ∈ R, p ∈ R N , M, N ∈ S N and l 1 , l 2 ∈ R,

F (x, u, p, M, l 1 ) ≤ F (x, u, p, N, l 2 ) when M ≥ N, l 1 ≥ l 2 , (1.3)
where M ≥ N has to be understood in the sense of the usual partial ordering on symmetric matrices. Our assumptions cover the cases of general linear (see example below) and nonlinear equations and, in particular, Bellman-Isaacs equations of control and game theory.

The equation is nonlocal because of its dependence on the nonlocal operator I[u] which we assume to be of Lévy-Ito type. For any smooth bounded function φ and for any x ∈ R N ,

I[φ](x) = R N
[φ(x + j(x, z)) -φ(x) -Dφ(x) • j(x, z)1 B (z)] dµ(z), (1.4) where B ⊂ R N is the unit ball, 1 B the indicator function of B, µ -the Lévy measure -is a positive Radon measure on R M \ {0} and j : R N × R M → R N is a function which is µ-measurable in z and continuous in x for µ-a.e. z, and there exists a constant c(j) > 0 such that A Taylor expansion shows that I[φ] is well-defined under (1.5). We will assume that (1.5) holds throughout this paper. The operator I is the generator of a stochastic jump process which solves a stochastic differential equation involving a general jump term/Poisson random measure, cf. [START_REF] Gihman | Stochastic Differential Equations[END_REF][START_REF] Pham | Optimal stopping of controlled jump diffusion processes: a viscosity solution approach[END_REF]. Included are generators of all pure jump Levy processes [START_REF] Applebaum | Lévy Processes and Stochastic Calculus[END_REF] as well most Levy models arising in Finance [START_REF] Cont | Financial modelling with jump processes[END_REF].

A typical example of Equation (1.1) is the linear equation a(-∆)

α 2 u -Tr(A(x)D 2 u) -b(x) • Du + λ(x)u = f (x) in Ω, (1.6)
where a ∈ R, A, b, λ, f are continuous functions on Ω, taking values respectively in S N , R N , (0, +∞) and R, and, for α ∈ (0, 2), I = -(-∆) α 2 is the Fractional Laplacian in R N , the generator of the symmetric α-stable processes. It is defined e.g. by (1.4) with j(x, z) ≡ z and dµ(z) = c α dz |z| N +α for some constant c α > 0. To satisfy the (degenerate) ellipticity condition (1.3), we must impose that both a ≥ 0 and A ≥ 0 in Ω.

It is the definition of I (cf. (1.4)) that requires u to be defined in all of R N , and hence that the boundary condition must be posed in all of Ω c .

For the boundary (or exterior) condition (1.2), we assume that (BC1) The functions γ : R N → R N and g : R N → R are bounded Lipschitz continuous functions, and there exists ν > 0 such that γ(x) • n(x) ≥ ν for any x ∈ ∂Ω.

(BC2) For any x ∈ Ω c , τ x := inf t>0 {X x (t) ∈ Ω} < +∞, where X x (•) solves X x (0) = x and Ẋx (t) = -γ(X x (t)) for t > 0.

(1.7)

Assumption (BC1) is sufficient for (1.2) to really play the role of a boundary condition in the case of local equations. Assumption (BC2) states that integral curves of the vector field -γ starting from any point x ∈ Ω c , will reach the boundary ∂Ω in finite time. This is a natural condition for a Neumann type boundary condition, and it is very closely related to the idea of the "penalization of the domain" method of Lions & Sznitman [START_REF] Lions | Stochastic Differential Equations with reflectiong Boundary conditions Com[END_REF] (see also [START_REF] Lions | Sznitman Construction de processus de diffusion réfléchis par pénalisation du domaine[END_REF][START_REF] Menaldi | Robin Reflected Diffusion Processes with Jumps[END_REF]). This method is based on the observation that in the limit κ → 0, the vector field -1 κ γ instantaneously returns the underlying stochastic process to Ω after an outside jump, and this is where (1.7) plays a role. We refer to Section 5 for more details in this direction. [START_REF] Lions | Stochastic Differential Equations with reflectiong Boundary conditions Com[END_REF], we use the notion of viscosity solutions. For nonlocal equations posed in full space, we refer to [START_REF] Barles | Imbert Second order elliptic integro-differential Equations: viscosity solutions's theory revisited[END_REF] (see also [START_REF] Barles | Backward stochastic differential equations and integral-partial differential equations[END_REF][START_REF] Jakobsen | A Maximum principle for semicontinuous functions applicable to integro-partial differential equations Nonlinear Differential Equations and Applications[END_REF][START_REF] Pham | Optimal stopping of controlled jump diffusion processes: a viscosity solution approach[END_REF]) and references therein for an account of this theory. A nonlocal Dirichlet problem was considered in [START_REF] Barles | On the Dirichlet Problem for Second-Order Elliptic Integro-Differential Equations[END_REF], where boundary conditions are given in all of Ω c in an analogous way as in this paper. Here we consider two different cases: (i) a very general class of equations with homogeneous Neumann boundary conditions posed in convex, possibly non-smooth and unbounded domains, and (ii) a less general class of equations with general oblique derivatives boundary conditions in smooth, bounded, and possibly non-convex domains. In each case we give appropriate definitions of viscosity solutions and prove uniqueness theorems for the corresponding boundary value problems. Here we want to point out that the extended oblique derivative condition (1.2) influences the behavior of the solutions at infinity and therefore interferes in the conditions which are needed to have a well-defined nonlocal operator; we discuss this point at the end of Section 2. We also show that our formulation follows from a sequence of problems posed in the whole space obatined from the penalization of the domain method in a similar way as in [START_REF] Lions | Stochastic Differential Equations with reflectiong Boundary conditions Com[END_REF]. As a consequence we also get some existence results for our problems.

As in Lions & Snizman

In a related paper [START_REF] Barles | On Neumann type problems for nonlocal equations set in a half space[END_REF], the authors along with E. Chasseigne investigate four different ways of understanding homogeneous Neumann boundary conditions for Levy-type nonlocal equations posed in the half space Ω = R N -1 × R + . Here, the simple geometry of the domain allows formulations where the nonlocal operators and equations are restricted to Ω. One of the cases, the one involving normal projection of outside jumps, is linked to the present work. The restriction to Ω formulation of [START_REF] Barles | On Neumann type problems for nonlocal equations set in a half space[END_REF] is technically more difficult to work with than the very natural full space formulation we use in this paper. This also explains why we now obtain much more general results for the normal/oblique projection type of models. Where we in [START_REF] Barles | On Neumann type problems for nonlocal equations set in a half space[END_REF] considered linear, non-degenerate problems with a restricted class of Levy operators on a simple domain, we can now treat very general equations, Levy operators, and domains, and inhomogeneous and even oblique boundary conditions.

In some cases covered in this paper, a probabilistic description of the reflection problems based on stochastic differential equations can be found in [START_REF] Mazumdar | Forward Equations for Reflected Diffusions with Jumps[END_REF]. Elsewhere in the literature similar problems have been investigated for Lévy operators where the measure µ x forces the underlaying process to stay in the domain either by a "smooth" restriction of its support or by "killing" all jumps leaving Ω. In these cases a Neumann boundary condition can be imposed only at the boundary ∂Ω, just as for local problems. The first type of problems is considered in [START_REF] Menaldi | Robin Reflected Diffusion Processes with Jumps[END_REF], see also the book [START_REF] Garroni | Second order elliptic integro-differential problems[END_REF], and the killing approach is linked to the α-censored process [START_REF] Bogdan | Censored stable processes[END_REF] and the regional fractional Laplacian [START_REF] Guan | Integration by parts formula for regional fractional Laplacian[END_REF][START_REF] Guan | Boundary problems for fractional Laplacian[END_REF][START_REF] Guan | Reflected symmetric α-stable processes and regional fractional Laplacian[END_REF].

When the underlying process is a symmetric α-stable processes (a subordinated Brownian motion), the above mentioned approaches follows after a "reflection" on the boundary: The processes can be constructed from a Brownian motion by first subordinating it and then reflecting it. Another possible way to construct a "reflected" process it to first reflect the Brownian motion and then subordinate the reflected process. This approach is related to Dirichlet-Neumann operator, and it have been described e.g. by Hsu [START_REF] Hsu | On the excursions of reflecting Brownian motion[END_REF] using probabilistic methods and by Caffarelli and Silvestre [START_REF] Caffarelli | An extension problem related to the fractional Laplacian[END_REF] by analytic PIDE methods. Especially the ideas of [START_REF] Caffarelli | An extension problem related to the fractional Laplacian[END_REF] have been used by many authors since.

In all these approaches, as well as in [START_REF] Barles | On Neumann type problems for nonlocal equations set in a half space[END_REF], the non-local operator is no longer the orginal R N -operator. In fact the operator and hence also the equation will depend on the domain, and different domains Ω yield different operators and different equations inside Ω.

Our paper is organized as follows: Section 2 is devoted to a key technical lemma which allows us to control the solutions outside the domain Ω. We also discuss the connections between the extended oblique derivative condition (1.2), the behavior of the solutions at infinity, and the conditions which are needed to have a well-defined nonlocal operator. In Section 3, we will focus on convex possibly nonsmooth and unbounded domains but restrict ourselves to homogeneous boundary conditions. We define the concept of viscosity solution and proof a comparison theorem. The key argument here is to obtain by convexity a contraction property that force maximum points of the test function to be in Ω. After this, the proof can be concluded in the standard (full space) way. In Section 4, we prove a comparison theorem in the case of general oblique derivative conditions and smooth bounded possibly nonconvex domains. The proof uses the complicated test function constructed by G. Barles in [START_REF] Barles | Nonlinear Neumann Boundary Conditions for Quasilinear Degenerate Elliptic Equations and Applications[END_REF], along with the technical lemma of Section 2. Finally, Section 5 is devoted to the analysis of an asymptotic result -the penalization of the domain method introduced by Lions and Sznitman. We prove that the above boundary value problems arise in the penalization of the domain limit of whole space problems and obtain as a corollary existence results for our Neumann problems.

Preliminary Results

In this section we state and prove two lemmas which play key roles in the proofs in the next sections. We recall from (BC2) that τ y := inf t>0 {X y (t) ∈ Ω} for y ∈ Ω c .

Lemma 2.1. Assume (BC1) and (BC2).

(a) If u is a locally bounded, usc function satisfying Du(x) • γ(x) ≤ g(x) in Ω c in the viscosity sense, then, for any y ∈ Ω c and for any t ≤ τ y , we have

u(y) ≤ t 0 g(X y (s))ds + u(X y (t)). (2.1) (b) If v is a locally bounded, lsc function satisfying Dv(x) • γ(x) ≥ g(x) in Ω c in
the viscosity sense, then, for any y ∈ Ω c and for any t ≤ τ y , we have

v(y) ≥ t 0 g(X y (s))ds + v(X y (t)). (2.2)
Proof. The proof is inspired by an argument of G. Barles, S. Mirrahimi, B. Perthame and P.E. Souganidis [START_REF] Barles | Singular Hamilton-Jacobi equation for the tail problem[END_REF]. We only prove (a) since the proof of (b) is similar. Let t ∈ (0, τ y ] and define w(s) := u(X y (ts)) for s ≥ 0.

If we can prove that the usc function w is a subsolution of

dw ds (s) = g(X y (t -s)) for s > 0, (2.3) Ω x X x (t) X x (τ x )
Figure 1. The curve X x (t) of (BC1) and the hitting time τ x .

then by the comparison principle w(s) ≤ w(0) + s 0 g(X y (tτ ))dτ for s > 0, since the right-hand side is the C 1 -solution of (2.3) with same initial data w(0). Now (2.1) follows by choosing s = t.

To prove that w is a supersolution of (2.3), we take any smooth test-function ϕ and any point s > 0 such that wϕ has a local strict maximum point at s. Note that X y (ts) ∈ Ω c since s > 0. Next we introduce the functions

φ(x, s) = |X y (t -s) -x| 2 ε 2 + ϕ(s) and ψ(x, s) = u(x) -φ(x, s).
For ε > 0 small enough, classical arguments show that ψ has a maximum point near (X y (ts), s) (depending on ε) that we also call (x, s). Moreover |X y (ts) -x| 2 ε 2 → 0 and s → s as ε → 0.

(2.4) Therefore x ∈ Ω c for ε small enough, and since u is a subsolution of (1.2) and uφ(•, t) has a local maximum at x,

- 2(X y (t -s) -x) ε 2 • γ(x) ≤ g(x).
Since τ → ψ(x, τ ) is a C 1 -function having a local maximum at τ = s, we also have

2(X y (t -s) -x) ε 2 • Ẋy (t -s) -φ(s) = ∂ψ ∂t (x, s) = 0,
and we can conclude using Ẋy (ts) = -γ(X y (ts)) and Lipschitz continuity of

γ that φ(s) = - 2(X y (t -s) -x) ε 2 • γ(X y (t -s)) ≤ - 2(X y (t -s) -x) ε 2 • γ(x) - 2(X y (t -s) -x) ε 2 • (γ(X y (t -s)) -γ(x)) ≤ g(x) + O |X y (t -s) -x| 2 ε 2 . (2.5)
In view of (2.4), we can send ε → 0 to find that φ(s) ≤ g(X y (ts)).

To allow for convex domains with corners and γ = n, we need to relax the Lipschitz assumption on γ in (BC1) and impose only a one-sided Lipschitz condition (BC1') The functions γ : Ω

c → R N and g : Ω c → R are bounded continuous functions, (γ(x) -γ(y)) • (x -y) ≥ -K|x -y| 2 , (2.6)
for some constant K and for all x, y ∈ Ω c , and there exists ν > 0 such that

γ(x) • n(x) ≥ ν for any x ∈ ∂Ω.
We state a slight generalization of Lemma 2.1.

Lemma 2.2. The results of Lemma 2.1 remain valid if we replace (BC1) by (BC1').

Proof. We first remark that (BC1') ensures the existence and uniqueness of the trajectory X y as long as it remains in Ω c , existence follows from Peano's Theorem while (2.6) provides the uniqueness. Therefore the trajectory exists on [0, τ y ) and can be extended by continuity to t = τ y .

To complete the proof, we let t < τ y and redo the proof of Lemma 2.1. The computations are exactly the same, e.g. to obtain (2.5), (BC1') is sufficient. To extend the result to t = τ y , we just remark that lim sup t↑τy u(X y (t)) ≤ u(X y (τ y )) , by the upper-semicontinuity of u.

We conclude this section by an important discussion on the consequences of Lemma If g ≡ 0 on Ω c , then u can be bounded under suitable assumptions on the (other) data. This is the case we face in Section 3 below for convex domains and extended homogeneous Neumann boundary conditions. But if g is not identically 0 on Ω c , then u can be unbounded and its growth is governed by the properties of g and τ y . The behavior of u at infinity is important in our framework to insure that the nonlocal terms are well-defined: We need some integrability property like e.g., for any x ∈ Ω and δ > 0,

2.1. If u is a solution of (1.1)-(1.2), then Du(x) • γ(x) = g(x) in Ω
|z|≥δ |u(x + j(x, z))| dµ(z) < +∞.
(2.7)

Such condition now connects the assumptions we have to place on τ x , g, j and µ.

To fix ideas, we are going to assume in Section 5.2 that: (BC3) Either the function g has a compact support in R N or there exists c > 0 such that for τ x from (BC2),

τ x ≤ c(1 + |x|) and sup x∈Ω |z|≥δ |j(x, z)| dµ(z) < ∞ for any δ > 0 . (2.8)
We briefly comment on this assumption. When g has compact support, the solutions are expected to be bounded by Lemma 2.1 and no additional assumption on j and µ is needed. On the contrary, if e.g. g ≡ 1, then the integral of g in (2.1) suggests that u and τ x have the same growth and (BC3) imposes a linear growth. Next one has to impose suitable hypothesis on j and µ to satisfy (2.7). This is obtained through the second part of (BC3) on the µ-integrability of j away from 0.

The homogeneous Neumann condition in convex non-smooth domains

In this section we consider the homogeneous Neumann problem, namely equation (1.1) and boundary condition (1.2) with g ≡ 0 and γ = n, the unit outward normal vector field in Ω c (see below)

Du(x) • n(x) = 0 in Ω c , (3.1)
in the case when Ω is a convex, possibly unbounded and non-smooth domain. At x ∈ ∂Ω, the set of outward normals N Ω (x) can be defined as

N Ω (x) := n ∈ R N : |n| = 1, n • (x -y) ≥ 0 for all y ∈ Ω .
This set is a singleton at any point where ∂Ω is C 1 and part of a convex cone where ∂Ω has a corner. Let d be the distance function to Ω, and note that d ≡ 0 in Ω and d > 0 in Ω c . Moreover, d is convex and belongs to

C 0 (R N ) ∩ C 1 (Ω c ) since Ω is a closed convex subset of R N .
In Ωc , we now define the outward unit normal vector n in the only natural way by setting n = D d. Note that the two definitions are consistent in the sense that

N Ω (x) = n ∈ R N : n = lim k→∞ D d(x k ) for some x k → x for all x ∈ ∂Ω,
and since d is convex, the function n satisfies (BC1') with K = 0.

To define the concept of viscosity solutions for this problem, we need the operators I δ , I δ , and F defined as follows

I δ [φ](x) = |z|<δ φ(x + j(x, z)) -φ(x) -Dφ(x) • j(x, z)1 B (z) dµ(z), I δ [u](x) = |z|≥δ u(x + j(x, z)) -u(x) -Dφ(x) • j(x, z)1 B (z) dµ(z), F[u, φ](x) = F (x, u(x), Dφ(x), D 2 φ(x), I δ [φ](x) + I δ [u](x)).
Under assumption (1.5), I δ [φ] is well-defined for φ ∈ C 2 . For I δ [u] we need some integrability condition a la (2.7), see the discussion at the end of Section 2. In this section g ≡ 0 on Ω c , so (2.7) will be automatically satisfied whenever u is bounded.

Definition 3.1. (i) A locally bounded, usc function u : R N → R is a viscosity subsolution of (1.1)-(1.
2) if it satisfies (2.7), and for any test function φ ∈ C 2 (R N ) and for any maximum point

x 0 ∈ R N of u -φ in B c(j)δ (x 0 ) where c(j) is defined in (1.5), we have            F[u, φ](x 0 ) ≤ 0 if x 0 ∈ Ω, min F[u, φ](x 0 ), inf n∈NΩ(x0) Dφ(x 0 ) • n ≤ 0 if x 0 ∈ ∂Ω, Dφ(x 0 ) • n(x 0 ) ≤ 0 if x 0 ∈ Ω c (ii) A locally bounded, lsc function v : R N → R is a viscosity supersolution of (1.1)-(1.
2) if it satisfies (2.7), and for any test function φ ∈ C 2 (R N ) and for any minimum point x 0 ∈ R N of the function uφ in B c(j)δ (x 0 ) where c(j) is defined in (1.5), we have

           F[u, φ](x 0 ) ≥ 0 if x 0 ∈ Ω, max F[u, φ](x 0 ), sup n∈NΩ(x0) Dφ(x 0 ) • n ≥ 0 if x 0 ∈ ∂Ω, Dφ(x 0 ) • n(x 0 ) ≥ 0 if x 0 ∈ Ω c (iii) A viscosity solution u of (1.1)-(1.
2) is a locally bounded function whose upper and lower semicontinuous envelopes are respectively sub-and supersolution of the problem.

This definition is a natural extension of the definition given in [START_REF] Barles | On the Dirichlet Problem for Second-Order Elliptic Integro-Differential Equations[END_REF] to the Neumann type boundary value problem. 2) In the subsolution definition, (Dφ(x 0 ), D 2 φ(x 0 )) can be replaced by elements (p, X) in the so-called super-jet

J + u(x 0 ) if X ≤ D 2 φ(x 0 ). In the definition of supersolutions, you can similarly use (q, Y ) ∈ J -u(x 0 ) if Y ≥ D 2 φ(x 0 )
. The second definition is useful for comparison proofs, and the proofs of these claims easily follow from the arguments for similar results in [START_REF] Barles | Imbert Second order elliptic integro-differential Equations: viscosity solutions's theory revisited[END_REF].

We now state the assumptions -remarking that the assumptions on F will be as general as for the whole space case Ω = R N without boundary conditions. For convenience we use the assumptions of [START_REF] Barles | Imbert Second order elliptic integro-differential Equations: viscosity solutions's theory revisited[END_REF], but see Remark 3.6 below for more general assumptions. For the nonlocal part we assume that (A1) Assumption (1.5) holds, and there is a constant c > 0 such that for all

x, y ∈ R N ,

R N |j(x, z) -j(y, z)| 2 |x -y| 2 + R N \B |j(x, z) -j(y, z)| |x -y| µ(dz) ≤ c.
The non-linearity F satisfies the following classical assumptions (A2) There exists λ 0 > 0 such that for any

x ∈ R d , u, v ∈ R, p ∈ R d , X ∈ S N and l ∈ R, F (x, u, p, X, l) -F (x, v, p, X, l) ≥ λ 0 (u -v) when u ≥ v.
(A3-1) F is continuous, and for any R > 0, there exist moduli of continuity ω, ω R such that, for any |x|, |y| ≤ R, |v| ≤ R, l ∈ R and for any X, Y ∈ S N satisfying

X 0 0 -Y ≤ 1 ε I -I -I I + r(β) I 0 0 I (3.2)
for some ε > 0 and r(β) → 0 as β → 0, then, if s i (β) → 0 as β → 0 for i = 1 and 2, we have

F (y, v, ε -1 (x -y) + s 1 (β), Y, l) -F (x, v, ε -1 (x -y) + s 2 (β), X, l) ≤ ω(β) + ω R (|x -y| + ε -1 |x -y| 2 ). (3.3) (A3-2) For any R > 0, F is uniformly continuous on R n × [-R, R] × B R × D R × R
where D R := {X ∈ S N ; |X| ≤ R} and there exist a modulus of continuity ω R such that, for any x, y ∈ R d , |v| ≤ R,l ∈ R and for any X, Y ∈ S N satisfying (3.2) and ε > 0, we have

F (y, v, ε -1 (x -y), Y, l) -F (x, v, ε -1 (x -y), X, l) ≤ ω R (ε -1 |x -y| 2 + |x -y|). (3.4) (A4) F (x, u, p, X, l
) is nondecreasing and Lipschitz continuous in l, uniformly with respect to all the other variables.

(A5) M F := sup x∈Ω |F (x, 0, 0, 0, 0)| < ∞.

Assumption (A3-1) and (A3-2) are two versions of assumption (3.14) in the Users' Guide [START_REF] Crandall | User's guide to viscosity solutions of second order Partial differential equations[END_REF] for possibly unbounded domains. These assumptions along with (A4) imply that equation (1.1) is degenerate elliptic. Assumptions (A3-1) allows more general x-dependence in the equation (e.g. HJB equations with at most linear growth in the derivatives and general x-depending coefficients), while (A3-2) allows more general gradient dependence in the equation (e.g. HJB equations with coefficients which are bounded in x but possibly with x-independent superlinear gradient terms).

To be more explicit, consider the linear equation (1.6). The above assumptions hold if a ≥ 0, A(x) = σ(x)σ T (x) for some matrix σ, and, for (A2), λ(x) ≥ λ 0 > 0 in R N . Assumptions (A3-1) and (A3-2) are the following two variants of conditions on σ, b, λ, f : (A3-1) is satisfied if σ and b are bounded and locally Lipschitz continuous and λ and f are continuous. For (A3-2), σ, b can have a linear growth but one needs the global Lipschitz continuity of σ and b, and the uniform continuity of λ and f .

In the local case with no I-dependence in the equation, assumptions (A2) -(A5) imply comparison, uniqueness, and existence (via Perron's method) of a bounded viscosity solution of (1.1)-(1.2), cf. e.g. [START_REF] Crandall | User's guide to viscosity solutions of second order Partial differential equations[END_REF]. In the nonlocal case when Ω = R N (and no Neumann conditions, Ω c = ∅), we have the following rather classical result which we will need later.

in Ω = R N satisfying |u(x)| ≤ M F λ 0 in R N . (3.5)
Part (a) was proved in [START_REF] Barles | Imbert Second order elliptic integro-differential Equations: viscosity solutions's theory revisited[END_REF] (see Section 5), and Part (b) follows from part (a) and Perron's method since M F /λ 0 and -M F /λ 0 are super and subsolutions of (1.1). Similar results have been given e.g. in [START_REF] Barles | Backward stochastic differential equations and integral-partial differential equations[END_REF][START_REF] Pham | Optimal stopping of controlled jump diffusion processes: a viscosity solution approach[END_REF][START_REF] Jakobsen | A Maximum principle for semicontinuous functions applicable to integro-partial differential equations Nonlinear Differential Equations and Applications[END_REF][START_REF] Barles | On the Dirichlet Problem for Second-Order Elliptic Integro-Differential Equations[END_REF].

Now we come to the first main result of this paper, a comparison result for the boundary value problem Uniqueness of solutions follow, and since ± M F λ0 are sub/super solutions of (1.1) when (A5) holds, we also get L ∞ -bounds. Corollary 3.5. Assume (A1), (A2), (A4) hold along with either (A3-1) or (A3-2).

(a) There is not more than one bounded solution of (1.1).

(b) If also (A5) holds, then any solution u of (1.1) satisfies (3.5).

Remark 3.6. Under assumption (A3-1), the above results also holds if assumption (A1) is replaced by the much more general assumption: (A1-2) Assumption (1.5) holds and there exists a constant c > 0 such that

B |j(x, z) -j(y, z)| 2 µ(dz) ≤ c|x -y| 2 .
The proof in Section 5 in [START_REF] Barles | Imbert Second order elliptic integro-differential Equations: viscosity solutions's theory revisited[END_REF] can be modified easily to cover this case by a clever trick which can be found e.g. in Section 6 in [START_REF] Jakobsen | A Maximum principle for semicontinuous functions applicable to integro-partial differential equations Nonlinear Differential Equations and Applications[END_REF]. Compared to assumption (A1), assumption (A1-2) allows more general dependences of j(x, z) in x. If we also relax (1.5) so that the constant c(j) is finite only for compact subsets of x ∈ Ω, then the above results also cover the case when j has linear growth in x. Note that the projection on the closed convex set Ω, P : Ω c → ∂Ω, is also given by

P y = X y (τ y ) for any y ∈ Ω c ,
where we recall that τ y = inf{t > 0 : X y (t) ∈ ∂Ω}. Since Ω is convex and |n| = 1, it follows that {X y (•)} y defines a family of constant speed, finite length, and non-intersecting paths in Ω c having the form

X y (t) = y -tn(y) for t ∈ [0, τ y ]. (3.7)
Obviously τ y < ∞ for all y so that (BC2) is trivially satisfied when γ = n.

We argue by contradiction assuming that

M := sup R N {u(x) -v(x)} > 0.
Since n satisfies (BC1'), Lemma 2.2 applies with g ≡ 0 and we find that u(x)v(x) ≤ u(P x ) -v(P x ) in Ω c , and hence that M = max

Ω {u(x) -v(x)}.
Let χ : R → R be a bounded smooth function such that χ(t) ≡ 0 for t ≤ 0, χ (t) > 0 for t > 0, and χ(t) ≥ 2(||u|| ∞ + ||v|| ∞ ) for t ≥ 1. We double the variables, introducing the function

Ψ(x, y) = u(x) -v(y) - |x -y| 2 ε 2 -χ(β(|x -x 0 | 2 + 1)) -χ(β(|y -x 0 | 2 + 1)),
where ε, β > 0, and x 0 is any given point in Ω. It is easy to see that, for β small enough, M ε,β = max R 2N Ψ(x, y) exists and is attained at some point (x, ȳ) ∈ R N × R N (that depends on ε and β). The crucial and new step in the proof is to show that (x, ȳ) ∈ Ω × Ω. If this was not the case, then two applications of Lemma 2.2 yields that

u(x) -v(ȳ) ≤ u(P x) -v(P ȳ ). (3.8)
Moreover, since Ω is convex and

x 0 ∈ Ω, |x -ȳ| ≥ |P x -P ȳ |, |x -x 0 | ≥ |P x -x 0 |, and |ȳ -x 0 | ≥ |P ȳ -x 0 |, (3.9)
and then, for β small enough, we have the contradiction

M ε,β = Ψ(x, ȳ) < Ψ(P x, P ȳ ). (3.10)
Since x, ȳ ∈ Ω, the rest of the proof follows classical arguments. Assume x ∈ ∂Ω and let

φ(x, y) = |x -y| 2 ε 2 + χ(β(|x -x 0 | 2 + 1)) + χ(β(|y -x 0 | 2 + 1)).
Note that by convexity of Ω,

(x -y) • n ≥ 0 for all y ∈ Ω, n ∈ N Ω (x).
Moreover, this inequality is strict if y ∈ Ω. Finally, since χ (t) > 0 for t > 0, we use the fact that x 0 ∈ Ω to find that

D x φ(x, ȳ) • n = 2 ε 2 (x -ȳ) • n + χ (β(|x -x 0 | 2 + 1))2β(x -x 0 ) • n > 0
for all n ∈ N Ω (x) and β > 0.

(3.11) Therefore, from Definition 3.1, the equation has to hold at x, i.e.

F[u, φ(•, ȳ)](x) ≤ 0. A similar argument shows that F[v, -φ(x, •)](ȳ) ≥ 0 if ȳ ∈ ∂Ω.
Now we are in the situation that x, ȳ ∈ Ω and that the equation is satisfied at these points. The conclusion of the proof is then exactly as for the R N case, and we omit the standard details. Under the present assumptions, essentially all the remaining details can be found in Section 5 in [START_REF] Barles | Imbert Second order elliptic integro-differential Equations: viscosity solutions's theory revisited[END_REF]. But see also [START_REF] Barles | Backward stochastic differential equations and integral-partial differential equations[END_REF][START_REF] Jakobsen | A Maximum principle for semicontinuous functions applicable to integro-partial differential equations Nonlinear Differential Equations and Applications[END_REF][START_REF] Pham | Optimal stopping of controlled jump diffusion processes: a viscosity solution approach[END_REF] for very similar results.

Remark 3.7. The key ingredients of the above proof are (i) Inequality (3.8) that comes from Lemma 2.1 or 2.2 and that allow us to compare values of u and v outside Ω with those on inside Ω.

(ii) Inequality (3.10) that comes from convexity and contraction properties (see (3.9)). In the above proof, the contraction property of the projection on the closed, convex set Ω was playing the key role (allowing us to use a very simple test function), but in general the contraction property comes from the control on the X y trajectories w.r.t. y.

(iii) As in the classical Neumann/oblique derivatives boundary conditions cases, the test-function has to be build in order to allow us to "avoid" the boundary condition (cf. (3.11)).

These three ingredients are the same in any proof but with different arguments to handle them. We are going to focus on these arguments.

Remark 3.8. If Ω is bounded, we can relax assumptions (A3-1) and (A3-2) in the standard way and the comparison result will still hold. E.g. since we no longer need to prevent maximum points from escaping to infinity, we can set all functions r, s 1 , s 2 and ω equal zero in (A3-1).

General oblique derivative conditions in non-convex smooth domains

In this section we consider the general oblique derivative problem of the form (1.1)-(1.2) on a bounded, possibly non-convex , C 2 -domain Ω. Compared to section 3, the domain and boundary condition and are more general, but the class of equations (see below) and the boundary regularity are more restricted.

Assuming that (1.5) and (BC1) hold, and we now have the following definition of viscosity solutions Definition 4.1. (i) A locally bounded, usc function u : R N → R is a viscosity subsolution of (1.1)-(1.2) if it satisfies (2.7), and for any test function φ ∈ C 2 (R N ) and for any maximum point x 0 ∈ R N of uφ in B c(j)δ (x 0 ) where c(j) is defined in (1.5),

           F[u, φ](x 0 ) ≤ 0 if x 0 ∈ Ω, min F[u, φ](x 0 ), Dφ(x 0 ) • γ(x 0 ) -g(x 0 ) ≤ 0 if x 0 ∈ ∂Ω, Dφ(x 0 ) • γ(x 0 ) ≤ g(x 0 ) if x 0 ∈ Ω c (ii) A locally bounded, lsc function v : R N → R is a viscosity supersolution of (1.1)-(1.
2) if it satisfies (2.7), and for any test function φ ∈ C 2 (R N ) and for any minimum point

x 0 ∈ R N of the function u -φ in B c(j)δ (x 0 ) where c(j) is defined in (1.5),            F[u, φ](x 0 ) ≥ 0 if x 0 ∈ Ω, max F[u, φ](x 0 ), Dφ(x 0 ) • γ(x 0 ) -g(x 0 ) ≥ 0 if x 0 ∈ ∂Ω, Dφ(x 0 ) • γ(x 0 ) ≥ g(x 0 ) if x 0 ∈ Ω c (iii) A viscosity solution u of (1.1)-(1.
2) is a locally bounded function whose upper and lower semicontinuous envelopes are respectively sub-and supersolution of the problem.

To handle non-convex domains and more general boundary conditions, we will use a rather complicated test-function which is no longer only a function of x-y plus small terms. For the proofs to work out we therefore need to replace assumption (A3-1) and (A3-2) by a more restrictive assumption similar to the one used in the local case [START_REF] Barles | Nonlinear Neumann Boundary Conditions for Quasilinear Degenerate Elliptic Equations and Applications[END_REF] (A3-3) For any R, K > 0, there exist moduli of continuity m R,K such that, for any

x, y ∈ Ω, |u| ≤ R, p, q ∈ R N , l ∈ R, and matrices X, Y ∈ S N satisfying |x -y| ≤ ηε, |p -q| ≤ Kηε(1 + |p| ∧ |q|), and
- K ε 2 Id ≤ X 0 0 -Y ≤ K ε 2 Id -Id -Id Id + KηId,
we have that

F (y, u, q, Y, l) -F (x, u, p, X, l) ≤ m R,K η + |x -y|(1 + |p| ∨ |q|) + |x -y| 2 ε 2 .
We have the following comparison result. This result will be proved in the subsections below. We start by introducing the test function we need for the proof.

4.1.

The test-function. As for local oblique derivative boundary conditions (see e.g. [START_REF] Barles | Nonlinear Neumann Boundary Conditions for Quasilinear Degenerate Elliptic Equations and Applications[END_REF] and references therein), the proof of our comparison result requires a rather complicated test-function. Fortunately there are no major differences between the test-function for the local and nonlocal cases, and we now recall a few facts about the test-function of [START_REF] Barles | Nonlinear Neumann Boundary Conditions for Quasilinear Degenerate Elliptic Equations and Applications[END_REF] and describe the adaptations we need to make here.

We start by changing our definition of the "distance to the boundary" d. Now d will be a bounded C 2 function which is equal to the signed distance function to ∂Ω in a neighborhood of ∂Ω (d > 0 in Ω and d < 0 in Ω c ) and where n(x) := -Dd(x) = 0 in Ω c . Note that n(x) is the outward unit normal vector to ∂Ω for any x ∈ ∂Ω.

The test-function ψ ε,η ∈ C 2 (R 2N ) of [START_REF] Barles | Nonlinear Neumann Boundary Conditions for Quasilinear Degenerate Elliptic Equations and Applications[END_REF] can then be defined as follows,

ψ ε,η (x, y) = e -K1[d(x)+d(y)] |x -y| 2 ε 2 -C ηε x + y 2 , e -K1[d(x)+d(y)] 2(x -y) ε 2 d(x) -d(y) (4.1) + e -K1[d(x)+d(y)] A d(x) -d(y) 2 ε 2 -K 2 ηε [d(x) + d(y)] ,
for parameters η, ε > 0 (small), constants A, K 1 , K 2 (large), and where the function C ηε (see [START_REF] Barles | Nonlinear Neumann Boundary Conditions for Quasilinear Degenerate Elliptic Equations and Applications[END_REF] page 214) is a suitable smooth approximation of a bounded Lipschitz extension of the solution t = C(x, p) of the equation

γ(x) • (p + tn(x)) -g(x) = 0 for p ∈ R N , x near ∂Ω.
The key properties of the test-function are given in the Lemma below.

Lemma 4.3. Assume (BC1) and let R > 0. If η, ε > 0 are small enough, then for A, K 1 , K 2 large enough, then the function ψ ε,η defined in (4.1) has the following properties (i) For any x, y ∈ R N ,

ψ ε,η (x, y) ≥ K -1 |x -y| 2 ε 2 -Kε 2 -K 2 ηε [d(x) + d(y)]. (4.2)
(ii) For ε, η ∈ (0, 1) and |x -y| ≤ ηε,

|D x ψ ε,η (x, y)| + |D y ψ ε,η (x, y)| ≥ -K + K -1 |x -y| ε 2 , |D x ψ ε,η (x, y)| + |D y ψ ε,η (x, y)| ≤ C |x -y| ε 2 + C 1 + η 2 K 1 e 2K1 d ∞ + εηK 2 , (4.3
)

|D x ψ ε,η (x, y) + D y ψ ε,η (x, y)| ≤ K |x -y| 2 ε 2 + K(ηε + ε 2 ), and
K ε 2 Id ≤ D 2 ψ ε,η (x, y) ≤ K ε 2 Id -Id -Id Id + KηId. (4.4)
(iii) There is δ > 0 such that for |x -y| ≤ δ and x, y in a neighborhood of ∂Ω,

γ(x) • D x ψ ε,η (x, y)) > g(x) if d(x) ≤ d(y) , (4.5)
-γ(y) • D y ψ ε,η (x, y)) < g(y) if d(y) ≤ d(x) , (4.6)
and if in addition |x -y| ≤ ηε, then

-γ(x) • D x ψ ε,η (x, y) + D y ψ ε,η (x, y) ≤ -K 1 ν 4 e -K1[d(x)+d(y)] |x -y| 2 ε 2 -K 2 ν 4 ηε. (4.7)
Except for (4.7), these estimates have essentially been proved in Section 5 in [START_REF] Barles | Nonlinear Neumann Boundary Conditions for Quasilinear Degenerate Elliptic Equations and Applications[END_REF]. Some new features that only marginally changes the proofs are: (i) x, y can now belong to Ω c , (ii) inequality (4.2) is slightly more accurate, and (iii) inequalities (4.5) and (4.6) are now given in a neighborhood and not only at ∂Ω. Moreover, the constants K will in general depend on K 1 and K 2 , and the precise dependence is not important except for the term (4.3). The importance of this dependence is both new and central to this paper (cf. the proof of Lemma 4.4 a)). We will therefore prove both (4.3) and (4.7) here.

Proof of (4.3) and (4.7). To simplify the computations, we write ψ ε,η in the following way

ψ ε,η (x, y) = χ(x -y, d(x) -d(y), x + y 2 , d(x) + d(y)) ,
where

χ(X, Y, Z, T ) := e -K1T X 2 ε 2 -C ηε Z, e -K1T 2X ε 2 Y + e -K1T AY 2 ε 2 -K 2 ηεT .
In this notation,

D x ψ ε,η (x, y) = χ X -χ Y n(x) + 1 2 χ Z -χ T n(x), D x ψ ε,η (x, y) + D y ψ ε,η (x, y) = -χ Y (n(x) -n(y)) + χ Z -χ T (n(x) + n(y)).
By the assumptions on γ and g and the construction of C ηε = C ηε (x, p) in [START_REF] Barles | Nonlinear Neumann Boundary Conditions for Quasilinear Degenerate Elliptic Equations and Applications[END_REF], there is a C > 0 such that

|C εη | + |D x C εη | ≤ C(1 + |p|) and |D p C εη | ≤ C.
Hence there are constants C 1 and C 2 such that

|χ X | + |χ Y | ≤ C 1 + C 2 e -K1T 2|X| ε 2 + 2(1 + A)|Y | ε 2 , |χ Z | ≤ C 1 + C 2 e -K1T 2|X| ε 2 |Y |, |χ T | ≤ K 1 e -K1T C 2 X 2 ε 2 + (1 + A)Y 2 ε 2 + K 2 ηε. Since |X|, |Y | ≤ C|x -y|, estimate (4.
3) now follows.

To prove (4.7), we note that by using Cauchy-Schwarz inequality on the D p C εηterm and taking A large enough,

χ T = -K 1 e -K1T X 2 ε 2 -D p C ηε • 2X ε 2 Y + AY 2 ε 2 -K 2 ηε ≤ - K 1 2 e -K1T X 2 ε 2 + AY 2 ε 2 -K 2 ηε.
Let W = {x : dist(x, ∂Ω) < r}, and let r > 0 be so small that γ • n ≥ ν 2 in W. Such a set exists by (BC1) and continuity of γ and n. After an easy computation based on the above estimates, the Lipschitz continuity of n (|n(x) -n(y)| ∼ |X|), the inequality γ • n ≥ ν 2 , Cauchy-Schwarz inequality, and finally, taking K 1 , K 2 large enough so that the χ T -term dominates, we conclude that (4.7) holds in W.

The next lemma plays a key role in the comparison proof. Lemma 4.4. Assume (BC1) and (BC2), let τ x be defined in Lemma 2.1, and τ := min(τ x , τ y ).

(a) For any K ≥ 0, there are constants K 1 , K 2 large enough, such that for any ε, η > 0 small enough, if x, y ∈ Ω c are close enough to ∂Ω and |x -y| ≤ ηε/2, then

ψ ε,η (X x (τ ), X y (τ )) ≤ ψ ε,η (x, y) -Kτ ηε. (4.8)
(b) For any η > 0, there are constants K 1 , K 2 large enough, such that for any ε > 0, if x, y ∈ Ω c are close enough to ∂Ω and τ y ≤ τ x , then

ψ ε,η (X x (τ x ), X y (τ )) ≤ ψ ε,η (X x (τ ), X y (τ )) - τx τy g(X x (t))dt. (4.9)
(c) For any η > 0, there are constants K 1 , K 2 large enough, such that for any ε > 0, if x ∈ Ω c and y ∈ Ω are close enough to ∂Ω, then

ψ ε,η (X x (τ x ), y) ≤ ψ ε,η (x, y) - τx 0 g(X x (t))dt.
Proof. Consider a neighborhood of ∂Ω, W r = {x : dist(x, ∂Ω) < r}, and let r > 0 be so small that (4.7) holds, d(x) = ± dist(x, ∂Ω), and γ • n ≥ ν 2 in W r . Such a set exists by the definition of d, (BC1), and continuity of γ and n. In the set W r ∩ Ω c the distance to boundary

f (t) = dist(X x (t), ∂Ω) = -d(X x (t)) is decreasing, ḟ (t) = -Dd(X x (t)) • Ẋx (t) = n(X x (t)) • (-γ(X x (t))) < - ν 2 , (4.10)
Ω ȳ x X ȳ (τ ȳ ) X x(τ ȳ ) X x(τ x)
Figure 3. Curves of (BC1) with different starting points in the oblique case. and hence X x (t) ∈ W r ∩ Ω c for all t ∈ [0, τ x ] and x ∈ W r ∩ Ω c .

Next we note that if L is the Lipschitz constant of γ, then by Grönwall's inequality,

|X x (t) -X y (t)| ≤ e Lt |x -y|. (4.11)
We estimate τ x , and hence also τ y and τ , by integrating (4.10) from t to τ x and noting that f

(τ x ) = 0 ν 2 (τ x -t) < f (t) = dist(X x (t), ∂Ω) ≤ dist(x, ∂Ω) for t ∈ [0, τ x ].
Hence if r is small, τ will also be small in W r ∩ Ω c . In the rest of the proof we take x, y ∈ W r ∩ Ω c , and then we take r so small that also |X x (t) -X y (t)| ≤ ηε for all t ∈ [0, τ ] and x, y ∈ W r ∩ Ω c such that |x -y| ≤ ηε 2 . We now prove part (a). We start by using the definition of X x (t) (see (BC2)) to show that

d dt [ψ ε,η (X x (t), X y (t))] = -D x ψ ε,η • γ(X x (t)) -D y ψ ε,η • γ(X y (t)) = -[D x ψ ε,η + D y ψ ε,η ] • γ(X x (t)) -D y ψ ε,η • [γ(X y (t)) -γ(X x (t))].
We may use (4.3) (check!) and the Lipschitz continuity of γ to have

|D y ψ ε,η • [γ(X y (t)) -γ(X x (t))]| ≤ L|X x (t) -X y (t)| • C |X x (t) -X y (t)| ε 2 + 1 + η 2 K 1 e K12 d ∞ + εηK 2 ≤ LC |X x (t) -X y (t)| 2 ε 2 + ηε 1 + εηK 2 + η 2 K 1 e 2K1 d ∞ ,
and by (4.7) we immediatly find that

-[D x ψ ε,η + D y ψ ε,η ] • γ(X x (t)) ≤ -K 1 ν 4 e -K1[d(Xx(t))+d(Xy(t))] |X x (t) -X y (t)| 2 ε 2 -K 2 ν 4 ηε. Since γ • n ≥ ν 2 , we then find that d dt [ψ ε,η (X x (t), X y (t))] ≤ LC -K 1 ν 4 e -K1[d(Xx(t))+d(Xy(t))] |X x (t) -X y (t)| 2 ε 2 + LC 1 + εηK 2 + η 2 K 1 e 2K1 d ∞ -K 2 ν 4 ηε ≤ -Kηε
for any given constant K since we can take first ε, η small enough and then K 1 and finally K 2 as large as we want. The conclusion follows by integrating from 0 to τ .

To prove (b), we notice that τ = τ y ≤ τ x . Since Ẋ = -γ(X) and d(X x (t)) ≤ d(X y (τ )) = 0 for τ = τ y ≤ t ≤ τ x , we can use (4.5) to find that

d dt [ψ ε,η (X x (t), X y (τ ))] = -D x ψ ε,η • γ(X x (t)) ≤ -g(X x (t)).
Part (b) now follows by integrating from τ y to τ x . The proof of (c) is just like the proof of (b) replacing X y (τ ) by y and setting τ = 0.

4.2. Proof of Theorem 4.2. In order to show that u(x) -v(x) ≤ 0 in R N , we first notice that, by (BC2) and Lemma 2.1,

u(x) -v(x) ≤ u(X x (τ x )) -v(X x (τ x )) for any x ∈ Ω c ,
and hence since X x (τ x ) ∈ ∂Ω, it follows that uv is bounded from above in R N and M = sup

R N {u(x) -v(x)} = max Ω {u(x) -v(x)}.
In the rest of the proof we argue by contradiction assuming that M > 0.

Then we define

w β (x) = u(x) -v(x) -2χ(-βd(x)),
where β > 0 (small), χ is the function we introduced in the proof of Theorem 3.4, and d is the signed distance function to ∂Ω (d < 0 in Ω c ). Since the χ-term vanishes on Ω and is strictly positive on Ω c , w β has maximum points only on Ω and these points are also maximum points of uv. Now we double the variables introducing the function

Φ(x, y) = u(x) -v(y) -ψ ε,η (x, y) -χ(-βd(x)) -χ(-βd(y)).
By standard arguments involving the definition of χ and the properties of ψ ε,η given in Lemma 4.3 (in particular (4.2)), this function achieves its maximum at a point (x, ȳ) ∈ R N × R N (depending on ε, η and β). Moreover, for fixed η and β, |x -ȳ| 2 ε 2 → 0 as ε → 0, and x, ȳ converges (along subsequences) to a maximum point x of w β (x), i.e. to a point in Ω. In particular, x, ȳ will be arbitrarily close to ∂Ω if ε close enough to 0. We will show that x, ȳ are in Ω when ε > 0 is small enough. Again we argue by contradiction assuming that x, ȳ are not both in Ω. Assume e.g. that x, ȳ ∈ Ω c and that τ ȳ ≤ τ x. We will get a contradiction to the maximum point property by showing that Φ(x, ȳ) < Φ(X x(τ x), X ȳ (τ ȳ )). To do this, we start by using Lemma 2.1 for both u and v to see that

Φ(x, ȳ) ≤ u(X x(τ x)) -v(X ȳ (τ ȳ )) + τȳ 0 g(X x(s)) -g(X ȳ (s)) ds + τx τȳ g(X x(t))dt -ψ ε,η (x, ȳ) -χ(-βd(x)) -χ(-βd(ȳ)).
But from Lemma 4.4, using first part (b) and then part (a),

τx τȳ g(X x(t)) dt ≤ ψ ε,η (X x(τ ȳ ), X ȳ (τ ȳ )) -ψ ε,η (X x(τ x), X ȳ (τ ȳ )) ≤ ψ ε,η (x, ȳ) -2 Kτ ηε -ψ ε,η (X x(τ x), X ȳ (τ ȳ )),
and by Lipschitz regularity of g and γ and the estimate (4.11), τȳ 0 g(X x(s)) -g(X ȳ (s)) ds ≤ τ ȳ L g e Lγ τȳ |x -ȳ|.

Hence we find that Φ(x, ȳ)

≤ Φ(X x(τ x), X ȳ (τ ȳ )) -2 Kτ ȳ ηε + τ ȳ L g e Lγ τȳ |x -ȳ| -χ(-βd(x)) -χ(-βd(ȳ)),
and since |x -ȳ| ≤ ηε, we get the contradiction by choosing K large enough. A similar argument covers the case when τ ȳ ≥ τ x, and we can conclude that at least one of x and ȳ belongs to Ω.

Next we show that it is not possible that e.g. x ∈ Ω c while ȳ ∈ Ω. This time we use Lemma 2.1 for only u to see that

Φ(x, ȳ) ≤ u(X x(τ x)) -v(ȳ) + τx 0 g(X x(t))dt -ψ ε,η (x, ȳ) -χ(-βd(x)).
But by Lemma 4.4(c),

τx 0 g(X x(t))dt -ψ ε,η (x, ȳ) ≤ -ψ ε,η (X x(τ x), ȳ),
and hence we find again a contradiction

Φ(x, ȳ) ≤ Φ(X x(τ x), ȳ) -χ(-βd(x)) < Φ(X x(τ x), ȳ).
The case that ȳ ∈ Ω c while x ∈ Ω gives a contraction in a similar way, and in view of previous arguments we can conclude that x, ȳ ∈ Ω, at least when ε > 0 is small enough. Since ψ ε,η satisfies by (4.5) and (4.6), it follows that the equation (the sub and supersolution inequalites), and not the boundary condition, has to hold if x or ȳ belongs to ∂Ω and hence for all x, ȳ ∈ Ω. By assumption, u, v are bounded on Ω so that assuption (A3-3) can be applied with R = max Ω (|u| + |v|). At this point we can conclude the proof as in the R N -case, sending first ε → 0, then η → 0, and finally β → 0. We omit the standard details only noting that under the present assumptions, essentially all the remaining details can be found in Section 5 in [START_REF] Barles | Imbert Second order elliptic integro-differential Equations: viscosity solutions's theory revisited[END_REF]. But see also [START_REF] Jakobsen | A Maximum principle for semicontinuous functions applicable to integro-partial differential equations Nonlinear Differential Equations and Applications[END_REF][START_REF] Barles | Backward stochastic differential equations and integral-partial differential equations[END_REF][START_REF] Pham | Optimal stopping of controlled jump diffusion processes: a viscosity solution approach[END_REF] for very similar results.

Note that F = F and

G(x, r, p, X, l) =      F (x, r, p, X, l) when x ∈ Ω, min{F (x, r, p, X, l), inf n∈NΩ(x) n • p} when x ∈ ∂Ω, n(x)p when x ∈ Ωc ,
and in a similar way we find that G is like G with max/sup replacing the min/inf. As a consequence of the stability of viscosity solutions, see e.g. Theorem 1 in [START_REF] Barles | Imbert Second order elliptic integro-differential Equations: viscosity solutions's theory revisited[END_REF], u = lim sup * u κ is a viscosity subsolution of

G(x, u, Du, D 2 u, I[u]) = 0 in R N , while u = lim sup * u κ is a viscosity supersolution of G(x, u, Du, D 2 u, I[u]) = 0 in R N .
By Definition 3.1 this means that u and u are sub-and supersolutions of (1.1)-(1.2), and hence by comparison, Theorem 3.4,

u ≤ u.
The opposite inequality is true by definition of u, and hence we have u = u =: u. It follows that u is continuous and u κ → u locally uniformly, as is standard in viscosity solution theory.

5.2.

Oblique boundary value problems in bounded smooth domains. In this section, we assume as in Section 4, that Ω is a bounded C 2 domain. We study the following equation for the penalization of the domain, cf. [START_REF] Lions | Stochastic Differential Equations with reflectiong Boundary conditions Com[END_REF]:

F (x, u κ , Du κ , D 2 u κ , I[u κ ](x)) + 1 κ d(x)[γ(x) • Du κ -g] = 0 in R N . (5.4)
where 0 < κ 1 and d is defined as in the previous section. We want to prove that we can obtain the oblique boundary value problem (1.1) from the penalized problem (5.4) in the limit as κ → 0. In (1.1) (Definition 4.1), only F 's values at Ω play any role, and we may modify equation (5.4) in Ωc and still obtain (1.1) from (5.4) in the limit as long as (A1)-(A5) still hold.

In order to avoid difficulties related to comparison results for sub and supersolutions, we assume that F (x, u, p, M, l) ≡ λ 0 u for x large enough, say for |x| ≥ R, where λ 0 is given by (A2). Taking into account the fact that the truncation on the distance function implies that d(x) ≡ 1 for x large enough, the equation outside a large enough ball reduces to

λ 0 u κ + 1 κ [γ(x) • Du κ -g] = 0 ,
which can be treated by a slight adaptation of the technics used in Section 2 as we will see it later on. For other extensions of F , additional conditions are typically needed to handle the growth (typically linear) of the solutions at infinity. Here it is unavoidable to impose additional assumptions on γ, g, j, µ to satisfy the integrability assumption (2.7), i.e. to balance the growth u(x + j(x, •)) with the decay of µ at infinity for solutions u of (1.1) and (5.4). We are going to use (BC3) and refer the reader to the discussion at the end of Section 2.

We just recall that, in the case when g has compact support, the solutions are expected to be bounded by Lemma 2.1 and no additional assumption on j and µ is needed. On the contrary, if, for example, g ≡ 1, then the integral of g in (2.1) suggests a linear growth and one has to impose suitable hypothesis on γ, j and µ in order to satisfy (2.7). Moreover, if we were considering more general extension of F , we would need a framework where we can compare sub and supersolutions with linear growth. Our restrictive extension allow us to avoid such (useless) technicalities.

Theorem 5.4. Assume that (A1)-(A5) and (BC1)-(BC3) hold. Then, for any κ > 0, there exists a unique continuous viscosity solution u κ of (5.4) which is uniformly locally bounded. Moreover, as κ → 0, u κ converges locally uniformly to the unique viscosity solution u of (1.1)-(1.2).

In the proof we use the following lemma. We prove this result after the proof of Theorem 5.4.

Proof of Theorem 5.4. We just sketch the proof of the existence and uniqueness of u κ when g is not compactly supported. This case involves the function θ of Lemma 5.5 while the other case is easier and involves a similarly defined but bounded function θ (where Dθ • γ > 1 only on a compact set).

The strong comparison principle (and hence uniqueness) for (5.4) holds by standard argument and a slight modification of the argument of Section 2 that we explain now. If u is a subsolution of (5.4) then we have κλ 0 u κ + γ(x) • Du κ -g = 0 in Bc R , where R is defined above, B R is the ball centered at 0 with the (large) radius R. A slight modification of the arguments of Section 2 shows that, if y ∈ Bc R and if X y (s) ∈ Bc R for s ∈ [0, t) then u κ (y) ≤ t 0 g(X y (s)) exp(-κλ 0 s)ds + u κ (X y (t)) exp(-κλ 0 t).

Using this result, we can reduce to the case where the maximum points are in a fixed compact subsets of R N and then classical comparison arguments apply. Using Lemma 5.5 and (A2), it is easy to check that, choosing first C 2 > 0 and then C 1 large enough, ±(C 1 + C 2 θ(x)) are respectively viscosity super and subsolutions of (5.4). Then we can apply Perron's method to obtain the exisitence of a solution u κ such that

-(C 1 + C 2 θ(x)) ≤ u κ (x) ≤ C 1 + C 2 θ(x) in R N .
Since the u κ 's are locally uniformly bounded, we can use the half-relaxed limits method. We rewrite (5.4) in the following equivalent way as G κ (x, u, Du, D 2 u, I[u](x)) = 0 in R N where G κ (x, r, p, X, l) = F (x, r, p, X, l), for x ∈ Ω, κ d(x) F (x, r, p, X, l) + γ(x)pg, for x ∈ Ωc .

we have to regularize locally and use a covering argument to build the global regularization of w. The covering argument is completely standard and will not be detailed here. we can take ε = ε K so small that

γ • Dw ε K ≥ 1 in K.
Finally, the bound on |θ| follows directly from a similar bound for w and a suitable (local) choice of ε. The bound for w is a direct consequence of Assumption (BC3) and Lemma 2.1.

R N |z| 2 ∧

 2 1 dµ(z) < ∞ and |j(x, z)| ≤ c(j)|z| for any |z| < 1, x ∈ R N . (1.5)

  y (s))ds + u(X y (τ y )).

Remark 3 . 2 .

 32 Two useful equivalent definitions can be given: (1) We can replace I δ [u] by I δ [φ] in the above definition if local maximum/minimum points are replaced by global ones. (

Proposition 3 . 3 (

 33 Results for Ω = R N ). Assume Ω = R N and (A1), (A2), (A4) hold along with either (A3-1) or (A3-2).(a) If u and v are respectively an usc bounded above subsolution and a lsc bounded below supersolution of(1.1) in Ω = R N , then u ≤ v in R N .(b) Assume also (A5) holds, then there exists a unique bounded viscosity solution u of (1.1)

  (1.1)-(3.1). Theorem 3.4 (Comparison I). Assume (A1), (A2), (A4) hold along with either (A3-1) or (A3-2). If u and v are respectively an usc bounded above subsolution and a lsc bounded below supersolution of (1.1)-(3.1), then u ≤ v in R N .

Ω y 1 y 2 Figure 2 .

 122 Figure 2. The curves of (BC1) with γ = n starting at y 1 and y 2 and ending at a corner point of ∂Ω.

Theorem 4 . 2 (

 42 Comparison II). Assume (A1), (A2), (A3-3), (A4), (BC1), and (BC2) hold. If u and v are respectively a locally bounded usc subsolution and a locally bounded lsc supersolution of (1.1)-(1.2), then u ≤ v in R N .

Lemma 5 . 5 .

 55 Assume (BC1)-(BC3). There exists a C∞ function θ : R N → R such that γ(x) • Dθ(x) ≥ 1 for x in a neighborhood W of Ω c . Moreover θ satisfies |θ(x)| ≤ c(1 + |x|) in R N , for some c > 0.

2 where 0 2 . 2 -δ 2 .

 20222 Locally we define w ε (x) = w * ρ ε (x) for x ∈ D δ < ε < δ 2 and ρ ε (x) is the standard mollifier, i.e. a positive C ∞ -function with mass one and support in |x| < ε. By the regularity of w, Dw exists a.e. and hence equation (5.5) holds a.e. It follows that (Dw • γ) * ρ ε = 2 in D δ By the definition of the convolution and of ρ ε , the Lipschitz continuity of γ, and the local boundedness of Dw, we are lead toDw ε • γ(x) = (Dw • γ) * ρ ε (x) + Dw(y) • (γ(y) -γ(x))ρ ε (xy) dy ≥ Dw L ∞ (B(x,ε)) L γ ε in DHence for any bounded subset K ⊂ D δ 2

This work was supported by the Research Council of Norway (NFR) through the project "Integro-PDEs: Numerical methods, Analysis, and Applications to Finance".

Penalization of the domain

In this section we show that our way of defining Neumann type boundary conditions is consistent with the so-called penalization of the domain method introduced by Lions and Sznitman in [START_REF] Lions | Stochastic Differential Equations with reflectiong Boundary conditions Com[END_REF]. We extend the results of [START_REF] Lions | Stochastic Differential Equations with reflectiong Boundary conditions Com[END_REF] to our non-local setting, proving the convergence of a sequence of solutions of penalized R N -problems to the solution of (1.1). We give separate results in the convex case of Section 3 and the oblique case of Section 4. (extended by 0 to Ω) satisfies (2.6) in R N . This property will play a key role below. Moreover, we assume that (A1)-(A5) hold, and if necessary, we extend the data and F to R N in a way that preseverves these properties. We study the following equation for the penalization of the domain, cf. [START_REF] Lions | Stochastic Differential Equations with reflectiong Boundary conditions Com[END_REF]

where 0 < κ 1. Since d(x)n(x) satisfies (2.6), Equation (5.1) with κ > 0 fixed satisfies (A1)-(A5) as long as F does.

Theorem 5.1. Assume that (A1), (A2), (A4), (A5) hold along with either (A3-1) or (A3-2). Then the viscosity solution u κ of (5.1) converge locally uniformly to a bounded continuous function u which is the viscosity solution of (1.1)-(1.2) according to Definition 3.1.

Remark 5.2. This result provides an existence result for (1.1)-(1.2). In contrast to the more difficult Dirichlet case in [START_REF] Barles | On the Dirichlet Problem for Second-Order Elliptic Integro-Differential Equations[END_REF], we have existence also when the there is loss of boundary conditions.

We need the following auxilliary result that follows from Proposition 3.3. Lemma 5.3. Assume that (A1), (A2), (A4), (A5) hold along with either (A3-1) or (A3-2). Then there exists a unique bounded viscosity solution u κ of (5.1) satisfying

Proof. Note that u κ is bounded uniformly in κ, and that we may rewrite (5.1) in the following equivalent way

where

(5.3)

Now we introduce the half relaxed limits

As in the proof of Theorem 5.1, we compute the half relaxed limits and find that

and that G is like G with a max replacing the min, and we find that u is a viscosity subsolution of the G(x, u, Du, D We prove that w is locally Lipschitz continous in D δ if δ is so small that by (BC1),

We first check that w is Lipschitz continuous in ∆ δ . Let x, y ∈ ∆ δ , f (t) := d(X x (t+ τ δ y )), and note that if τ x > τ y , then

• n(X x (t + τ δ y )) for t ∈ (0, τ x -τ y ). We integrate from 0 to τ δ x -τ δ y and use (BC1) to find that

where the last inequality is a consequence of the definition of the distance of the point X x (τ δ y ) to the boundary. Then if L is the Lipschitz constant of γ, inequality (4.11) holds and we may use e.g. (BC3) to obtain that

where R = max x∈∆ δ |x|. It follows that w is Lipschitz in ∆ δ . Let x, y ∈ D δ \ ∆ δ be near one another and take a T > 0 such that X x (T ) ∈ ∆ δ . Such T exists and T ≤ c(1 + |x|) by (BC3). By inequality (4.11), we can (and do) take y close enough to x so that also X y (T ) ∈ ∆ δ . Then τ δ x = T + τ δ Xx(T ) and τ δ y = T + τ δ Xy(T ) , and hence by (BC3) and inequalities (5.


A Bennis

R Niemiec

C Brousseau

K Mahdjoubi

O Emile

Flat plate for OAM generation in the millimeter band

Keywords: orbital, angular, momentum, phaseplate, millimeter frequency band I

The use of electromagnetic waves carrying orbital angular momentum in the radio frequency range is a growing subject, and so is the design of antennas able to produce such waves. The aim of this paper is to present a new type of antenna (a flat plate) in order to generate an electromagnetic wave carrying orbital angular momentum in the millimeter frequency band.

INTRODUCTION

An ElectroMagnetic (EM) wave is usually defined by its magnitude, its wave vector, its frequency and its angular momentum. The latter can be decomposed in two distinct components [START_REF] Jackson | Classical Electrodynamics[END_REF]: the Spin Angular Momentum (SAM) which corresponds to the wave polarization and the Orbital Angular Momentum (OAM) which is associated to the spatial distribution of the phase of the fields. So, an EM wave carrying OAM is characterized by an azimuthal phase dependency noted e -jl , with l being an integer value called the topological charge and corresponding to the phase rotational mode, and the azimuthal angle. Furthermore, the phase distribution leads to a destructive interference at the beam axis, and creates a vortex.

Where polarization states are limited to only two states, each l integer value is associated to an OAM mode. Thus there is theoretically an infinite number of OAM modes. Furthermore, all these modes are orthogonal to each other and independents to the wave polarization. This orthogonality has been proposed as a new possible diversity, e.g. in telecommunications, assuming the existence of dedicated suitable transmissions systems to generate these OAM modes. This idea has received recently some attention for radio communications [START_REF] Thidé | Utilization of photon orbital angular momentum in the low-frequency radio Domain[END_REF][START_REF] Mohammadi | Orbital Angular Momentum in Radio -A System Study[END_REF][START_REF] Tamburini | Encoding many channels on the same frequency through radio vorticity: first experimental test[END_REF], to create "multiple sub-channels" distinguished by the twisting degree of the EM field structure to encode information. Since, many intense debates have been conducted comparing OAM coding and MIMO techniques, and concluding that the "OAM sub-channels" is a subset of the solutions offered by MIMO [START_REF] Edfors | Is orbital angular momentum (OAM) based radio communications an unexploited area?[END_REF]. At the same time, some interference tests between an OAM beam and an unknown object have been realized in the optical domain [START_REF] Uribe-Patarroyo | Object Identification Using Correlated Orbital Angular Momentum (OAM) States[END_REF] to estimate rotational symmetries, opening new perspectives in radar identification in the radio frequency band.

In this paper, we present a new type of antenna aimed to generate an OAM EM wave, using a flat phase plate with a variable permittivity, at a frequency f = 30 GHz and for an OAM of order l = 1.

II. ANTENNAS GENERATING WAVES CARRYING OAM

In the radio frequency bands, circular short dipole arrays have been proposed to generate OAM of order |l| < N/2, N being the number of antennas [START_REF] Mohammadi | Orbital Angular Momentum in Radio -A System Study[END_REF]. Each array element has a 2 l/N phase shift compared with its neighbor. This leads to a 2 l phase rotation around the beam axis. Others antennas have been used to generated OAM in these bands, such as a turnstile antenna [START_REF] Niemiec | Orbital angular momentum induced torque on a macroscopic system[END_REF]. The difference in this case is that OAM lies in the plane of the turnstile antenna only, and not ahead of it. On the other hand, spiral phase plates have been used in optics for some years, and more recently in the 100 GHz regime [START_REF] Turnbull | The generation of free-space Laguerre-Gaussian modes at millimetrewave frequencies by use of a spiral phaseplate[END_REF]. The spiral surface introduces a linear phase variation along the azimuthal angle , depending on the helical thickness of the plate.

Leaving aside the theoretical aspect, the first experimental demonstration of the orthogonality of OAM modes in radio frequency and in natural environment has been reported last year [START_REF] Tamburini | Encoding many channels on the same frequency through radio vorticity: first experimental test[END_REF]. Besides the confirmation that OAM can indeed be used for telecommunications, the relevant point in this article relies on the antenna the authors used to generate an EM wave carrying OAM. Starting from a regular parabolic reflector dish, they "twisted" it in order to induce the azimuthal phase distribution. This antenna is the first one ever used in a telecommunication demonstration system involving OAM.

III. FLAT PHASE PLATE WITH ANGULAR VARIATION OF THE PERMITTIVITY

A. Design of the flat phase plate

This approach we used to design this phase plate is similar to that of the spiral phase plate, in the sense that the wave is subject to a phase delay upon transmission through the phase plate. The innovation here is the use of a flat plate which induces a phase delay by an angular variation of the permittivity due to holes drilled in matter, as seen in Fig. 1, instead of being due to an angular width variation. This concept has been used for the first time for spherical lenses, using radials drilled holes [START_REF] Zimmerman | Luneberg lens and method of constructing same[END_REF], and then for similar spherical lenses reconstructed with stepped plate lenses [START_REF] Rondineau | A sliced spheriacl Luneburg lens[END_REF]. Here we use only one flat plate, with a totally different hole distribution.

The relation between the relative permittivity r and the azimuthal angle is represented in Fig. 2 and defined in [START_REF] Jackson | Classical Electrodynamics[END_REF], where n is the variation of the refractive index and n m is the refractive index of the raw material (Teflon, in our case): In our case, to have an OAM topological charge |l| = 1 corresponding to a rotational phase of ±2 on the wave front, the drilled holes have to induce a refractive index varying from 1.35 to 1.45 as a function of the azimuthal angle . Equation [START_REF] Thidé | Utilization of photon orbital angular momentum in the low-frequency radio Domain[END_REF] shows the relation between the volume with the drilled holes V h and the relative permittivity r ( ) as a function of the azimuthal angle :

r ( ) = 2
V h ( ) = ( r ( ) -n m 2 ) / (1 -n m 2 )
The major challenge is to design a hole distribution that induces the expected variation in permittivity but that can be mechanically done.

B. Mecanichal conception of the phase plate.

As we choose to perform numerical simulations at a frequency of f = 30 GHz, this means that each drilled hole should have a diameter of the order of d = /10 = 1 mm. In our case, the plate has a radius of R = 5 = 50 mm and a length of L = 104 mm. Furthermore, as can be seen in Fig. 1, the number of holes is quite important, of the order of a thousand. This induces strong constraints on the raw material as it is manufactured. Another difficulty would be to have a drill with a diameter d and a length L. The problem is that an usual drill with a diameter d has a length of the order of L/10 ~ 10 mm.

To overcome this, we had to make the architecture of the flat phase plate more complex, as can be seen on Fig. 3. The flat phase plate is decomposed into smaller flat sub-plates, the hole distribution being carried on from a sub-plate to another.

To keep a comfortable safety margin, we choose to use 2 sub-plates with a length of L s1 = 4 mm and 16 sub-plates with a length of L s2 = 6 mm. In principle, as long as the hole distribution is respected the order in which these sub-plates are arranged does not matter even if the holes vary in depth, as the important thing is the volume extracted in each angular sector, not the continuity of each hole along the length of the flat phase plate. Nevertheless, we choose to conserve this continuity to add a visual verification (the number of holes on each sub-plate) when combining the sub-plates.

IV. RESULTS

A. Numerical Simulations

We have numerically simulated the total EM field generated by this flat phase plate, illuminated by a horn antenna. Fig. 4 presents the simulated transmitted beam in the far-field. Panel (a) on the left is a visualization of the directivity of the beam. We can see the specific "donut" shape of the intensity profile and the low intensity zone (vortex zone) around the axis of the beam. Panel (b) on the right is a visualization of the phase of the beam. As expected, we can see a well-defined rotational phase distribution from 0 to 2 , on the white circle corresponding to the maximum intensity of the EM field. This simulation has been carried out for both polarization of the horn antenna in regards to the flat phase plate, for similar results. For clarity, only one set of results will be presented in this paper.

B. Comparison with other antennas

We choose to consider only the spiral lens and the "twisted" parabolic antenna arrays out of the already existing antennas in the following comparison, as the circular short dipole does not possess a single radiating element only. The first has been numerically simulated at a frequency of f = 30 GHz and results are visible in Fig. 5. The second has also been numerically simulated at a frequency of f = 30 GHz, but for visualization concerns (as a beam is much more focalized with a parabolic antenna) it has also been carried out at a frequency of f = 2.41 GHz (as the antenna used in [START_REF] Tamburini | Encoding many channels on the same frequency through radio vorticity: first experimental test[END_REF]) for similar results which are visible in Fig. 6. In both case we can see on Panel (a), the specific "donut" shape of the intensity profile and the vortex zone around the axis of the beam, and on Panel (b), a well-defined rotational phase distribution from 0 to 2 on the white circle corresponding to the maximum intensity of the EM field.

We also choose to disregard differences in directivity and to focus our attention purely on the OAM carried by the generated EM wave. But this information on the intensity of the field is still important as the information on the phase is only relevant when there is enough energy, i.e. on the "donut shape". Thus, to compare the three antennas, we have to study the rotational phase variation on the white circle corresponding to the maximum intensity of the EM field. The criterion chosen is the mode purity of the OAM EM wave generated.

To obtain this mode purity, the phase gradient has been exploited based on the second method presented in [START_REF] Mohammadi | Orbital Angular Momentum in Radio -Measurement methods[END_REF]: the phase is extracted along the white circle perimeter. Mode decomposition is then obtained directly with the help of a Fourier Transform, using (3) with l being the distribution of angular momentum and ( ) the extracted phases values [START_REF] Yao | Fourier relationship between angular position and optical angular momentum[END_REF]:

l = (1/2 ) d ( )exp(-jl )
Results for the three antennas are presented in Fig 7 . The ideal case would be OAM mode with a proportion close to 1. Indeed, we can see that there is a clearly predominant OAM mode which correspond to l = -1 but with a proportion a little higher than 0.5 for the spiral lens and the "twisted" parabolic antenna and a little higher than 0.4 for the flat phase plate. This can be explained by the variation of the amplitude on the white circle as it is a constant on the ideal case. Nevertheless, we consider these results as quite good since the predominant OAM mode has a proportion 3.5 times greater than the secondary mode in the case of the flat phase lens, up to 7 in the case of the "twisted" parabolic antenna and even 10 in the case of the spiral lens.

The intensity distribution is counterbalanced by several advantages of this plate, namely the ability to change the drilling law, and a higher versatility.

V. CONCLUSION

In this paper, we presented a new way to generate an EM wave carrying OAM in the millimeter frequency band. A flat phase plate is designed to have an OAM order equals to 1, using drilling holes in a Teflon flat plate. Details on the techniques used to manufacture the flat phase plate are presented, as well as some simulated results to held a comparative study between our antenna and the already existing ones (parabolic dish antenna and spiral phase plate).

n 2 /

 2 (2 l) 2 -.n m . n / l + n m 2 This work is supported by the Université de Rennes 1, the French Ministry of Defense (DGA) and the council of Brittany (Région Bretagne).

Figure 1 .

 1 Figure 1. Front view of the phase plate. Permittivity is controlled by drilling holes in a Teflon flat plate. Distribution of the drilled holes is a function of the azimuthal angle .

Figure 2 .

 2 Figure 2. Refractive index as a function of the azimuthal angle .

Figure 3 .

 3 Figure 3. Side view (left) and top view (right) of the flat phase plate. The sub-plates are maintained together with the help of four nylon stick to stenghten the global structure.

Figure 4 .Figure 5 .

 45 Figure 4. Farfield in front of the flat phase plate as obtained by numerical simulations. The white circle corresponds to the maximum intensity of the EM field. Panel (a) is a visualization of the directivity of the beam, with the characteristical "donut" shape, and panel (b) is a visualization of the rotational phase distribution

Figure 6 .

 6 Figure 6. Farfield in front of the twisted parabolic antenna as obtained by numerical simulations. The white circle corresponds to the maximum intensity of the EM field. Panel (a) is a visualization of the directivity of the beam, with the characteristical "donut" shape, and panel (b) is a visualization of the rotational phase distribution.

Figure 7 .

 7 Figure 7. Mode decomposition for the flat phase plate (a), the spiral lens (b) and the "twisted" parabolic antenna (c). In each case the reference circle is represented with a white circle.