Generalized Maxwell boundary conditions for transport equations

Abstract

We prove the existence of a solution of a transport equation satisfying a generalized (non linear) Maxwell boundary condition.

1 Introduction

Let Ω be a regular open subset of IR^{N}. We denote by ν_{x} the outward unit normal vector at point x ∈ ∂Ω. We are concerned with the problem

λu + v · ∇_{x}u = q + Au in Ω_{x} × IR_{v}^{N}  (1.1)

where λ ∈ IR, q ∈ L^{p}(Ω × IR^{N}), 1 ≤ p < ∞ are given and A is a bounded linear operator on L^{p}(Ω×IR^{N}). Equation (1.1) is completed with a boundary condition of reflection type which leads to a connection between the outgoing/incoming traces u_{±} of the unknown u on the sets

Σ^{±} = {(x, v) ∈ ∂Ω × IR^{N}, ±v · ν_{x} > 0}.

We set L^{p,±} = L^{p}(Σ^{±}, | v · ν_{x} | dvdσ_{x}) with dσ_{x} the Lebesgue measure on ∂Ω. Let R be a linear operator, bounded from L^{p,+} to L^{p,−}. Then, u satisfies the boundary condition

u_{−} = Ru_{+} on Σ^{−}.  (1.2)

The operator R contains the details of the reflection law we prescribe at the boundary. A model currently used is given by the following Maxwell law, [17]

u_{−}(x, v) = a(x)u_{+}^{∗}(x, v) + (1 − a(x))ũ_{+}(x)M_{−}(x, v) = R_{a}u_{+}  (1.3)

on Σ^{−} where

i) 0 ≤ a(x) ≤ 1 a.e. ∂Ω,
ii) u_{+}^{∗}(x, v) = u_{+}(x, v − 2(v · ν_{x})ν_{x}),
iii) the map u_{+} ⟼ ũ_{+} satisfies
| ũ_{+}(x) |=| ∫_{v·νx>0} M_{+}^{p−1}(x, v)u_{+}(x, v)v · ν_{x}dv |≤ (∫_{v·νx>0} | u_{+}(x, v) |^{p} v · ν_{x}dv)^{\frac{1}{p}} ,
iv) M satisfies (1.3) and the normalization condition
∫_{v·νx<0} | M_{−}(x, v) |^{p}| v · ν_{x} | dv = 1 = ∫_{v·νx>0} | M_{+}(x, v) |^{p} v · ν_{x}dv.

In pratice, M is a Maxwellian M(x, v) = \frac{c}{Θ(x)} exp(−\frac{v^{2}}{Θ(x)}) where Θ depends on the physical features of the boundary. The coefficient a, usually called accomodation coefficient, splits the reflexion into a specular part and a diffuse part described by M. The hypotheses i-iv) imply that R_{a} is a bounded operator from L^{p,+} to L^{p,−} of norm 1.

The equation that we choose to study may be seen as a simple model for various problems arising in rarefied gas dynamics, neutron transport, radiative transfer or vacuum technology... We look more closely at the boundary conditions for these problems and this work is an attempt to deal with a generalization of the reflection law (1.3). Indeed, the boundary condition is related to interaction phenomena at the boundary ∂Ω. Then, considering very complicated situations, the condition (1.3) does not remain linear and the accomodation coefficient should depend on macroscopic quantities like ũ_{+}. We refer the reader to [16], [6], [7], [5] for a discussion on these phenomena.

Let α be a function from IR to [0, 1]. Our purpose is to prove the existence of solutions of the problem (1.1), satisfying the boundary condition (1.3) with an accomodation coefficient given by

a(x) = α(ũ_{+}(x)).  (1.4)

Let us briefly point out the difficulties. On the one hand, the linear operator R_{a} (with a fixed) defined by (1.3) still has norm unity. Then, in general, we have no bound on the traces : it is known that (1.1-1.3) has a solution u ∈ L^{p}(Ω × IR^{N}), however we do not know if the traces lie in L^{p,±}, see [4]. This lack of information about the traces is at the origine of the relaxation of the boundary condition encountered when studying the Boltzmann equation [15], [2]. On the other hand, the Maxwell law mixes specular and diffuse reflections and the monotonicity methods break down [14]. In this paper, we wish to investigate an example of situation where we are able to obtain the existence of a solution of (1.1) satisfying the non linear Maxwell boundary condition (1.3-1.4). We also mention that a study of the stationary Boltzmann equation equipped with this boundary condition is in progress and will appear in a forthcoming paper written in collaboration with K. Hamdache.

This work is organized as follows. In Section 2, we have compiled some of the standard facts on the equation (1.1- 1.2). Our main result is stated at the end of Section 2. Namely, we establish an existence result for (1.1-1.4) in the L^{2} functional framework under the fundamental condition that the set Ω of positions is the exterior of a closed and bounded convex subset of IR^{N}. Section 3, which constitutes the heart of the matter, is devoted to the proof, developing a Shauder’s fixed point method.

2 Preliminaries

2.1 Functional spaces

We introduce the following functional spaces, commonly used in transport theory,

W^{p} = {u ∈ L^{p}(Ω × IR^{N}), v · ∇_{x}u ∈ L^{p}(Ω × IR^{N})}.

By extension of the mappings u ∈ C^{∞}(\overline{Ω} × IR^{N}) ⟼ u_{±} = u_{|Σ^{±}}, we define the traces u_{±} on Σ^{±} of the elements u of W^{p}. However, in general, u_{±} does not belong to L^{p,±}, see [18], [8], [4]. We set

W̃^{p} = {u ∈ W^{p}, u_{±} ∈ L^{p,±}}.

A ”Green’s formula” holds on W̃^{p} and consequently, the following equality [18], [4]

λp ∥ u ∥p_{L^{p}} + ∥ u_{+} ∥p_{L^{p,+}}=∥ u_{−} ∥p_{L^{p,−}} +p ∫ | u |^{p−1} sgn(u)(λ + v · ∇_{x})udxdv.  (2.1)

is satisfied for u ∈ W̃^{p} and λ ∈ IR. We also recall the following existence result [4].

Theorem 1 Let q ∈ L^{p}(Ω×IR^{N}) and let A be a bounded linear operator on L^{p}(Ω×IR^{N}), λ >∥ A ∥. If ∥ R ∥< 1, then, there exists a unique solution u ∈ W̃^{p} of (1.1-1.2). If A ≥ 0, q ≥ 0 and R ≥ 0 satisfies ∥ R ∥= 1, then, there exists a solution u ∈ W^{p} of (1.1) satisfying (1.2) a.e. (there exists at most one solution satisfying u_{±} ∈ L^{p,±}).

2.2 Exterior problem

From now on, it is required that

Ω =^{c} \overline{O}, with O a convex bounded subset of IR^{N}.

Throughout the paper, we refer to this condition by (H). We present now some consequences of (H).

For (x, v) ∈ \overline{Ω} × IR^{N}, we define the staying times by

t^{±}(x, v) = sup{t > 0, x ± sv ∈ Ω, 0 ≤ s ≤ t}.

In particular, (H) implies that the following assertions hold

- if (x, v) ∈ Σ^{±}, then t^{±}(x, v) = 0 and t^{∓}(x, v) = +∞. In other words, the trajectory {x∓sv, s > 0} leaving ∂Ω with ±v ·ν_{x} > 0, never returns back to the boundary ∂Ω again.

- if (x, v) ∈ Ω × IR^{N}, then either the trajectory starting from x with the direction v never hits ∂Ω, i.e. {x + sv, s ∈ IR} ∩ ∂Ω = ∅, or there exists a unique τ ∈ IR such that x + τv ∈ ∂Ω (if ±τ > 0, then (x + τv, v) belongs to Σ^{±}).

Therefore, assuming that (H) holds, the map

Φ_{v} : {y ∈ Ω, ∃x ∈ ∂Ω, ∃s > 0, y = x ± sv, ±v · ν_{x} > 0} ⟼ {x ∈ ∂Ω, ±v · ν_{x} > 0}

is a bijection with jacobian | v · ν_{x} |, see [3]. Finally, we formulate the following lemma on the traces [8], [11], [3].

Lemma 1 We assume that (H) holds. Then, the trace operator

u ⟼ (u_{+}, u_{−})

is continuous, surjective from W^{p} to L^{p,+} × L^{p,−} with continuous inverse.

We are thus led to the following strengthening of Theorem 1.

Proposition 1 We assume that (H) holds. Let q ∈ L^{p}(Ω × IR^{N}), A ∈ L(L^{p}(Ω × IR^{N})), λ >∥ A ∥ and R ∈ L(L^{p,+}, L^{p,−}) with norm ∥ R ∥≤ 1. Then, there exists a unique solution u ∈ W̃^{p} of (1.1-1.2).

Proof. Let us first examine the problem

\{ \begin{matrix}λu + v · ∇_{x}u = h in Ω_{x} × IR_{v}^{N} \\ u_{−} = g_{−} on Σ^{−} \end{matrix}  (2.2)

where we regard, temporarily, h ∈ L^{p}(Ω × IR^{N}) and g_{−} ∈ L^{p,−} as fixed. For this purpose, we set, for (x, v) ∈ \overline{Ω} × IR^{N},

u^{♯}(s) = u(x − sv, v)

which satisfies

\frac{d}{ds}(e^{−λs}u^{♯}(s)) = −e^{−λs}(λu + v · ∇_{x}u)^{♯}(s) = −e^{−λs}h^{♯}(s).

It follows that

e^{−λt_{1}}u(x − t_{1}v, v) = e^{−λt_{2}}u(x − t_{2}v, v) + ∫_{t_{1}}^{t_{2}} e^{−λσ}h(x − σv, v)dσ

with x − σv ∈ Ω for t_{1} ≤ σ ≤ t_{2}. We are thus led to the following discussion - if {x − σv, σ > 0} ∩ ∂Ω = ∅, then (choosing t_{1} = 0, t_{2} = ∞)

u(x, v) = ∫_{0}^{∞} e^{−λσ}h(x − σv, v)dσ.  (2.3)

In particular, this formula still holds if (x, v) ∈ Σ^{+}, since the trajectory {x − σv, σ > 0} never hits the boundary. Then, the outgoing trace is given by

u_{+}(x, v) = ∫_{0}^{∞} e^{−λσ}h(x − σv, v)dσ.  (2.4)

- if there exists t^{−}(x, v) < ∞ such that (x − t^{−}(x, v)v, v) ∈ Σ^{−} (by (H) such a t^{−} is unique), then, integrating between t_{1} = 0, t_{2} = t^{−}(x, v), one has

u(x, v) = e^{−λt^{−}(x,v)}u_{−}(x − t^{−}(x, v)v, v) + ∫_{0}^{t^{−}(x,v)} e^{−λσ}h(x − σv, v)dσ

= e^{−λt^{−}(x,v)}g_{−}(x − t^{−}(x, v)v, v) + ∫_{0}^{t^{−}(x,v)} e^{−λσ}h(x − σv, v)dσ.  (2.5)

The reflection boundary condition (1.2) is satisfied if g_{−} = Ru_{+}. Hence, according to (2.4), we write

u_{−}(x, v) = g_{−}(x, v) = Ru_{+}(x, v) = R(∫_{0}^{∞} e^{−λσ}h(x − σv, v)dσ).  (2.6)

Finally, the unique solution of (2.2) satisfying (1.2) is determined by (2.3-2.6). One remarks that

∫_{Σ^{+}} | u_{+} |^{p}| v · ν_{x} | dvdσ_{x}
≤ ∫_{Σ^{+}} \{∫_{0}^{∞} e^{−λσ}dσ\}^{\frac{p}{p′}} \{∫_{0}^{∞} e^{−λσ} | h(x − σv, v) |^{p} dσ\} | v · ν_{x} | dvdσ_{x}

which follows from e^{−λσ} = e^{− \frac{λσ}{p}} e^{− \frac{λσ}{p′}} and Holder’s inequality. Assuming (H), one gets

∥ u_{+} ∥p_{L^{p,+}}≤ \frac{1}{λ^{p−1}} ∫_{Σ^{+}} ∫_{0}^{∞} e^{−λσ} | h(x − σv, v) |^{p} dσ | v · ν_{x} | dvdσ_{x}

≤ \frac{1}{λ^{p−1}} ∫_{Ω×IR^{N}} | h(y, v) |^{p} dvdy = \frac{1}{λ^{p−1}} ∥ h ∥p_{L^{p}} .

Hence u_{+} belongs to L^{p,+} and u ∈ W̃^{p}.

Let u = S_{λ}(h) ∈ W̃^{p} stand for the solution of

\{ \begin{matrix}λu + v · ∇_{x}u = h in Ω × IR^{N} \\ u_{−} = Ru_{+} on Σ^{−} \end{matrix}

that we have just obtained. Green’s formula (2.1) leads to

λp ∥ u ∥p_{L^{p}}≤ p ∫ | u |^{p−1}| h | dxdv

since ∥ R ∥≤ 1. By using the elementary convexity inequality ab ≤ \frac{a^{p}}{p} + \frac{b^{p′}}{p′} with a =

λ^{− \frac{1}{p′}} | h |, b = λ^{\frac{1}{p′}} | u |^{p−1}, we get

λp ∥ u ∥p_{L^{p}}≤ \frac{1}{λ^{p−1}} ∥ h ∥p_{L^{p}} +(p − 1)λ ∥ u ∥p_{L^{p}}

and finally

∥ u ∥_{L^{p}}=∥ S_{λ}(h) ∥_{L^{p}}≤ \frac{1}{λ} ∥ h ∥_{L^{p}} .

Thus, since R is a linear reflection law, S_{λ} is a bounded linear operator on L^{p}(Ω × IR^{N}) whose norm satisfies ≤ \frac{1}{λ}. We look for a solution u of (1.1-1.2) having the form u = S_{λ}(h_{∗}) where h_{∗} is to be determined. We have

(λ + v · ∇_{x})u = (λ + v · ∇_{x})S_{λ}(h_{∗}) = q + Au = q + A(S_{λ}(h_{∗})) = h_{∗}

or, in other words, h_{∗} satisfies

(I − AS_{λ})h_{∗} = q ∈ L^{p}(Ω × IR^{N}).

However, the assumptions on A and λ imply that

∥ AS_{λ} ∥≤∥ A ∥∥ S_{λ} ∥≤ \frac{1}{λ} ∥ A ∥< 1.

Hence we conclude that I − AS_{λ} is invertible and the proof is complete.

Remark 1 Let us assume that A = 0. Relation (2.4) shows that the outgoing trace of the solution u of (1.1-1.2) only depends on the right hand side q. Consequently, ũ_{+}(x) = g_{q}(x) is a function of L^{p}(∂Ω) only determined by q. Then, u satisfies the non linear Maxwell boundary condition (1.3-1.4) if the accomodation coefficient is given by, a.e. x ∈ ∂Ω,

a(x) = α(g_{q}(x)) = α(∫_{v·νx>0} | v · ν_{x} | M_{+}^{p−1}(x, v) ∫_{0}^{∞} e^{−λσ}q(x − σv, v)dσdv).

2.3 Main result

First of all, we give some precision on the problem (1.1-1.4) under consideration. The linear operator A is of (weighted) velocity average type

Au(x, v) = ∫ p(v, v_{∗})u(x, v_{∗})dv_{∗}  (2.7)

with p ∈ C_{0}^{∞}(IR^{N} × IR^{N}). Next, the boundary condition (1.3) is defined as follows

u_{−}(x, v) = R_{a}u_{+}(x, v)

= a(x)u_{+}(x, v − 2(v · ν_{x})ν_{x}) + (1 − a(x))ũ_{+}(x)M_{−}(x, v)  (2.8)

on Σ^{−} with

\{ \begin{matrix}0 ≤ a(x) ≤ 1 a.e. ∂Ω \\ ũ_{+}(x) = ∫_{v·νx>0} M_{+}(x, v)u_{+}(x, v)v · ν_{x}dv \\ ∫_{v·νx<0} M_{−}^{2}(x, v) | v · ν_{x} | dv = 1 = ∫_{v·νx>0} M_{+}^{2}(x, v)v · ν_{x}dv. \end{matrix}  (2.9)

We can now formulate our main result, which is proved in the remainder of the paper.

Theorem 2 We assume that (2.7-2.9) and (H) hold. Let q ∈ L^{2}(Ω × IR^{N}) and let λ >∥ p ∥_{L^{2}(IR^{2N})}. Let α be a continuous function from IR to [0, 1]. Then, there exists a solution u ∈ W̃^{2} of (1.1) satisfying the non linear boundary condition (2.8) with

a(x) = α(ũ_{+}(x))

a.e. ∂Ω.

3 Fixed point method

This section is devoted to the proof of Theorem 2, by using a fixed point method. Before going into the technical details, let us explain the essential ideas. The particular form of the operator A, given by (2.7) leads to some gain in regularity on the term Au, applying the velocity averaging lemma [13], [9], [12],... Then, assumption (H) is exploited to propagate, in a certain sense, this gain in regularity up to the boundary. These arguments give the key to apply the Schauder fixed point theorem on the mapping

T : j ∈ L^{2}(∂Ω) ⟼ ũ_{+}  (3.1)

where u is a solution of (1.1) satisfying the boundary condition (2.8) with

a(x) = α(j(x)) a.e.∂Ω.  (3.2)

We shall show that T has a fixed point into a certain ball B(0, R) of L^{2}(∂Ω). The proof falls naturally into the following three lemmas.

Lemma 2 There exists R > 0, depending on λ, p and q such that

T (B(0, R)) ⊂ B(0, R).

Proof. Let a ∈ L^{∞}(∂Ω) such that 0 ≤ a(x) ≤ 1, a.e. ∂Ω. Let u be a solution of (1.1) satisfying (2.8) on Σ^{−}, obtained in Proposition 1. It suffices to prove that

∥ ũ_{+} ∥_{L^{2}(∂Ω)}≤ R = C_{λ,p} ∥ q ∥_{L^{2}}

where C_{λ,p} depends on λ, p. Green’s formula (2.1) yields

2λ ∥ u ∥2_{L^{2}} + ∥ u_{+} ∥2_{L^{2,+}}=∥ u_{−} ∥2_{L^{2,−}} +2 ∫ | u | sgn(u)(q + Au)dxdv

≤∥ R_{a}u_{+} ∥2_{L^{2,−}} +λ ∫ | u |^{2} dxdv + \frac{1}{λ} ∫ | q + Au |^{2} dxdv

estimating 2ab ≤ a^{2} + b^{2} with a = \sqrt{λ} | u |, b = \frac{|q+Au|}{\sqrt{λ}}. Since ∥ R_{a} ∥= 1, it follows that

λ ∥ u ∥2_{L^{2}}≤ \frac{1}{λ} ∥ q + Au ∥2_{L^{2}} .

Hence, obviously, we have

∥ u ∥_{L^{2}}≤ \frac{1}{λ− ∥ A ∥} ∥ q ∥_{L^{2}}  (3.3)

where λ >∥ A ∥. Furthermore, by using (1.1), one deduces that

∥ v · ∇_{x}u ∥_{L^{2}}=∥ q + Au − λu ∥_{L^{2}}≤ (1 + \frac{λ+ ∥ A ∥}{λ− ∥ A ∥}) ∥ q ∥_{L^{2}} .  (3.4)

Then, Lemma 1 leads to

∥ u_{±} ∥_{L^{2,±}}≤ C ∥ u ∥_{W^{2}}≤ C_{λ,p} ∥ q ∥_{L^{2}} .  (3.5)

By using the Holder inequality and (2.9), one gets

∫_{∂Ω} | ũ_{+}(x) |^{2} dσ_{x} = ∫_{∂Ω} | ∫_{v·ν_{x}>0} M_{+}(x, v)u_{+}(x, v)v · ν_{x}dv |^{2} dσ_{x}

≤ ∫_{∂Ω} ∫_{v·ν_{x}>0} M_{+}^{2}v · ν_{x}dv ∫_{v·ν_{x}>0} | u_{+} |^{2} v · ν_{x}dvdσ_{x} =∥ u_{+} ∥2_{L^{2,+}}≤ C_{λ,p}^{2} ∥ q ∥2_{L^{2}} .

Lemma 3 Let α be a continuous function from IR to [0, 1]. Then T is continuous on L^{2}(∂Ω).

Proof. Let j_{n} be a sequence in L^{2}(∂Ω) satisfying

j_{n} ⟶ j (strongly) in L^{2}(∂Ω).  (3.6)

We consider the associated reflection operators R_{n} = R_{α(j_{n})} (resp. R = R_{α(j)}) defined by (2.8). We denote by u_{n} (resp. u), the sequence of solutions of (1.1), satisfying (2.8) with R_{n} (resp. R), obtained in Proposition 1. Possibly at the cost of extracting a subsequence, we assume that j_{n} converges a.e.∂Ω to j, and, since α is continuous,

a_{n}(x) = α(j_{n}(x)) ⟶ a(x) = α(j(x)) a.e. ∂Ω.  (3.7)

We will consider the behaviour of the sequence w_{n} = u_{n} − u. We have

\{ \begin{matrix}(λ + v · ∇_{x})w_{n} = Aw_{n} in Ω × IR^{N} \\ w_{n,−} = R_{n}w_{n,+} + g_{n,−} on Σ^{−} \end{matrix}  (3.8)

where g_{n,−} stands for

(R_{n} − R)u_{+} = (a_{n} − a)u_{+}^{∗} + (a − a_{n})ũ_{+}M_{−}.  (3.9)

Lebesgue’s theorem and (3.7) yield

g_{n,−} ⟶ 0 in L^{2,−}.  (3.10)

By using Lemma 1, it may be constructed G_{n} ∈ W̃^{2} satisfying

\{ \begin{matrix}G_{n,|Σ^{−}} = g_{n,−} ; G_{n,|Σ^{+}} = 0, \\ ∥ G_{n} ∥_{W^{2}}≤ C ∥ g_{n,−} ∥_{L^{2,−}} . \end{matrix}  (3.11)

Let Γ_{n} = (λ + v · ∇_{x})G_{n}. From (3.10-3.11), one has

G_{n} ⟶ 0 ; Γ_{n} ⟶ 0 in L^{2}(Ω × IR^{N}).  (3.12)

We set z_{n} = w_{n} − G_{n} which is a solution of

\{ \begin{matrix}(λ + v · ∇_{x})z_{n} = Aw_{n} − Γ_{n} in Ω × IR^{N} \\ z_{n,−} = R_{n}z_{n,+} on Σ^{−} \end{matrix}  (3.13)

Thus, (2.1) yields

2λ ∥ z_{n} ∥2_{L^{2}} + ∥ z_{n,+} ∥2_{L^{2,+}}=∥ z_{n,−} ∥2_{L^{2,−}} +2 ∫ | z_{n} | sgn(z_{n})(Az_{n} + AG_{n} − Γ_{n})dxdv

≤∥ R_{n}z_{n,+} ∥2_{L^{2,−}} +λ ∥ z_{n} ∥2_{L^{2}} +\frac{1}{λ} ∥ Az_{n} + AG_{n} − Γ_{n} ∥2_{L^{2}}

proceeding as in the proof of Lemma 2. Since ∥ R_{n} ∥≤ 1, this forces

∥ z_{n} ∥_{L^{2}}≤ \frac{1}{λ} ∥ Az_{n} + AG_{n} − Γ_{n} ∥_{L^{2}}

and finally

∥ z_{n} ∥_{L^{2}}≤ \frac{1}{λ− ∥ A ∥} ∥ AG_{n} − Γ_{n} ∥_{L^{2}}

which tends to 0 by (3.12). By using (3.13), a similar conclusion is obtained for

∥ v · ∇_{x}z_{n} ∥_{L^{2}}≤ (λ+ ∥ A ∥) ∥ z_{n} ∥_{L^{2}} + ∥ AG_{n} − Γ_{n} ∥_{L^{2}} .

Consequently, w_{n} = z_{n} + G_{n} is the sum of two sequences which converge to 0 in W^{2} and, by applying Lemma 1, one deduces that

∥ ũ_{n,+} − ũ_{+} ∥_{L^{2}(∂Ω)}=∥ w̃_{n,+} ∥_{L^{2}(∂Ω)}≤∥ w_{n,+} ∥_{L^{2,+}}≤ C ∥ w_{n} ∥_{W^{2}} .

Therefore, T (j_{n}) = ũ_{n,+} ⟶ T (j) = ũ_{+} in L^{2}(∂Ω). Moreover, Proposition 1 guarantees the uniqueness of the solution u ∈ W^{2} of (1.1, 2.7) with a(x) = α(j(x)) a.e. ∂Ω. We conclude that, in fact, the whole sequence ũ_{n,+} converges to ũ_{+}.

To achieve the proof of Theorem 1 it suffices to apply the Schauder fixed point theorem, since one has

Lemma 4 The mapping T is compact on L^{2}(∂Ω).

Proof. Let j_{n} be a bounded sequence in L^{2}(∂Ω). Our purpose is to prove that T (j_{n}) = ũ_{n} is compact in L^{2}(∂Ω), following the previous notations. In fact, we prove more, namely that

u_{n,+} is compact in L^{2,+}.  (3.14)

The proof is adapted from arguments used in [11] about an homogeneization problem. As a consequence of assumption (H), the outgoing trace is smoothed by the gain in regularity induced by averaging lemma on Au_{n} far from ∂Ω. The proof will be divided into three steps.

- Step 1. Comparison between u_{n} and the solution of a problem with a modified right hand side. First of all, by (3.3-3.4), one observes that u_{n} and v · ∇_{x}u_{n} are bounded in L^{2}(Ω × IR^{N}). Then, averaging lemmas [13], [9], [12] imply that

Au_{n} is bounded in H^{\frac{1}{2}} (Ω × IR^{N}).  (3.15)

Let δ > 0. We introduce a cut off function ψ^{δ} ∈ C^{∞}(IR^{N}) satisfying

0 ≤ ψ^{δ} ≤ 1

ψ^{δ}(x) = \{ \begin{matrix}0 if d(x, ∂Ω) < δ or x ∉ Ω \\ 1 if d(x, ∂Ω) > 2δ and x ∈ Ω. \end{matrix}

Let q^{δ} ∈ C_{0}^{∞}(Ω × IR^{N}) approaching q in L^{2}(Ω × IR^{N}) when δ goes to 0. We compare u_{n} with w_{n}^{δ}, solution of the problem

\{ \begin{matrix}(λ + v · ∇_{x})w_{n}^{δ} = q^{δ} + ψ^{δ}Au_{n} in Ω × IR^{N} \\ w_{n,−}^{δ} = R_{n}w_{n,+}^{δ} on Σ^{−}. \end{matrix}  (3.16)

Thus, it follows that

\{ \begin{matrix}(λ + v · ∇_{x})(u_{n} − w_{n}^{δ}) = q − q^{δ} + (1 − ψ^{δ})Au_{n} in Ω × IR^{N} \\ (u_{n,−} − w_{n,−}^{δ}) = R_{n}(u_{n,+} − w_{n,+}^{δ}) on Σ^{−} \end{matrix}

and (2.1) leads to

∥ u_{n} − w_{n}^{δ} ∥_{L^{2}}≤ \frac{1}{λ}(∥ q − q^{δ} ∥_{L^{2}} + ∥ (1 − ψ^{δ})Au_{n} ∥_{L^{2}}).

Then, remarking that supp_{x}((1 − ψ^{δ})Au_{n}) lies in {x ∈ Ω, d(x, ∂Ω) < 2δ} ⊂ B(0, ρ) for a certain ρ large enough, one obtains, by using (3.15-3.16),

lim_{δ→0} sup_{n} ∥ u_{n} − w_{n}^{δ} ∥_{L^{2}}, ∥ v · ∇_{x}(u_{n} − w_{n}^{δ}) ∥_{L^{2}}= 0.  (3.17)

-Step 2. Comparison between w_{n}^{δ} and the solution of the equation in IR^{N} ×IR^{N} having the same right hand side. Precisely, let W_{n}^{δ} be the solution of

(λ + v · ∇_{x})W_{n}^{δ} = Q^{δ} + S_{n}^{δ} in IR^{N} × IR^{N}  (3.18)

where Q^{δ} (resp. S_{n}^{δ}) stands for q^{δ} (resp. ψ^{δ}Au_{n}) extended by 0 in O =^{c} \overline{Ω}. Then, by (3.15), the right hand side of (3.18) is bounded in H^{\frac{1}{2}} (IR^{N} × IR^{N}) (with respect to n, the bound possibly depending on δ...). Classical energy estimates and L^{2}/H^{1} interpolation results, see [1], give

(W_{n}^{δ})_{n}, (v · ∇_{x}W_{n}^{δ})_{n} are bounded in H^{\frac{1}{2}} (IR^{N} × IR^{N}).

Furthermore, the right hand side in (3.18) has a compact support with respect to v. Thus, by Rellich’s theorem, one deduces that

(W_{n}^{δ})_{n}, (v · ∇_{x}W_{n}^{δ})_{n} are compact in L^{2}(IR_{loc}^{N} × IR^{N}).  (3.19)

-Step 3. Conclusion. Assumption (H) implies that the outgoing trace of w_{n}^{δ} is not influenced by the boundary and one has

w_{n,+}^{δ} = W_{n,|Σ^{+}}^{δ}.  (3.20)

Combining the compactness property (3.19) and Lemma 1, we see that

w_{n,+}^{δ} ∈ K^{δ}, compact set of L^{2,+}.  (3.21)

Therefore, u_{n,+} = w_{n,+}^{δ} + (u_{n,+} − w_{n,+}^{δ}) belongs to K^{δ} + B(0, C_{δ})_{L^{2,+}} where, by (3.18) and Lemma 1, lim_{δ→0} C_{δ} = 0. Then, classical arguments of integration, [10], leads to (3.14).

Remark 2 \underline{On the one dimensional case} Let Ω =]0, ∞[. We write

\{ \begin{matrix}u_{in}(0, v) = u(x, v)_{|x=0,ξ>0} \\ u_{out}(0, v) = u(x, v)_{|x=0,ξ<0} \end{matrix}

where v = (ξ, v̂) ∈ IR × IR^{2}. Within this framework, we are able to discuss the case, physically more natural, of solutions in L^{1}(Ω × IR^{3}), with weaker assumptions on the continuous operator A. Indeed, let q ∈ L^{1}(Ω × IR^{3}) and let α be a continuous function from IR to [0, 1]. We consider M_{0} satisfying

∫_{ξ>0} M_{0}(v) | ξ | dv = 1.

Then, we introduce the mapping

T : j ⟼ ∫_{ξ<0} u_{out}(0, v) | ξ | dv,

u being the solution of the following (linear) problem

\{ \begin{matrix}λu + ξ∂_{x}u = q + Au in Ω × IR^{3} \\ u_{in}(0, ξ, v̂) = α(j)u_{out}(0, −ξ, v̂) \\ +(1 − α(j)) ∫_{ξ′<0} u_{out}(0, v′) | ξ′ | dv′M_{0}(v) on x = 0, ξ > 0. \end{matrix}

T is a simple application from IR to IR. Some slight modifications in the previous proofs (see Lemma 2-3) show that T is continuous from [−R, +R] into [−R, +R], R depending on ∥ q ∥_{L^{1}}, λ and ∥ A ∥_{L(L^{1})}. Therefore, by Brouwer’s theorem, T has a fixed point which leads to a solution u ∈ W̃^{1} of (1.1) satisfying the generalized Maxwell boundary condition (1.3-1.4).

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