On Some Transmission Problems Set in a Biological Cell, Analysis and Resolution

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On Some Transmission Problems Set in a Biological Cell, Analysis and Resolution

Rabah Labbas, Keddour Lemrabet, Kheira Limam, Ahmed Medeghri, Maëlis Meisner

To cite this version:

Rabah Labbas, Keddour Lemrabet, Kheira Limam, Ahmed Medeghri, Maëlis Meisner. On Some

Transmission Problems Set in a Biological Cell, Analysis and Resolution. Journal of Differential

Equations, Elsevier, 2015, pp.In Press. �10.1016/j.jde.2015.04.002�. �hal-01138558�

On Some Transmission Problems Set in a Biological Cell, Analysis and Resolution ^∗

Rabah Labbas

, Keddour Lemrabet

, Kheira Limam

, Ahmed Medeghri

and Ma¨ elis Meisner

aLaboratoire de Math´ematiques Appliqu´ees, Universit´e du Havre,

B.P. 540, 76058 Le Havre Cedex, France.

[email protected] [email protected]

bLaboratoire AMNEDP, Facult´e de maths, USTHB,

B.P. 32, El Alia Bab Ezzouar, 16111 Alger, Alg´erie.

[email protected]

cLaboratoire de Math´ematiques Pures et Appliqu´ees,

Universit´e de Mostaganem, 27000 Mostaganem, Alg´erie.

limam [email protected] [email protected]

Abstract

Some transmission problems are set in bodies with a crown of small thicknessε >0. For instance, those concerning the conductivity in the biological cell. By a natural change of variables, we transform them in transmission problems set in two cylindrical bodies ]−∞,0[×]−π, π[ and ]0, δ[×]−π, π[, (where δ = ln (1 +ε)) and then, in some general elliptic abstract differential equations P^δ

δ>0. The goal of this first work is

to give a complete study of these problems P^δ

δ>0 for every δ > 0.

Existence, uniqueness and maximal regularity results are obtained for the classical solutions essentially by using the semigroups theory.

Key Words and Phrases. Biological cells, Transmission problems, Thin layers, Analytic semigroups.

AMS subject Classifications. 34K10, 34K30, 35J25, 35J40, 47A60.

1 Introduction

Let us consider the model of a biological cell, constituted of an homogeneous cytoplasm Ω^∗₋ (of boundary Γ^∗) centered at (0,0) with radius one micrometer surrounded by a thin membrane Ω^∗ε₊ (of boundary Γ^∗ε₊) with thickness of few nanometersε >0. The electric potential in this cell Ω^∗ε= Ω^∗₋∪Ω^∗ε₊ verifies the

∗This work has been supported by Ministry for Higher Education and Research, Algeria, under the PNR Program.

following problem

(P^ε)















∇.(µ∇w^ε) =µh^ε in Ω^∗ε

∂w^ε

∂n =l₊^ε on Γ^∗ε₊ Z

Γ^∗

w^ε(σ)dσ= 0, where

µ=

( µ− in Ω^∗₋ µ+ in Ω^∗ε₊,

(typically about 1 S/m (Siemens per meter), 5×10⁻⁷S/m respectively) are the conductivity positive coefficients of the two bodies Ω^∗₋, Ω^∗ε₊ depending possibly onε, and the electric charge density

h^ε=

( h₋ in Ω^∗₋ h^ε₊ in Ω^∗ε₊,

is taken, for instance, in space L^p(Ω^∗ε), 1 < p < ∞, that is h₋ ∈ L^p Ω^∗₋ , h^ε₊ ∈ L^p Ω^∗ε₊

and ∂/∂n denotes the outward normal derivative, l₊^ε is the electric field imposed on the boundary Γ^∗ε₊. The Neumann boundary condition on Γ^∗ε₊ implies the following compatibility condition onl₊^ε and (µh^ε)

Z

Ω^∗ε

(µh^ε) (x, y)dxdy+ Z

Γ^∗ε₊

µ+l₊^ε (σ)dσ= 0.

We will see that the gauge condition Z

Γ^∗

w^ε(σ)dσ= 0,

is imposed to have the uniqueness of the solution. Problem (P^ε) can be written in the complete form

P_x,y^ε

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

(eq.1) ∆w₋^ε (x, y) =h₋(x, y) in Ω^∗₋ (eq.2) ∆w₊^ε (x, y) =h^ε₊(x, y) in Ω^∗ε₊ (t.c.) w^ε₋=w^ε₊, µ₋∂w^ε₋

∂n =µ₊∂w^ε₊

∂n on Γ^∗ (g.c.)

Z

Γ^∗

w₋^ε (σ)dσ= Z

Γ^∗

w^ε₊(σ)dσ= 0 (b.c.) ∂w^ε₊

∂n =l^ε₊ on Γ^∗ε₊, under the compatibility condition

(CC_x,y^ε )









 Z

Ω^∗₋

(µ−h−) (x, y)dxdy+ Z

Ω^∗ε₊

µ+h^ε₊

(x, y)dxdy +

Z

Γ^∗ε₊

µ₊l₊^ε (σ)dσ= 0.

Several authors were interested in the study of transmission problems in different spaces with different boundary conditions in Hilbert spaces (see, for

instance, Calozet al. [3] and Nicaise [16]). In [8], Faviniet al. have considered some transmission problems set on a bounded domain for a second member in an interpolation space. They have used the Da Prato-Grisvard sum theory of linear operators.

A very interesting work is given in Doreet al. [5], where the authors have considered some transmission problems on cylindrical domains for second mem- bers only in L^p,p >1.

Many other authors have worked on this subject in many concrete biological situations. We cite, for example, Fear and Stuchly [9], [10], where the cyto- plasm is considered an homogeneous material. In Poignard [18], the method of asymptotic expansions is used to model this kind of problems.

In this first work our approach is quite different and uses the theory of operational differential equations set in L^p- spaces and the celebrated Dore- Venni Theorem.

Note that the small thickness of the membrane surrounding the cytoplasm leads to many numerical difficulties. Therefore, the important question is how to handle them and take into account the effect of this thin layer. So, our aim consists to solve (P^ε) for every smallε >0 and then, in a forthcoming work, to letε→0 in order to find the limiting problem well-posed in the cytoplasm Ω^∗₋ with a good impedance boundary condition on Γ^∗. This impedance condition will describe exactly the limiting effect of the thin membrane and will answer our question.

In order to give our main result in this work, let us recall the definition of the following Besov space for 0< s <1:

B_p,p^s (Γ^∗ε₊) =B^s_p(Γ^∗ε₊) =W^s,p(Γ^∗ε₊) :=

(

ψ∈L^p Γ^∗ε₊ :

Z

Γ^∗ε₊

Z

Γ^∗ε₊

|ψ(θ1)−ψ(θ₂)|^p

|θ1−θ2|^sp+1 dθ₁dθ₂<∞ )

, see Grisvard [11], p. 680 (here the dimension of Γ^∗ε₊ is one).

Theorem 1. Let l^ε₊ ∈ B¹⁻

1

p p Γ^∗ε₊

and h₋ ∈ L^p Ω^∗₋

, h^ε₊ ∈ L^p Ω^∗ε₊ , 1 <

p <∞with p6= 2, satisfying the above compatibility condition (CC_x,y^ε ). Then, problem P_x,y^ε

has a unique solution

w^ε=

( w^ε₋ inΩ^∗₋ w^ε₊ inΩ^∗ε₊ such that

w^ε₋∈W^2,p Ω^∗₋

, w^ε₊∈W^2,p Ω^∗ε₊

. (1)

This maximal L^p-regularity is very important to solve many quasilinear parabolic evolution equations corresponding to (P^ε). In fact, the use of the fixed point theorem to solve these nonlinear problems requires necessarily opti- mal regularities such (1).

The organization of the paper is as follows.

In the next section, we show that our model problem can be transformed by natural changes of variables into a particular abstract second order differential problem set in an unbounded cylindrical body. We then give an important

theorem (see Theorem 7) on this problem. In section 3, we collect some useful basic lemmas. In section 4, we solve two auxiliary problems in order to get the representation formula of the abstract solution. We then prove Theorem 7.

Finally, in section 5, we go back to our first problem in the biological cell in order to prove Theorem 1.

2 Operational formulation and main results

From the polar coordinatesx=rcosθ,y=rsinθ, we consider the changes ( v^ε₋(r, θ) =w^ε₋(rcosθ, rsinθ) andv₊^ε(r, θ) =w^ε₊(rcosθ, rsinθ)

k−(r, θ) =h−(rcosθ, rsinθ) andk^ε₊(r, θ) =h^ε₊(rcosθ, rsinθ), and for anyσ∈Γ^∗ε₊, we writeσ= (1 +ε) (cosθ,sinθ),θ∈[−π,+π[, and

l^ε₊(σ) =l^ε₊((1 +ε) (cosθ,sinθ)) :=L^ε₊(θ), (2) then, we have

Z

Γ^∗ε₊

l^ε₊(σ)dσ= Z +π

−π

l^ε₊((1 +ε) (cosθ,sinθ)) q

(1 +ε)²dθ

= (1 +ε) Z +π

−π

L^ε₊(θ)dθ, (3)

v^ε_±(r,−π) =w_±^ε(−r,0), v^ε_±(r, π) =w^ε_±(−r,0), and

∂v^ε_±

∂θ (r,−π) =−r∂w_±^ε

∂y (−r,0), ∂v_±^ε

∂θ (r, π) =−r∂w_±^ε

∂y (−r,0). It follows that

(P BC_r^ε)







v^ε₋(r,−π) =v^ε₋(r, π), ∂v₋^ε

∂θ (r,−π) =∂v₋^ε

∂θ (r, π), r∈(0,1) v^ε₊(r,−π) =v^ε₊(r, π), ∂v^ε₊

∂θ (r,−π) = ∂v^ε₊

∂θ (r, π), r∈(1,1 +ε).

Therefore, problem P_x,y^ε

becomes

P_r,θ^ε





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





















(eq.1) (r∂_r)²v^ε₋(r, θ) +∂_θ²v₋^ε (r, θ) =r²k₋(r, θ) inΩb₋ (eq.2) (r∂r)²v^ε₊(r, θ) +∂_θ²v₊^ε(r, θ) =r²k₊^ε(r, θ) inΩb^ε₊ (t.c.)







v₋^ε (1, θ) =v₊^ε (1, θ), θ∈(−π, π) µ₋∂v₋^ε

∂r (1, θ) =µ+

∂v₊^ε

∂r (1, θ), θ∈(−π, π) (g.c.)

Z π

−π

v^ε₋(1, θ)dθ= Z π

−π

v^ε₊(1, θ)dθ= 0 (b.c.) ∂v₊^ε

∂r (1 +ε, θ) =L^ε₊(θ), θ∈(−π, π),

with the periodic boundary conditions (P BC_r^ε) and the compatibility condition

(CC_r,θ^ε )









 Z

Ωb−

(µ₋k₋) (r, θ)rdrdθ+ Z

bΩ^ε₊

µ+k₊^ε

(r, θ)rdrdθ + (1 +ε)

Z +π

−π

µ₊L^ε₊(θ)dθ= 0;

here

Ωb₋:= (0,1)×(−π, π) ;Ωb^ε₊:= (1,1 +ε)×(−π, π). Using the following change of variables

Φ : (−∞, δ)×(−π, π) → (0,1 +ε)×(−π, π) (t, θ) 7→ Φ (t, θ) = (e^t, θ) = (r, θ), whereδ= ln(1 +ε), and the functions

u^δ₋(t, θ) =v^ε₋(r, θ), u^δ₊(t, θ) =v^ε₊(r, θ) g₋(t, θ) =r²k₋(r, θ), g^δ₊(t, θ) =r²k^ε₊(r, θ), our problem

P_r,θ^ε

becomes

P_t,θ^δ





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

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





(eq.1) ∆u^δ₋(t, θ) =g₋(t, θ) in Ω₋ (eq.2) ∆u^δ₊(t, θ) =g₊^δ (t, θ) in Ω^δ₊ (t.c.)







u^δ₋(0, θ) =u^δ₊(0, θ), θ∈(−π, π) µ₋∂u^δ₋

∂t (0, θ) =µ₊∂u^δ₊

∂t (0, θ), θ∈(−π, π) (g.c.)

Z π

−π

u^δ₋(0, θ)dθ= Z π

−π

u^δ₊(0, θ)dθ= 0 (b.c.) ∂u^δ₊

∂t (δ, θ) =f₊^δ (θ), θ∈(−π, π), with the periodic boundary conditions

(P BC_t^δ)











u^δ₋(t,−π) =u^δ₋(t, π), ∂u^δ₋

∂θ (t,−π) =∂u^δ₋

∂θ (t, π), t∈(−∞,0) u^δ₊(t,−π) =u^δ₊(t, π), ∂u^δ₊

∂θ (t,−π) = ∂u^δ₊

∂θ (t, π), t∈(0, δ), and the compatibility condition

(CC_t,θ^δ ) Z

Ω−

(µ₋g₋) (t, θ)dtdθ+ Z

Ω^δ₊

µ₊g^δ₊

(t, θ)dtdθ+ Z +π

−π

µ₊f₊^δ(θ)dθ= 0;

here

f₊^δ (θ) :=e^δL^ε₊(θ),Ω₋:= (−∞,0)×(−π, π),Ω^δ₊:= (0, δ)×(−π, π). Recall that we want to findw^ε_±inW^2,p Ω^∗ε_±, dxdy

for problem P_x,y^ε with h^ε_±∈L^p Ω^∗ε_±, dxdy

. This last assumption implies thatg^δ_± belong to weighted

spaces. Indeed we have Z

Ω^∗ε_±

h^ε_±(x, y)

pdxdy= Z

Ωb^ε_±

k^ε_±(r, θ)

prdrdθ

= Z

Ω^δ_±

k^ε_± e^t, θ

pe^2tdtdθ

= Z

Ω^δ_±

g^δ_±(t, θ)

pe^2(1−p)tdtdθ.

In other words the good assumption is the following g_±∈L^p_e(−2+2/p)t Ω^δ_±, dtdθ . Let us translate the regularities ofw_±^ε onu^δ_±. We have

w_±^ε ∈L^p Ω^∗ε_±, dxdy

⇔u^δ_±∈L^p_e(2/p)t Ω^δ_±, dtdθ . Then, using the fact that

∂w_±^ε

∂x (x, y) = cosθ∂v_±^ε

∂r (r, θ)−1

rsinθ∂v^ε_±

∂θ (r, θ)

∂w_±^ε

∂y (x, y) = sinθ∂v^ε_±

∂r (r, θ) +1

rcosθ∂v^ε_±

∂θ (r, θ), and after multiplying by cosθand sinθ, we deduce that

∂w^ε_±

∂x ,∂w^ε_±

∂y ∈L^p Ω^∗ε_±, dxdy , if and only if

∂u^δ_±

∂t ,∂u^δ_±

∂θ ∈L^p_e(−1+2/p)t Ω^δ_±, dtdθ . By the same way we get

∂²w^ε_±

∂x² ,∂²w^ε_±

∂y∂x,∂²w_±^ε

∂y² ∈L^p Ω^∗ε_±, dxdy , if and only if

∂²u^δ_±

∂t² , ∂²u^δ_±

∂t∂θ, ∂²u^δ_±

∂θ² ∈L^p_e(−2+2/p)t Ω^δ_±, dtdθ . Summarizing, the natural space foru^δ_± is

E^2,p Ω^δ_±, dtdθ

=n

u^δ_± :e(−|α|+2/p)t∂^αu^δ_±∈L^p Ω^δ_±, dtdθ

for |α|62o . Remark 2. Note that the exponent

$:=−2 + 2/p

is precisely the opposite of the Sobolev exponent of the space W^2,p in two vari- ables.

It would be difficult to use the weighted spaces mentioned above. This is why we define new unknown functions

U_±^δ(t, θ) =e^$tu^δ_±(t, θ), and new right hand sides

G^δ_±(t, θ) =e^$tg^δ_±(t, θ).

Remark 3. We observe that, if U_±^δ is in the space W^2,p Ω^δ_±, dtdθ

then u^δ_± belongs to E^2,p Ω^δ_±, dtdθ

. Moreover,G^δ_± ∈L^p Ω^δ_±, dtdθ

if and only if g^δ_± ∈ L^p_e$t Ω^δ_±, dtdθ

.

Then the equations verified byU_±^δ are

U P_t,θ^δ





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





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





















(eq.1) ∂_t²U₋^δ (t, θ)−2$∂tU₋^δ (t, θ) +$²U₋^δ (t, θ) +∂_θ²U₋^δ (t, θ)

=G₋(t, θ) in Ω₋

(eq.2) ∂_t²U₊^δ (t, θ)−2$∂tU₊^δ (t, θ) +$²U₊^δ (t, θ) +∂_θ²U₊^δ(t, θ)

=G^δ₊(t, θ) in Ω^δ₊

(t.c.)











U₋^δ (0, θ) =U₊^δ(0, θ), θ∈(−π, π) µ₋∂U₋^δ

∂t (0, θ)−µ+

∂U₊^δ

∂t (0, θ)

=$(µ₋−µ₊)U_±^δ (0, θ), θ∈(−π, π) (g.c.)

Z π

−π

U₋^δ (0, θ)dθ= Z π

−π

U₊^δ (0, θ)dθ= 0 (b.c.) ∂U₊^δ

∂t (δ, θ)−$U₊^δ (δ, θ) =F₊^δ (θ), θ∈(−π, π), with the periodic boundary conditions

(U P BC_t^δ)











U₋^δ (t,−π) =U₋^δ (t, π), ∂U₋^δ

∂θ (t,−π) =∂U₋^δ

∂θ (t, π), t∈(−∞,0) U₊^δ (t,−π) =U₊^δ(t, π), ∂U₊^δ

∂θ (t,−π) = ∂U₊^δ

∂θ (t, π), t∈(0, δ), and the compatibility condition

(U CC_t,θ^δ )









 µ₋

Z

Ω−

e^−$tG₋(t, θ)dtdθ+µ+

Z

Ω^δ₊

e^−$tG^δ₊(t, θ)dtdθ +µ₊e^−$δ

Z +π

−π

F₊^δ(θ)dθ= 0, whereF₊^δ(θ) :=e^$δf₊^δ (θ).

Recall thatE:=L^p_#(−π, π) is the space of 2π-periodic measurable functions onRwith locally summablep-th power, with norm

kψk_E= Z π

−π

|ψ(θ)|^pdθ ^1/p

.

LetP be the projection operator defined by P : L^p_#(−π, π) → R

ψ 7→ 1

2π Z π

−π

ψ(θ)dθ.

Now, set

E0:=L^p_#,0(−π, π) :=n

ψ∈L^p_#(−π, π) ;P ψ = 0o

=Ker(P), then

E=L^p_#(−π, π) =E₀⊕E₁, where

E₁:=R.f₁={λf₁, λ∈R}, f1(θ) = 1 for θ∈[−π, π[.

It is known that every closed subspaces of UMD Banach space is UMD; thus E0 is UMD (we recall that a Banach spaceX is a UMD space if and only if for some p >1 the Hilbert transform is continuous fromL^p(R;X) into itself, see Bourgain [1] and Burkholder [2]).

We define the following operators

D(A) ={ψ∈E:ψ⁰∈E, ψ⁰⁰∈E}:=W_#^2,p(−π, π) (Aψ)(θ) =ψ⁰⁰(θ), ψ∈D(A),

D(A0) ={ψ∈E0:ψ⁰∈E0, ψ⁰⁰∈E0}:=W_#,0^2,p(−π, π)

(A0ψ)(θ) =ψ⁰⁰(θ), ψ∈D(A0). (4) Operator Ais not injective inE butA₀is bijective inE₀.

We have

A(I−P)ψ=A0ψ, AP ψ= 0.

Setting

U_±^δ = (I−P)U_±^δ ⊕P U_±^δ :=W_±^δ +V_±^δ, and

H−:= (I−P)G−, H₊^δ := (I−P)G^δ₊, M₊^δ := (I−P)F₊^δ, problem

U P_t,θ^δ

with the periodic boundary conditions U P BC_t^δ

splits into the two following problems (respectively in E0 for the first and in E1 for the second)

W P_t^δ











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

















(eq.1) W₋^δ00

(t)−2$ W₋^δ0

(t) +$²W₋^δ(t) +A0W₋^δ(t)

=H−(t) on (−∞,0) (eq.2) W₊^δ00

(t)−2$ W₊^δ0

(t) +$²W₊^δ (t) +A0W₊^δ(t)

=H₊^δ (t) on (0, δ) (t.c.)

( W₋^δ(0) =W₊^δ(0) µ₋ W₋^δ0

(0)−µ+ W₊^δ0

(0) =$(µ₋−µ+)W_±^δ(0) (b.c.) W₊^δ0

(δ)−$W₊^δ(δ) =M₊^δ,

and

V P_t^δ



































(eq.1) V₋^δ00

(t)−2$ V₋^δ0

(t) +$²V₋^δ(t) =P G₋(t) on (−∞,0)

(eq.2) V₊^δ00

(t)−2$ V₊^δ0

(t) +$²V₊^δ(t) =P G^δ₊(t) on (0, δ)

(t.c.)

V₋^δ(0) =V₊^δ(0) µ₋ V₋^δ0

(0)−µ₊ V₊^δ0

(0) =$(µ₋−µ₊)V_±^δ(0) (g.c.) V₋^δ(0) =V₊^δ(0) = 0

(b.c.) V₊^δ0

(δ)−$V₊^δ(δ) =P F₊^δ, under the compatibility condition

V CC_t^δ µ₋

Z 0

−∞

e^−$tP G₋(t)dt+µ+

Z δ 0

e^−$tP G^δ₊(t)dt−µ+e^−$δF₊^δ = 0, with H₋ ∈L^p(−∞,0;E0), H₊^δ ∈L^p(0, δ;E0), 1< p < ∞, M₊^δ is some given element inE0.

For problem V P_t^δ

, we have the following proposition.

Proposition 4. Let F₊^δ ∈ E and G− ∈ L^p(−∞,0;E), G^δ₊ ∈ L^p(0, δ;E), 1< p <∞, satisfying V CC_t^δ

. Then, problem V P_t^δ

has a unique solution

V^δ =

V₋^δ in ]−∞,0[

V₊^δ in ]0, δ[, such that

V₋^δ ∈W^2,p(−∞,0;E1), V₊^δ ∈W^2,p(0, δ;E1). Proof. The solution is given by

V₋^δ(t) = Z 0

−∞

se^$(t−s)P G₋(s)ds+ Z t

−∞

(t−s)e^$(t−s)P G₋(s)ds,

V₊^δ(t) =te^$(t−δ)P F₊^δ + Z δ

0

se^$(t−s)P G^δ₊(s)ds− Z δ

t

(t−s)e^$(t−s)P G^δ₊(s)ds.

Now, let us specify the essential spectral properties of operator (A₀, D(A₀)).

Proposition 5. OperatorA₀defined by (4) is linear closed densely defined and boundedly invertible in E0 with the following properties

σ(A0) =

−n²:n∈N\ {0} ,

∃C >0,∀λ /∈]−∞,0[,

(A0−λI)⁻¹ _L(E

0)6 C 1 +|λ|, ( ∀s∈R,(−A0)^is∈L(E0) and

∃C >1,∃α∈]0, π[,∀s∈R,

(−A0)^is _L(E

0)6Ce^α|s|.

Proof. First, it is not difficult to have the invertibility ofA₀ with the following explicit representation of its inverse inE₀

A⁻¹₀ (ψ)(θ) = 1 2π

Z π

−π

[(θ+π)t−t²/2]ψ(t)dt+ Z θ

−π

(θ−t)ψ(t)dt, θ∈(−π, π), which gives obviously the continuity ofA⁻¹₀ onE0 and thusA0is closed. Since the inclusion D(A0)⊂W^2,p(−π, π) is compact we deduce by composition that A⁻¹₀ is compact, therefore the spectrumσ(A0) consists entirely of isolated eigen- values with finite multiplicities; see Kato [13], Theorem 6.29, p. 187. Now, the equation

A0ψ=λψ, means







ψ⁰⁰=λψ ψ(−π) =ψ(π) ψ⁰(−π) =ψ⁰(π);

then, necessarilyψ∈C²[−π, π] and ψ(θ) =C₁sinh√

λθ+C₂cosh√ λθ.

We have two cases: the first one is C1 =C2 = 0 which gives ψ ≡0, then no eigenvectors; the second one is

sinh√

λπ= 0 = sini√ λπ, that is√

λ=−inand soλ=−n²,n∈N; but the case n= 0 must be omitted since it corresponds toψ=C2 and the conditionP ψ= 0 leads toψ≡0 which cannot be an eigenvector. Finally

σ(A0) =

−n²:n∈N\ {0} .

Now an explicit calculus for the spectral equation A₀ϕ−λϕ = ψ in E₀, with λ /∈]−∞,0], gives

ϕ(θ) =

(A0−λI)⁻¹ψ (θ) = Z π

−π

K^√_λ(s, θ)ψ(s)ds, where

K^√_λ(s, θ) =−











cosh√

λ(π−s+θ) 2√

λsinh√

λπ ifθ6s6π cosh√

λ(π−θ+s) 2√

λsinh√

λπ if −π6s6θ.

It is not difficult to verify thatϕ∈D(A₀) and for any λ /∈]−∞,0],ψ∈E₀, we have, by the well known Schur Lemma

(A0−λI)⁻¹ψ

6 sup

θ∈[−π,π]

Z π

−π

K^√_λ(s, θ) ds

! kψk,

and Z π

−π

K^√_λ(s, θ)

ds6 1 2|λ|^1/2

sinh√

λπ

Z θ

−π

coshRe√

λ(π+s−θ)ds

+ 1

2|λ|^1/2 sinh√

λπ

Z π θ

coshRe√

λ(π−s+θ)ds

6sinhRe√

λπ−sinhRe√ λθ

|λ|^1/2Re√ λ

sinh√

λπ 6 sinhRe√

λπ

|λ|^1/2Re√ λ

sinh√

λπ

6 1

|λ|cos(^arg(λ)₂ ).

SinceE₀ is a UMD space, we deduce from the previous inequality thatD(A₀) is dense inE₀; see Haase [12], Proposition 2.1.1, pp. 18-19.

The property of bounded imaginary powers ofA₀is due to Pr¨uss-Sohr [19], Theorem C, p. 166.

Therefore, it is well known that

B:=−(−A0)^1/2 generates an analytic semigroup, e^tB

t>0, and in virtue of the spectral mapping theorem (see Haase [12], p. 56), we deduce thatσ(B) =−N\ {0}.On the other hand, there exist two positive constants a, M such that, for all t >0, for all m∈N\ {0}, we have

e^tB _L(E

0)6M e^−at,

B^me^tB _L(E

0)6M t^−me^−at; (5) see Pazy [17], Theorem 6.13, p. 74.

Moreover, it is well known that operator I+e^2δB−1

∈L(E₀); see Lunardi [15], Proposition 2.3.6, p. 60.

We will consider the two following natural operators B_$⁺:=B+$I, B_$⁻:=B−$I.

Note that, B_$⁺ with domainD(B_$⁺) =D(B) generates an analytic semigroup;

see Engel-Nagel [7], Proposition 1.12, p. 164. It is boundedly invertible since for allp∈]1,∞[,−$ is never equal to−n, n∈N\ {0}.

It is not the same for the second operatorB⁻_$; since for the unique particular casep= 2, $coincides with the spectral value

$=−2 + 2 p=−1.

This is why we will assume in all this work that p∈]1,∞[ andp6= 2.

Remark 6. Note that the situationp= 2corresponds to hilbertian case which can be treated by the usual variational method.

We will study existence and uniqueness for a natural solution of W P_t^δ , that is a function

W^δ =

( W₋^δ in ]−∞,0[

W₊^δ in ]0, δ[, such that

W₋^δ ∈W^2,p(−∞,0;E0)∩L^p(−∞,0;D(A0)),

W₊^δ ∈W^2,p(0, δ;E0)∩L^p(0, δ;D(A0)), (6) and verifying W P_t^δ

. Then for our problem W P_t^δ

, we have the following.

Theorem 7. Let H− ∈ L^p(−∞,0;E0), H₊^δ ∈ L^p(0, δ;E0), 1 < p < ∞ with p6= 2. Then, problem W P_t^δ

has a unique solution W^δ satisfying (6) if and only ifM₊^δ ∈(D(A₀), E₀)1

2p+¹₂,p.

We recall that for allα∈]0,1[ andq∈[1,∞], the space (E0, D(A0))_α,q= (D(A0), E0)_1−α,q

is a real interpolation space between D(A0) and E0, defined for instance in Triebel [20], p. 96.

This theorem is proved in section 5. Our techniques are essentially based on the Dunford operational calculus, the analytic semigroup theory and the celebrated Dore-Venni Theorem.

Remark 8. In this work, we have considered problem W P_t^δ

withA0 defined by (4). However, we can also study it when A0 is some closed linear opera- tor of domain D(A0) in some complex Banach space E0 under the following assumptions

E0 is a UMD space,

ρ(−A0) ={λj}_j>1 with0< λ1< λ2< λ3...,

∀j>1,2−2 p6=p

λj,

∃C >0,∀λ∈[0,+∞[,

(A0−λI)⁻¹ _L(E

0)6 C 1 +|λ|, ( ∀s∈R,(−A0)^is∈L(E0) and

∃C >1,∃α∈]0, π[,∀s∈R,

(−A₀)^is _L(E

0)6Ce^α|s|.

Remark 9. If we have consideredΩ^∗₋as the unit ball ofR^m, then by analogous calculus we obtain the equations

W P_t^δ,m































W₋^δ00

(t)−(m−2−2$) W₋^δ0

(t) +$($−m+ 2)W₋^δ(t) +A0W₋^δ(t) =H₋(t) on (−∞,0)

W₊^δ00

(t)−(m−2−2$) W₊^δ0

(t) +$($−m+ 2)W₊^δ(t) +A₀W₊^δ(t) =H₊^δ (t) on (0, δ)

W₋^δ(0) =W₊^δ(0) µ₋ W₋^δ0

(0)−µ₊ W₊^δ0

(0) = (µ₋−µ₊)$W_±^δ(0), W₊^δ0

(δ)−$W₊^δ(δ) =M₊^δ,

where$=−2+m/pandA₀is the Laplace-Beltrami operator on the unit sphere S^m−1. Here, we must assume for allj>1 that

$($−m+ 2) = m

p −2 m p −m

6=λ_j. (7) Consider in the Banach space X =L^p(−∞, δ;E), the following operators











D(Λ1) ={W^δ = (W₋^δ, W₊^δ)∈L^p(−∞,0;E)×L^p(0, δ;E) : W₋^δ (0) =W₊^δ(0), µ₋ W₋^δ0

(0)−µ+ W₊^δ0

(0) = (µ₋−µ+)$W_±^δ(0) and W₊^δ0

(δ)−$W₊^δ(δ) = 0}

Λ1W^δ = (W^δ)⁰⁰−(m−2−2$) W^δ0

+$($−m+ 2)W^δ,







D(Λ2) ={W^δ= (W₋^δ, W₊^δ)∈L^p(−∞,0;E)×L^p(0, δ;E) : W^δ(t)∈D(A0)a.e. t∈(−∞, δ)}

(Λ₂W^δ)(t) =A₀W^δ(t).

We can verify that operators −Λ1 and −Λ2 are sectorial and that their resol- vents commute. We then know that if the spectra σ(Λ₁) and σ(−Λ₂) do not intersect, then Λ₁+ Λ₂ is closable. This last condition means that (7) is satis- fied. Therefore, we can conjecture that in the critical cases (that is when (7) is not satisfied), the sum Λ1+ Λ2 is not closable.

3 Technical lemmas

Set, for allf ∈L^p(a, b;E₀), witha < b, and for a.e. t∈(a, b) F(f) (t) :=

Z b a

f(s) s+tds.

SinceE0 is a UMD space, we have

F ∈L(L^p(a, b;E0)). (8)

In fact, we have forf ∈L^p(a, b;E0) and for a.e. t∈(a, b) F(f) (t) =

Z b a

f(s) s+tds=

Z

R

g(s) s+tds=

Z

R

g(τ−t) τ dτ,

whereg=I[a,b]f (Iis a characteristic function). On the other hand, we have F(f) (t) = lim

ε→0

Z

ε6|τ|6¹_ε

g(τ−t)

τ dτ =V_p 1

τ

∗g(−·) =H(g(−·)) (t), where V_p is the Cauchy principal value and H is the Hilbert transform. The desired result follows from the fact that E₀is a UMD space.

We need the following lemmas.

Lemma 10. Let Q ∈ {B, B_$⁻, B_$⁺} and f ∈ L^p(0, δ;E₀), 1 < p < ∞ with p6= 2. Then, the following applications

t7→Q Z t

0

e^(t−s)Qf(s)ds, t7→Q

Z δ t

e^(s−t)Qf(s)ds, t7→Q

Z δ 0

e^(t+s)Qf(s)ds,

are well defined for a.e. t∈(0, δ)and belong toL^p(0, δ;E0).

Proof. For the first and the second applications, it is a consequence of the Dore- Venni Theorem [6]. For the third, it suffices to write for a.e. t∈(0, δ)

Q Z δ

0

e^(t+s)Qf(s)ds=Q Z t

0

e^(t+s)Qf(s)ds+Q Z δ

t

e^(t+s)Qf(s)ds

=Q Z t

0

e^(t−s)Q(e^2sQf(s))ds+e^2tQQ Z δ

t

e^(s−t)Qf(s)ds.

Lemma 11. Let Q∈ {B, B_$⁻, B_$⁺} andf ∈L^p(−∞,0;E₀), 1 < p < ∞ with p6= 2. Then, the following applications

t7→Q Z 0

t

e^(s−t)Qf(s)ds, t7→Q

Z t

−∞

e^(t−s)Qf(s)ds, t7→Q

Z 0

−∞

e^−(t+s)Qf(s)ds,

are well defined for a.e. t∈(−∞,0)and belong toL^p(−∞,0;E0).

Proof. Write, for a.e. t∈(−∞,0), Q

Z 0 t

e^(s−t)Qf(s)ds=Q Z −t

0

e^(−t−τ)Qf(−τ)dτ and consider the change −t=xwithx∈(0,+∞). Then

Q Z 0

t

e^(s−t)Qf(s)ds=Q Z x

0

e^(x−τ)Qf(−τ)dτ, wheret7→f(−t)∈L^p(0,+∞;E0). Then, it is clear that

t7→Q Z 0

t

e^(s−t)Qf(s)ds∈L^p(−∞,0;E0), if and only if

t7→Q Z t

0

e^(t−τ)Qf(−τ)dτ ∈L^p(0,∞;E0).

In virtue of the Dore-Venni Theorem [6], we have t7→Q

Z t 0

e^(t−τ)Qf(−τ)dτ ∈L^p(0, T;E0), for allT >0 and by Theorem 2.4, p. 28 in Dore [4], we get

t7→Q Z t

0

e^(t−τ)Qf(−τ)dτ ∈L^p(0,∞;E0).

For the second application, the proof is not obvious, the idea follows from the method used in Dore [4], pp. 28-29. We have for a.e. t∈(−∞,0),

Q Z t

−∞

e^(t−s)Qf(s)ds=Q Z t−1

−∞

e^(t−s)Qf(s)ds+Q Z t

t−1

e^(t−s)Qf(s)ds.

For the first integral, we use (5) to get Z 0

−∞

Z t−1

−∞

Qe^(t−s)Qf(s)ds

p

dt 6M^p

Z 0

−∞

Z t−1

−∞

1

t−se^−(t−s)akf(s)kds ^p

dt 6M^p

Z 0

−∞

Z t−1

−∞

e^−(t−s)akf(s)kds ^p

dt.

Now, we can write Z t−1

−∞

e^−(t−s)akf(s)kds= (h∗Ψ) (t), where

h(ξ) =

0 ifξ61 e^−ξa ifξ >1, Ψ(ω) =

kf(ω)k ifω60 0 ifω >0,

it is clear that Ψ∈L^p(R) andh∈L¹(R). Then the Young’s inequality implies h∗Ψ∈L^p(R), that is

Z 0

−∞

Q

Z t−1

−∞

e^(t−s)Qf(s)ds

p

dt <∞.

It remains to estimate Z 0

−∞

Q

Z t t−1

e^(t−s)Qf(s)ds

p

dt.

Consider, for j∈N, the truncated functions defined by f_−j =I[−j−1,−j[f,

where I[−j−1,−j[ denotes the characteristic function of the set [−j−1,−j[.

Then, we have Z 0

−∞

Q

Z t t−1

e^(t−s)Qf(s)ds

p

dt

=

∞

X

j=0

Z −j

−j−1

Q

Z −j−1 t−1

e^(t−s)Qf−j−1(s)ds+Q Z t

−j−1

e^(t−s)Qf−j(s)ds

p

dt

62^p−1

∞

X

j=0

Z −j

−j−1

Q

Z −j−1 t−1

e^(t−s)Qf_−j−1(s)ds

p

dt

+ 2^p−1

∞

X

j=0

Z −j

−j−1

Q

Z t

−j−1

e^(t−s)Qf_−j(s)ds

p

dt.

Set 







 Ij =

Z −j

−j−1

Q Z t

−j−1

e^(t−s)Qf−j(s)ds

p

dt, Jj =

Z −j

−j−1

Q

Z −j−1 t−1

e^(t−s)Qf−j−1(s)ds

p

dt.

Then, by the changes of variablesτ=−1−t−j andσ=−s−j−1, we obtain Ij =

Z 0

−1

Q

Z −j−1−τ

−1−j

e(−1−j−τ−s)Qf_−j(s)ds

p

dτ

= Z 0

−1

Q

Z 0 τ

e^(σ−τ)Qf−j(−σ−j−1)dσ

p

dτ, therefore, in virtue of Dore-Venni [6], there existsC1>0 such that

I_j 6C₁^pkf_−j(−1−j− ·))k^p_Lp(−1,0;E

0)6C₁^pkf_−jk^p_{Lp(−j−1,−j;E}

0). (9) For J_j, we use the changes of variables τ =−1−t−j and y = s+j+ 1 to obtain

Jj= Z 0

−1

Q

Z −j−1

−2−τ−j

e(−1−τ−j−s)Qf_−j−1(s)ds

p

dτ

= Z 0

−1

Q

Z 0

−1−τ

e^(−τ−y)Qf−j−1(y−j−1)dy

p

dτ 6

Z 0

−1

Z 0

−1

Qe^(−τ−y)Qf−j−1(y−j−1) dy

^p dτ 6M^p

Z 0

−1

Z 0

−1

−1

τ+ykf−j−1(y−j−1)kdy ^p

dτ.

Now, by (8) the kernel −1

τ+y defines a bounded operator onL^p(−1,0;R), then J_j 6C₂M^pkf_−j−1(· −j−1))k^p_Lp(−1,0;E0) (10)

6C₂M^pkf−j−1k^p_Lp(−j−2,−j−1;E0).

Using (9) and (10), we conclude that Z 0

−∞

kQ Z t

t−1

e^(t−s)Qf(s)dsk^pdt6C22^p−1M^p

∞

X

j=0

kf_−j−1k^p_Lp(−j−2,−j−1;E0)

+ 2^p−1C₁^p

∞

X

j=0

kf_−jk^p_Lp(−j−1,−j;E₀).

Since















∞

X

j=0

kf_−jk^p_Lp(−j−1,−j;E₀)=kfk^p_Lp(−∞,0;E0),

∞

X

j=0

kf_−j−1k^p_Lp(−j−2,−j−1;E0)6kfk^p_Lp(−∞,0;E0), we get

Z 0

−∞

kQ Z t

t−1

e^(t−s)Qf(s)dsk^pdt62^pmax (C₁^p, C₂M^p)kfk^p_Lp(−∞,0;E0). For the third application, we write for a.e. t∈(−∞,0)

Q Z 0

−∞

e^−(t+s)Qf(s)ds

=e^−2tQQ Z t

−∞

e^(t−s)Qf(s)ds+Q Z 0

t

e^(s−t)Q(e^−2sQf(s))ds, and apply the previous results.

Lemma 12. Let Q ∈ {B, B_$⁻, B_$⁺} and f ∈ L^p(0, δ;E₀), 1 < p < ∞ with p6= 2. Then

t7→Q Z 0

−∞

e^(t−s)Qf(s)ds∈L^p(0, δ;E0). Proof. We have by the change of variablest=−x

Z δ 0

Q

Z 0

−∞

e^(t−s)Qf(s)ds

p

dt= Z 0

−δ

Q

Z 0

−∞

e^(−x−s)Qf(s)ds

p

dx 6

Z 0

−∞

Q

Z 0

−∞

e^−(x+s)Qf(s)ds

p

dx <∞, see Lemma 11.

Lemma 13. Let Q∈ {B, B_$⁻, B_$⁺} andf ∈L^p(−∞,0;E0), 1 < p < ∞ with p6= 2. Then

t7→Q Z δ

0

e^(s−t)Qf(s)ds∈L^p(−∞,0;E0).