Singular boundary conditions and regularity for the biharmonic problem in the half-space

HAL Id: hal-00206217

https://hal.archives-ouvertes.fr/hal-00206217

Preprint submitted on 16 Jan 2008

HAL

is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire

destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Singular boundary conditions and regularity for the biharmonic problem in the half-space

Chérif Amrouche, Yves Raudin

To cite this version:

Chérif Amrouche, Yves Raudin. Singular boundary conditions and regularity for the biharmonic

problem in the half-space. 2008. �hal-00206217�

AIMS’ Journals

VolumeX, Number0X, XX200X _pp.X–XX

SINGULAR BOUNDARY CONDITIONS AND REGULARITY FOR THE BIHARMONIC PROBLEM IN THE HALF-SPACE

Ch´erif Amrouche, Yves Raudin

Laboratoire de Math´ematiques Appliqu´ees Universit´e de Pau et des Pays de l’Adour

IPRA, Avenue de l’Universit´e 64000 Pau, France

Abstract. In this paper, we are interested in some aspects of the biharmonic equation in the half-spaceR^N₊, withN≥2. We study the regularity of gener- alized solutions in weighted Sobolev spaces, then we consider the question of singular boundary conditions. To finish, we envisage other sorts of boundary conditions.

1. Introduction. The purpose of this paper is the resolution of the biharmonic problem with nonhomogeneous boundary conditions

(P)







∆²u=f inR^N₊, u=g0 on Γ =R^N−1,

∂Nu=g1 on Γ.

Since this problem is posed in the half-space, it is important to specify the behaviour at infinity for the data and solutions. We have chosen to impose such conditions by setting our problem in weighted Sobolev spaces, where the growth or decay of func- tions at infinity are expressed by means of weights. These weighted Sobolev spaces provide a correct functional setting for unbounded domains, in particular because the functions in these spaces satisfy an optimal weighted Poincar´e-type inequality.

Our analysis is based on the isomorphism properties of the biharmonic operator in the whole space and the resolution of the Dirichlet and Neumann problems for the Laplacian in the half-space. This last one is itself based on the isomorphism properties of the Laplace operator in the whole space and also on the reflection principle inherent in the half-space. Note here the double difficulty arising from the unboundedness of the domain in any direction and from the unboundedness of the boundary itself.

In a previous work (see [7]), we established the existence of generalized solutions to problem (P), i.e. solutions which belong to weighted Sobolev spaces of type W_l^{2, p}(R^N₊). For the sake of convenience, we shall recall the necessary results of this paper in Section 2.4. Here, we are interested both in the existence of more regular solutions, as for instance strong solutions which belong to spaces of typeW_l+2^{4, p}(R^N₊), and singular solutions which belong toW_l−2^{0, p}(R^N₊) in the casef = 0 with singular

2000Mathematics Subject Classification. 35D05, 35D10, 35J35, 35J40.

Key words and phrases. Biharmonic problem, Half-space, Weighted Sobolev spaces.

1

2 CH´ERIF AMROUCHE, YVES RAUDIN

boundary conditions. We also establish the existence of solutions which belong to intermediate spaces as for exampleW_l+1^{3, p}(R^N₊).

As we saw (see [7]), it turns out that the use of classical Sobolev spaces is inadequate in this case, contrary to the study of elliptic problems of type:

(Q) u+ ∆²u=f inR^N₊, u=g0 and ∂Nu=g1 on Γ,

where it is more reasonable to consider data and solutions in standard Sobolev spaces. For example, if f ∈ L²(R^N₊), g0 ∈ H^7/2(R^N−1) and g1 ∈ H^5/2(R^N−1), problem (Q) admits an unique solutionu∈H⁴(R^N₊). In the case of problem (P), if we assume thatf ∈L²(R^N₊), the solutionucan not be better than inW₀^4,2(R^N₊) and its traces u|Γ and ∂Nu|Γ respectively in W₀^7/2,2(R^N−1) and W₀^5/2,2(R^N−1).

Moreover, we can observe thatH⁴(R^N₊) is a subspace of W₀^4,2(R^N₊) and the same remark holds for the traces spaces.

On the one hand, we can find in the literature an approach via homogeneous spaces. For instance, when f ∈L²(R^N₊), that consists in finding solutions to (P) satisfying∇⁴u∈L²(R^N₊)^N

4

, but that gives no information on the other derivatives, nor specifes the behavior at infinity for the data and solutions.

On the other hand, Boulmezaoud has established (see [9]) in a Hilbertian frame- work, the existence of solutions u ∈ H_l+1^3,2(R^N₊) for data f ∈ H_l+1^−1,2(R^N₊) and the corresponding regularity result. However, owing to some critical cases, this frame- work excludes in particular the dimensions 2 and 4 (see Section 3.3).

To reduce the set of critical values, we have used a special class of weighted Sobolev spaces with logarithmic factors (see Section 2.2 and Remark 2.1). We shall show the part of these logarithmic factors in the weights particularly in the question of singular boundary conditions.

This paper is organized as follows. Section 2 is devoted to the notations, func- tional setting and main results of our previous paper on the biharmonic operator, which is the source of the present work. In Section 3, we study the regularity of the solutions to problem (P) according to the data. After that, we give a panorama of basic cases and we recall the Boulmezaoud theorem to throw light on our contribu- tion in this study. In Section 4, we come back to the homogeneous problem (f = 0) with singular boundary conditions. Lastly, in Section 5, we shall consider the bihar- monic equation, but with different types of boundary conditions. The main results of this paper are Theorem 3.1 for the regularity and Theorem 4.4 for the singular boundary conditions. In a forthcoming work, we shall use these results to solve the Stokes system.

2. Notations, functional framework and useful results.

2.1. Notations. For any real numberp > 1, we always takep^′ to be the H¨older conjugate ofp,i.e.

1 p+ 1

p^′ = 1.

Let Ω be an open set ofR^N,N≥2. Writing a typical pointx∈R^N asx= (x^′, xN), wherex^′ = (x1, . . . , xN−1)∈R^N−1andxN ∈R, we will especially look on the upper half-spaceR^N₊ ={x∈R^N; xN >0}. We letR^N₊ denote the closure ofR^N₊ inR^N and let Γ ={x∈R^N; xN = 0} ≡R^N−1denote its boundary. Let|x|= (x²₁+· · ·+x²_N)^1/2

denote the Euclidean norm ofx, we will use two basic weights

̺= (1 +|x|²)^1/2 and lg̺= ln(2 +|x|²).

We denote by ∂i the partial derivative ∂

∂xi

, similarly ∂_i² = ∂i◦∂i = ∂²

∂x²_i, ∂_ij² =

∂i◦∂j = ∂²

∂xi∂xj

, . . . More generally, ifλ= (λ1, . . . , λN)∈N^N is a multi-index, then

∂^λ=∂₁^λ1· · ·∂_N^λN = ∂^|λ|

∂x^λ₁¹· · ·∂x^λ_N^N , where|λ|=λ1+· · ·+λN. In the sequel, for any integerq, we shall use the following polynomial spaces:

—P_q is the space of polynomials of degree smaller than or equal toq;

—P_q^∆ is the subspace of harmonic polynomials ofP_q;

—P_q^∆2 is the subspace of biharmonic polynomials ofP_q;

—A_q^∆ is the subspace of polynomials ofP_q^∆, odd with respect toxN, or equiva- lently, which satisfy the conditionϕ(x^′,0) = 0;

—N ^∆

q is the subspace of polynomials ofP^∆

q , even with respect toxN, or equiva- lently, which satisfy the condition∂Nϕ(x^′,0) = 0;

with the convention that these spaces are reduced to{0}ifq <0.

For any real numbers, we denote by [s] the integer part ofs.

Given a Banach space B, with dual space B^′ and a closed subspaceX of B, we denote byB^′⊥X the subspace ofB^′ orthogonal toX,i.e.

B^′⊥X={f ∈B^′; ∀v∈X, hf, vi= 0}= (B/X)^′.

Lastly, ifk∈Z, we shall constantly use the notation{1, . . . , k}for the set of the first kpositive integers, with the convention that this set is empty ifkis nonpositive.

2.2. Weighted Sobolev spaces. For any nonnegative integer m, real numbers p >1,αandβ, we define the following space (see [3], Section 7):

W_{α, β}^{m, p}(Ω) =n

u∈D^′(Ω); 0≤ |λ| ≤k, ̺^α−m+|λ|(lg̺)^β−1∂^λu∈L^p(Ω);

k+ 1≤ |λ| ≤m, ̺^α−m+|λ|(lg̺)^β∂^λu∈L^p(Ω)o ,

(1) where

k=







−1 if ^N_p +α /∈ {1, . . . , m}, m−N

p −α if ^N_p +α∈ {1, . . . , m}.

Note thatW_{α, β}^{m, p}(Ω) is a reflexive Banach space equipped with its natural norm:

kuk_W^{m, p}

α, β(Ω)= X

0≤|λ|≤k

k̺^α−m+|λ|(lg̺)^β−1∂^λuk^p_Lp(Ω)

+ X

k+1≤|λ|≤m

k̺^α−m+|λ|(lg̺)^β∂^λuk^p_Lp(Ω)

1/p

.

We also define the semi-norm:

|u|_W^{m, p}

α, β(Ω)= X

|λ|=m

k̺^α(lg̺)^β∂^λuk^p_Lp(Ω)

1/p

.

4 CH´ERIF AMROUCHE, YVES RAUDIN

The weights in the definition (1) are chosen so that the corresponding space satisfies two fundamental properties. On the one hand,D R^N

+

is dense inW_{α, β}^{m, p}(R^N

+). On the other hand, the following Poincar´e-type inequality holds inW_{α, β}^{m, p}(R^N₊) (see [5], Theorem 1.1): if

N

p +α /∈ {1, . . . , m} or (β−1)p6=−1, (2) then the semi-norm| · |_W^{m, p}

α, β(R^N

+)defines onW_{α, β}^{m, p}(R^N₊)/P_q′ a norm which is equiv- alent to the quotient norm,

∀u∈W_{α, β}^{m, p}(R^N

+), kuk_W^{m, p}

α, β(R^N

+)/P_q′ ≤C|u|_W^{m, p}

α, β(R^N

+), (3)

withq^′ = inf(q, m−1), whereq is the highest degree of the polynomials contained inW_{α, β}^{m, p}(R^N₊). Now, we define the space

W◦^{m, p}_{α, β}(R^N₊) =D(R^N₊)^{k·kWα, βm, p(RN+)};

which will be characterized in Lemma 2.2 as the subspace of functions with null traces in W_{α, β}^{m, p}(R^N₊). From that, we can introduce the spaceW_−α,^{−m, p}_−β^′(R^N₊) as the dual space of W^{◦m, p}α, β(R^N₊). In addition, under the assumption (2),| · |_W^{m, p}

α, β(R^N

+)is a norm onW^{◦m, p}_{α, β}(R^N₊) which is equivalent to the full normk · k_W^{m, p}

α, β(R^N

+).

We will still recall some properties of the weighted Sobolev spacesW_{α, β}^{m, p}(R^N₊). We have the algebraic and topological imbeddings:

W_{α, β}^{m, p}(R^N₊)֒→W_{α−1, β}^{m−1, p}(R^N₊)֒→ · · ·֒→W_{α−m, β}^{0, p} (R^N₊) if N

p +α /∈ {1, . . . , m}.

When N

p +α=j∈ {1, . . . , m}, then we have:

W_{α, β}^{m, p}֒→ · · ·֒→W_{α−j+1, β}^{m−j+1, p}֒→W_{α−j, β−1}^{m−j, p} ֒→ · · ·֒→W_{α−m, β−1}^{0, p} . Note that in the first case, for any γ ∈Rsuch that N

p +α−γ /∈ {1, . . . , m} and m∈N, the mapping

u∈W_{α, β}^{m, p}(R^N₊)7−→̺^γu∈W_{α−γ, β}^{m, p} (R^N₊)

is an isomorphism. In both cases and for any multi-indexλ∈N^N, the mapping u∈W_{α, β}^{m, p}(R^N₊)7−→∂^λu∈W_{α, β}^{m−|λ|, p}(R^N₊)

is continuous. Finally, it can be readily checked that the highest degree q of the polynomials contained inW_{α, β}^{m, p}(R^N₊) is given by

q=













 m−

N p +α

−1, if







N

p +α∈ {1, . . . , m} and (β−1)p≥ −1, or

N

p +α∈ {j∈Z; j≤0} and βp≥ −1,

m− N

p +α

, otherwise.

(4)

Remark 2.1. In the caseβ = 0, we simply denote the spaceW_α,^{m, p}₀ (Ω) byW_α^{m, p}(Ω).

In [11], Hanouzet introduced a class of weighted Sobolev spaces without logarith- mic factors, with the same notation W_α^{m, p}(Ω). We recall his definition under the notationH_α^{m, p}(Ω):

H_α^{m, p}(Ω) =n

u∈D^′(Ω); 0≤ |λ| ≤m, ̺^α−m+|λ|∂^λu∈L^p(Ω)o .

It is clear that if N

p +α /∈ {1, . . . , m}, we have W_α^{m, p}(Ω) = H_α^{m, p}(Ω). The fun- damental difference between these two families of spaces is that the assumption (2) and thus the Poincar´e-type inequality (3), hold for any value of (N, p, α) in W_α^{m, p}(Ω), but not inH_α^{m, p}(Ω) if N

p +α∈ {1, . . . , m}.

2.3. The spaces of traces. In order to define the traces of functions ofW_α^{m, p}(R^N₊) (here we don’t consider the caseβ6= 0), for any σ∈]0,1[, we introduce the space:

W₀^{σ, p}(R^N) =

u∈D^′(R^N); w^−σu∈L^p(R^N) and ∀i= 1, . . . , N, Z +∞

0

t^−1−σpdt Z

RN

|u(x+tei)−u(x)|^pdx <∞

,

(5)

where w = ̺ if N/p 6= σ and w = ̺(lg̺)^1/σ if N/p = σ, and e1, . . . , eN is the canonical basis of R^N. It is a reflexive Banach space equipped with its natural norm:

kuk_W^{σ, p}

0 (R^N)=

u w^σ

p

L^p(RN)+

N

X

i=1

Z +∞

0

t^−1−σpdt Z

R^N

|u(x+tei)−u(x)|^pdx ^1/p

which is equivalent to the norm

u w^σ

p

L^p(RN)+ Z

R^N×R^N

|u(x)−u(y)|^p

|x−y|^N+σp dxdy ^1/p

.

Similarly, for any real numberα∈R, we define the space:

W_α^{σ, p}(R^N) =

u∈D^′(R^N); w^α−σu∈L^p(R^N), Z

R^N×R^N

|̺^α(x)u(x)−̺^α(y)u(y)|^p

|x−y|^N+σp dxdy <∞

,

where w = ̺ if N/p+α 6= σ and w = ̺(lg̺)^1/(σ−α) if N/p+α = σ. For any s∈R⁺, we set

W_α^{s, p}(R^N) =n

u∈D^′(R^N); 0≤ |λ| ≤k, ̺^α−s+|λ|(lg̺)⁻¹∂^λu∈L^p(R^N);

k+ 1≤ |λ| ≤[s]−1, ̺^α−s+|λ|∂^λu∈L^p(R^N); |λ|= [s], ∂^λu∈W_α^{σ, p}(R^N)o ,

6 CH´ERIF AMROUCHE, YVES RAUDIN

wherek=s−N/p−αifN/p+α∈ {σ, . . . , σ+ [s]}, withσ=s−[s] andk=−1 otherwise. It is a reflexive Banach space equipped with the norm:

kuk_Wα^{s, p}₍RN)= X

0≤|λ|≤k

k̺^α−s+|λ|(lg̺)⁻¹∂^λuk^p_Lp(RN)

+ X

k+1≤|λ|≤[s]−1

k̺^α−s+|λ|∂^λuk^p_Lp(RN)

^1/p

+ X

|λ|=[s]

k∂^λuk_Wα^{σ, p}₍RN).

We can similarly define, for any real numberβ, the space:

W_{α, β}^{s, p}(R^N) =n

v∈D^′(R^N); (lg̺)^βv∈W_α^{s, p}(R^N)o .

We can prove some properties of the weighted Sobolev spacesW_{α, β}^{s, p}(R^N). We have the algebraic and topological imbeddings ifN/p+α /∈ {σ, . . . , σ+ [s]−1}:

W_{α, β}^{s, p}(R^N)֒→W_{α−1, β}^{s−1, p}(R^N)֒→ · · ·֒→W_{α−[s], β}^{σ, p} (R^N), W_{α, β}^{s, p}(R^N)֒→W_{α+[s]−s, β}^{[s], p} (R^N)֒→ · · ·֒→W_{α−s, β}^{0, p} (R^N).

WhenN/p+α=j∈ {σ, . . . , σ+ [s]−1}, then we have:

W_{α, β}^{s, p} ֒→ · · ·֒→W_{α−j+1, β}^{s−j+1, p} ֒→W_{α−j, β−1}^{s−j, p} ֒→ · · ·֒→W_{α−[s], β−1}^{σ, p} ,

W_{α, β}^{s, p} ֒→W_{α+[s]−s, β}^{[s], p} ֒→ · · ·֒→W_{α−σ−j+1, β}^{[s]−j+1, p} ֒→W_{α−σ−j, β−1}^{[s]−j, p} ֒→ · · ·֒→W_{α−s, β−1}^{0, p} . Ifuis a function onR^N₊, we denote its trace of orderj on the hyperplane Γ by:

∀j∈N, γju:x^′ ∈R^N−17−→∂^j_Nu(x^′,0).

Let’s recall the following trace lemma due to Hanouzet (see [11]) for the functions of H_α^{m, p}(R^N

+) and extended by Amrouche-Neˇcasov´a (see [5]) to the class of weighted Sobolev spacesW_α^{m, p}(R^N

+):

Lemma 2.2. For any integerm≥1 and real number α, the mapping γ= (γ0, γ1, . . . , γm−1) : D R^N₊

−→

m−1

Y

j=0

D(R^N−1), can be extended to a linear continuous mapping, still denoted byγ,

γ: W_α^{m, p}(R^N₊)−→

m−1

Y

j=0

W_α^{m−j−1/p, p}(R^N−1).

Moreover γ is surjective andKerγ=W^{◦m, p}α (R^N₊).

2.4. Previous results on∆². First note that if we replaceR^N₊ by a bounded open set Ω ofR^N in problem (P), the natural functional framework in which we find the solutionuis the one of classical Sobolev spaces. But as the things are well made, in this case the weighted spaces coincide with the classical spaces. Concerning the regularity results on bounded domains, we can refer to Luckhaus (see [12]) for Dirichlet boundary conditions and to the Appendix B in the paper by R. van der Vorst (see [15]) for boundary conditions on (u,∆u) that we shall see in Section 5.

In unbounded domains, we must recall the mains results of our previous paper on the biharmonic equation (see [7]), which are the start point of this study. First, let’s give the global result for the biharmonic operator in the whole space:

Theorem 2.3 (Amrouche-Raudin [7]). Let l∈Zandm∈Nand assume that N

p^′ ∈ {1, . . . , l/ + min{m,2}} and N

p ∈ {1, . . . ,/ −l−m}, (6) then the biharmonic operator

∆²:W_m+l^{m+2, p}(R^N)/P^∆2

[2−l−N/p] −→W_m+l^{m−2, p}(R^N)⊥P^∆2

[2+l−N/p^′]

is an isomorphism.

About the half-space, the first point concerns the kernel K^m of the operator (∆², γ0, γ1) in W_m+l^{m+2, p}(R^N₊). For any q ∈ Z, we introduce the space B_q as a subspace ofP^∆2

q :

B_q =n

u∈P_q^∆2; u=∂Nu= 0 on Γo . Then we define the two operators ΠDand ΠN by:

∀r∈A^∆

k , ΠDr=1 2

Z xN

0

t r(x^′, t) dt,

∀s∈N^∆

k , ΠNs= 1 2xN

Z xN

0

s(x^′, t) dt, satisfying the following properties:

∀r∈A^∆

k , ∆ΠDr=r inR^N₊, ΠDr=∂NΠDr= 0 on Γ,

∀s∈N_k^∆, ∆ΠNs=s in R^N₊, ΠNs=∂NΠNs= 0 on Γ.

So we have the following characterizations for this kernel (see [7], Lemma 4.4): let l∈Zandm∈Nand assume that

N

p ∈ {1, . . . ,/ −l−m}, (7) then

K^m=B_{[2−l−N/p]} = ΠDA^∆

[−l−N/p]⊕ΠNN ^∆

[−l−N/p]. (8)

Everywhere in the sequel, we shall denote this kernel by B_{[2−l−N/p]}. Now, the fundamental result that we showed and which is the pivot of the present work, concerns the generalized solutions to problem (P):

Theorem 2.4 (Amrouche-Raudin [7]). Let l∈Zand assume that N

p^′ ∈ {1, . . . , l}/ and N

p ∈ {1, . . . ,/ −l}. (9) For any f ∈ W_l^{−2, p}(R^N₊), g0 ∈ W_l^{2−1/p, p}(Γ) and g1 ∈W_l^{1−1/p, p}(Γ) satisfying the compatibility condition

∀ϕ∈B_{[2+l−N/p′]}, hf, ϕi

W_l^{−2, p}(R^N

+)×W^{◦2, p′}_−l (R^N

+)+hg1,∆ϕi_Γ− hg0, ∂N∆ϕi_Γ= 0, (10) problem(P)has a solutionu∈W_l^{2, p}(R^N₊), unique up to an element ofB_{[2−l−N/p]}, with the estimate

q∈B_{[2−l−N/p]}inf ku+qk_W^{2, p}

l (R^N

+) ≤ C

kfk_W−2, p l (R^N

+)+kg0k_W^{2−1/p, p}

l (Γ)+kg1k_W^{1−1/p, p}

l (Γ)

.

8 CH´ERIF AMROUCHE, YVES RAUDIN

We also established a global result for the homogeneous problem in the half-space:

(P⁰)







∆²u= 0 inR^N₊, u=g0 on Γ,

∂Nu=g1 on Γ.

Proposition 2.5 (Amrouche-Raudin [7]). Let l∈Zandm∈Nand assume that N

p^′ ∈ {1, . . . , l}/ and N

p ∈ {1, . . . ,/ −l−m}. (11) For anyg0∈W_m+l^{m+2−1/p, p}(Γ)andg1∈W_m+l^{m+1−1/p, p}(Γ), satisfying the compatibility condition

∀ϕ∈B_{[2+l−N/p′]} , hg1,∆ϕi_Γ− hg0, ∂N∆ϕi_Γ= 0, (12) problem (P⁰) admits a solution u ∈ W_m+l^{m+2, p}(R^N₊), unique up to an element of B_{[2−l−N/p]}, with the estimate

q∈B_{[2−l−N/p]}inf ku+qk_W^{m+2, p}

m+l (R^N

+) ≤ C

kg0k_W^m+2−1/p, p

m+l (Γ)+kg1k_W^m+1−1/p, p

m+l (Γ)

.

3. Weak solutions, strong solutions and regularity. The first part of the present work consists in the study of solutions to problem (P) for more regular data. We shall now establish a global result which extends our previous result to different types of data.

3.1. Extension of Theorem 2.4.

Theorem 3.1. Let l ∈ Z and m ∈ N. Under hypothesis (6), for any f ∈ W_m+l^{m−2, p}(R^N

+), g0 ∈W_m+l^{m+2−1/p, p}(Γ) andg1∈W_m+l^{m+1−1/p, p}(Γ) satisfying the com- patibility condition (10), problem (P)has a solution u∈W_m+l^{m+2, p}(R^N₊), unique up to an element ofB_{[2−l−N/p]}, with the estimate

q∈B_{[2−l−N/p]}inf ku+qk_W^{m+2, p}

m+l (R^N

+)

≤ C

kfk_W^m−2, p m+l (R^N

+)+kg0k_W^{m+2−1/p, p}

m+l (Γ)+kg1k_W^{m+1−1/p, p}

m+l (Γ)

. Proof. Note at first that ifm= 0, we find Theorem 2.4. The kernel has been globally characterized by (8) (see [7], Lemma 4.4). Let’s recall that this kernel is reduced to {0} ifl ≥0 and symmetrically the compatibility condition (10) vanishes ifl ≤ 0. Moreover under hypothesis (6), the imbeddings W_m+l^{m−2, p}(R^N₊) ֒→ W_l^{−2, p}(R^N₊), W_m+l^{m+2−1/p, p}(Γ)֒→W_l^{2−1/p, p}(Γ) andW_m+l^{m+1−1/p, p}(Γ)֒→W_l^{1−1/p, p}(Γ) hold for all l ≥ 1, hence the necessity of (10) for anym∈N. So it suffices to show the existence of a solution. By Lemma 2.2, we can consider the problem with homogeneous boundary conditions

(P^⋆)







∆²u=f in R^N₊, u= 0 on Γ,

∂Nu= 0 on Γ,

with f ⊥B_{[2+l−N/p′]}. This orthogonality condition naturally corresponds to the compatibility condition (10). For more details on these questions, see [7].

Let’s now give the plan of the proof of the existence form≥1:

i) Ifl≤ −2, we establish globally the existence of a solution.

ii) Ifl≥ −1 andm= 1, we show that by a direct construction.

iii) Ifl≥ −1 andm≥1, we show that by induction onmfrom the previous case (m= 1), thanks to a regularity argument.

i) Assume that l ≤ −2. Then hypothesis (6) is reduced to (7). Let f ∈ W_m+l^{m−2, p}(R^N₊). Let’s first suppose m ≥ 2. We know that there exists a con- tinuous linear extension operator from W_m+l^{m−2, p}(R^N₊) to W_m+l^{m−2, p}(R^N) and thus f˜∈ W_m+l^{m−2, p}(R^N) which extends f to R^N. Then we use Theorem 2.3 to obtain

˜

z ∈ W_m+l^{m+2, p}(R^N) such that ˜f = ∆²z˜ in R^N and thus f = ∆²z in R^N₊, with z= ˜z|R^N

+ ∈W_m+l^{m+2, p}(R^N₊). Then Proposition 2.5 asserts the existence of a solution v∈W_m+l^{m+2, p}(R^N₊) to the homogeneous problem

∆²v= 0 inR^N₊, v=z and ∂Nv=∂Nz on Γ,

with z|_Γ ∈W_m+l^{m+2−1/p, p}(Γ) and ∂Nz|_Γ ∈ W_m+l^{m+1−1/p, p}(Γ). We remark again that B_{[2+l−N/p′]} ={0} because l ≤0, thus there is no compatibility condition. Then the functionu=z−vanswer to problem (P^⋆) in this case.

Let’s now consider the case m = 1, i.e. f ∈ W_l+1^{−1, p}(R^N₊). As we did for the distributions ofW_l^{−2, p}(R^N₊) in [7], we can show that there existsF = (Fi)_1≤i≤N ∈ W_l+1^{0, p}(R^N₊)^N such thatf = divF=

N

X

i=1

∂iFi, with the estimate

N

X

i=1

kFik_W^{0, p}

l+1(R^N

+) ≤ Ckfk_W^{−1, p}

l+1 (R^N

+). Let’s denote by ˜F ∈W_l+1^{0, p}(R^N)^N the extension by 0 ofF to R^N. Since N

p ∈ {1, . . . ,/ −l−1}, by Theorem 2.3, there exists ˜Ψ∈W_l+1^{4, p}(R^N)^N such that F˜ = ∆²Ψ in˜ R^N. Setting ˜ψ= div ˜Ψ andψ= ˜ψ|R^N

+, so we haveψ∈W_l+1^{3, p}(R^N

+) and by Proposition 2.5, there existsv∈W_l+1^{3, p}(R^N₊) such that

∆²v= 0 inR^N₊, v=ψ and ∂Nv=∂Nψ on Γ.

The functionu=ψ−v∈W_l+1^{3, p}(R^N₊) is a solution to Problem (P^⋆) in this case.

ii) Assume that l≥ −1 andm= 1. Note that the distributionf ∈W_l+1^{−1, p}(R^N₊) defines the linear functionalLonA^∆

[l+2−N/p^′] by L: r7−→ hf, ri

W_l+1^{−1, p}(R^N

+)×W^{◦1, p′}_−l−1(R^N

+), and introduce the inner product Φ onA^∆

[l+2−N/p^′]×A^∆

[l+2−N/p^′] defined by Φ : (µ, r)7−→

Z

Γ

̺′−2l−1−N/p+N/p^′∂Nµ ∂Nrdx^′. Note that

r∈A^∆

[l+2−N/p^′] =⇒ ̺^{′−l−1+1/p′}∂Nr∈L^p′(Γ), and

µ∈A^∆

[l+2−N/p^′]=A^∆

[l+2−N/p^′+N/p−N/p] =⇒ ̺′−l−1−N/p+N/p^′+1/p∂Nµ∈L^p(Γ).

Thus, thanks to H¨older inequality, Φ is well-defined. Then, there exists an unique µ∈A^∆

[l+2−N/p^′] such that

∀r∈A^∆

[l+2−N/p^′], L(r) = Φ(µ, r),

10 CH ´ERIF AMROUCHE, YVES RAUDIN

i.e.

∀r∈A^∆

[l+2−N/p^′], hf, ri

W_l+1^{−1, p}(R^N

+)×W^{◦1, p′}₋l−1(R^N

+)= Z

Γ

̺′−2l−1−N/p+N/p^′∂Nµ ∂Nrdx^′. (13) Let’s set ξ0 = ̺′−2l−1−N/p+N/p^′∂Nµ, then we have ξ0 ∈ W_l+1^{1−1/p, p}(Γ) and (13) becomes

∀r∈ A^∆

[l+2−N/p^′], hf, ri

W_l+1^{−1, p}(R^N

+)×W^{◦1, p′}_−l−1(R^N

+)=hξ0, ∂Nri

W_l+1^{1−1/p, p}(Γ)×W₋⁻_l^{1/p′, p′}₋₁ (Γ). (14) That is precisely the compatibility condition of the Dirichlet problem

(Q)

(∆ξ=f inR^N₊, ξ=ξ0 on Γ,

Thus (see [5], Theorem 3.1), (Q) admits a solutionξ∈W_l+1^{1, p}(R^N₊) under hypothesis (6). Here we shall use the characterization (8):

B_{[2+l−N/p′]}= ΠDA^∆

[l−N/p^′]⊕ΠNN ^∆

[l−N/p^′]. Sincef ⊥B_{[2+l−N/p′]}, we have

∀r∈A^∆

[l−N/p^′], h∆ξ,ΠDri

W_l+1^{−1, p}(R^N

+)×W^{◦1, p′}_−l−1(R^N

+)=hf,ΠDri= 0.

By a Green formula, we can deduce that

∀r∈A^∆

[l−N/p^′], hξ,∆ΠDri

W_l−1^{−1, p}(R^N

+)×W^{◦1, p′}_−l+1(R^N

+)= 0, becauseW_l+1^{1, p}(R^N₊)֒→W_l−1^{−1, p}(R^N₊) unlessN

p =−lor N

p^′ =l. The second possibility is excluded by (6), and sincel≥ −1, the only problematic case isl=−1. But then [l−N/p^′]<0 and the condition vanishes. Thus, we have

∀r∈A^∆

[l−N/p^′], hξ, ri

W_l^{−1, p}

−1 (R^N

+)×W^{◦1, p′}_−l+1(R^N

+)= 0, which is the compatibility condition for the Dirichlet problem

(R^⋆)

(∆ϑ=ξ in R^N₊, ϑ= 0 on Γ.

Thus (see [5], Corollary 3.4), (R^⋆) admits a solutionϑ∈W_l+1^{3, p}(R^N₊) under hypoth- esis (6). Similarly we have

∀s∈N ^∆

[l−N/p^′], h∆ξ,ΠNsi

W_l+1^{−1, p}(R^N

+)×W^{◦1, p′}_−l−1(R^N

+)=hf,ΠNsi= 0, therefore as previously, we have

∀s∈N_[l−N/p^∆ _′], hξ, si

W_l−1^{−1, p}(R^N

+)×W^{◦1, p′}_−l+1(R^N

+)= 0, which is the compatibility condition for Neumann problem

(S^⋆)

(∆ζ=ξ inR^N₊,

∂Nζ= 0 on Γ.

As for problem (R^⋆), we can show that (S^⋆) admits a solutionζ∈W_l+1^{3, p}(R^N₊) under hypothesis (6) (see [7], Theorem 2.8). Then the function defined by

u=xN∂N(ζ−ϑ) +ϑ (15)

is a solution to (P^⋆). It remains to show thatu∈W_l+1^{3, p}(R^N

+).

If N

p 6= −l, then we have the imbedding W_l+1^{3, p}(R^N₊) ֒→ W_l^{2, p}(R^N₊) therefore u∈W_l^{2, p}(R^N₊). Moreover usatisfies the system

(T^⋆)

(∆u= 2∂²_N(ζ−ϑ) +ξ inR^N

+,

u= 0 on Γ,

with 2∂_N²(ζ−ϑ) +ξ∈W_l+1^{1, p}(R^N₊). As for (R^⋆), we know that problem (T^⋆) has a solutiony∈W_l+1^{3, p}(R^N₊). We can deduce thatu−y∈A^∆

[2−l−N/p] ⊂W_l+1^{3, p}(R^N₊),i.e.

u∈W_l+1^{3, p}(R^N₊).

If N

p = −l, then we have necessary l = −1 and moreover the imbedding W₀^{3, p}(R^N₊) ֒→ W_−1,^{2, p}₋₁(R^N₊) with l−N

p^′ = l+ N

p −N = −N < 0, therefore no compatibility condition for (T^⋆). So we can still deduce that u∈W_l+1^{3, p}(R^N₊).

iii) Assume that l≥ −1 andm≥1. Consider f ∈W_m+l^{m−2, p}(R^N

+)⊥B_{[2+l−N/p′]}. Remark at first that we have the imbedding

W_m+l^{m−2, p}(R^N₊)֒→W_l+1^{−1, p}(R^N₊) if N

p^′ 6=l+ 2 or m= 1.

Then, thanks to the previous step, there exists a solutionu∈W_l+1^{3, p}(R^N₊) to problem (P^⋆). Let’s prove by induction that, under hypothesis (6),

f ∈W_m+l^{m−2, p}(R^N₊) =⇒ u∈W_m+l^{m+2, p}(R^N₊). (16) Form= 1, (16) is true. Assume that (16) is true for 1,2, . . . , m and suppose that f ∈ W_m+1+l^{m−1, p}(R^N₊). Let’s prove that u ∈ W_m+1+l^{m+3, p}(R^N₊). Let’s first observe that W_m+1+l^{m−1, p}(R^N₊) ֒→ W_m+l^{m−2, p}(R^N₊), hence u belongs to W_m+l^{m+2, p}(R^N₊) thanks to the induction hypothesis. Now, for anyi∈ {1, . . . , N−1},

∆(̺ ∂iu) =̺∆∂iu+2

̺x.∇∂iu+

N−1

̺ + 1

̺³

∂iu.

Then let’s set vi = 2

̺x.∇∂iu+

N−1

̺ + 1

̺³

∂iu. We can remark that vi ∈ W_m+l^{m, p}(R^N₊) and moreover we can write

∆²(̺ ∂iu) = ∆(̺∆∂iu) + ∆vi, with ∆vi∈W_m+l^{m−2, p}(R^N₊). It remains to see the first term,i.e.

∆(̺ ∂i∆u) =̺ ∂if +2

̺x.∇∂i∆u+

N−1

̺ + 1

̺³

∂i∆u.

We can see that ∆(̺ ∂i∆u) ∈ W_m+l^{m−2, p}(R^N₊), hence ∆²(̺ ∂iu) ∈ W_m+l^{m−2, p}(R^N₊).

Let’s set zi = ̺ ∂iuand fi = ∆²zi ∈ W_m+l^{m−2, p}(R^N₊). A priori, we only havezi ∈ W_m−1+l^{m+1, p}(R^N₊). However, we know that γ0u=γ1u= 0, then we can deduce that

γ0zi= (̺ ∂iu)|_Γ = 0 and γ1zi= (∂N̺ ∂iu+̺ ∂_iN² u)|_Γ = 0 sincei6=N.

12 CH ´ERIF AMROUCHE, YVES RAUDIN

Therefore

∆²zi=fi in R^N₊, zi=∂Nzi = 0 on Γ,

withfi ∈W_m+l^{m−2, p}(R^N₊)֒→W_l^{−2, p}(R^N₊) under hypothesis (6). Moreover, thanks to the Green formula, we have for anyϕ∈B_{[2+l−N/p′]}:

∆²zi, ϕ

W_l^{−2, p}(R^N

+)×W^{◦2, p′}_−l (R^N

+)=

zi,∆²ϕ

W◦^{2, p}_l (R^N

+)×W^{−2, p′}

−l (R^N

+)= 0.

So the orthogonality conditionfi⊥B_{[2+l−N/p′]} is satisfied for the problem

(Qi)







∆²ζi=fi inR^N₊, ζi= 0 on Γ,

∂Nζi= 0 on Γ,

which admits, by the induction hypothesis, a solution ζi ∈ W_m+l^{m+2, p}(R^N₊), unique up to an element of B_{[2−l−N/p]}. Thus zi−ζi ∈B_{[2−l−N/p]}, hence we can deduce thatzi∈W_m+l^{m+2, p}(R^N₊). Sincezi=̺ ∂iu, that implies

∀i∈ {1, . . . , N−1}, ∂iu∈W_m+1+l^{m+2, p}(R^N₊) (17) and consequently, for any (i, j, k)∈ {1, . . . , N}²× {1, . . . , N −1},

∂³_ijk(∂Nu) =∂_ijN³ (∂ku) ∈W_m+1+l^{m−1, p}(R^N₊). (18) Furthermore, (17) gives us

∀(i, j)∈ {1, . . . , N} × {1, . . . , N−1}, ∂_i²∂_j²u∈W_m+1+l^{m−1, p}(R^N₊), which implies

∂_N⁴u=f−

N

X

i,j=1 (i,j)6=(N,N)

∂_i²∂_j²u ∈W_m+1+l^{m−1, p}(R^N₊). (19)

Then combining (18) and (19), we obtain that ∇³(∂Nu) ∈ W_m+1+l^{m−1, p}(R^N₊)^N

3

and knowing that ∂Nu∈W_m+l^{m+1, p}(R^N₊), we can deduce that ∂Nu∈ W_m+1+l^{m+2, p}(R^N₊), be- causem≥1. Adding this last point to (17), we have∇u∈W_m+1+l^{m+2, p}(R^N₊)^N, hence we can conclude thatu∈W_m+1+l^{m+3, p}(R^N₊).

3.2. Panorama of basic cases. The purpose of this part is to extract the ba- sic cases included in Theorem 3.1. We give them for the lifted problem (P^⋆).

There is no orthogonality condition in these cases because l ∈ {−2,−1,0}, hence B_{[2+l−N/p′]}={0}. Form≥3, we introduce the notation

⋆

W^{m, p}l (R^N₊) =

u∈W_l^{m, p}(R^N₊); u=∂Nu= 0 on Γ . Corollary 3.2. The following biharmonic operators are isomorphisms:

i) For l= 0

∆²: W^{◦2, p}0 (R^N₊) −→ W₀^{−2, p}(R^N₊).

∆²: W^{⋆3, p}1 (R^N₊) −→ W₁^{−1, p}(R^N₊), ifN/p^′ 6= 1.

∆²: W^{⋆4, p}2 (R^N₊) −→ W₂^{0, p}(R^N₊), ifN/p^′ ∈ {1,/ 2}.