126 (2001)
MATHEMATICA BOHEMICA
No. 2, 265–274
LAPLACE EQUATION IN THE HALF-SPACE WITH
A NONHOMOGENEOUS DIRICHLET BOUNDARY CONDITION
Cherif Amrouche, Pau, Šárka Nečasová, Praha
Dedicated to Prof. J. Nečas on the occasion of his 70th birthday
Abstract. We deal with the Laplace equation in the half space. The use of a special
family of weigted Sobolev spaces as a framework is at the heart of our approach. A complete
class of existence, uniqueness and regularity results is obtained for inhomogeneous Dirichlet
problem.
Keywords: the Laplace equation, weighted Sobolev spaces, the half space, existence,
uniqueness, regularity
MSC 2000 : 35J05, 58J10
1. Introduction
The purpose of this paper is to solve the problem
(P)
−∆u = f
u=g
in
ÊN+ ,
on Γ = ÊN −1 ,
with the Dirichlet boundary condition on Γ. The approach is based on the use of
a special class of weighted Sobolev spaces for describing the behavior at infinity.
Many authors have studied the Laplace equation in the whole space ÊN or in an
exterior domain. The main difference is due to the nature of the boundary and one
of difficulties is to obtain the appropriate spaces of traces. However, the half-space
has a useful symmetric property.
Second author would like to thank the Grant Agency of the Czech Republic No. 201/99/
0267 and both authors to the Barrande project between the Czech Republic and France.
265
Problem (P) has been investigated in weighted Sobolev spaces by several authors,
but only in the Hilbert cases (p = 2) and without the critical cases corresponding to
the logarithmic factor (cf. [2], [4]). We can also mention the book by Simader, Sohr
[5] where the Dirichlet problem for the Laplacian is investigated.
Let Ω be an open subset of ÊN , N 2. Let x = (x1 , . . . , xN ) be a typical point
of ÊN and r = |x| = (x21 + . . . + x2N )1/2 . We use two basic weights:
= (1 + r2 )1/2
and lg = ln(2 + r2 ).
As usual, D(ÊN ) denotes the spaces of indefinitely differentiable functions with a
compact support and D (ÊN ) denotes its dual space, called the space of distributions.
For any nonnegative integers n and m, real numbers p > 1, α and β, setting
−1
if Np + α ∈
/ {1, . . . , m},
k = k(m, N, p, α) =
m − Np − α
if Np + α ∈ {1, . . . , m},
we define the following space:
(1.1)
m,p
Wα,β
(Ω) = {u ∈ D (Ω); 0 |λ| k, α−m+|λ| (lg )β−1 Dλ u ∈ Lp (Ω);
k + 1 |λ| m, α−m+|λ| (lg )β Dλ u ∈ Lp (Ω)}.
m,p
In case β = 0, we simply denote the space by Wαm,p (Ω). Note that Wα,β
(Ω) is a
reflexive Banach space equipped with its natural norm
m,p
uWα,β
α−m+|λ| (lg )β−1 Dλ upLp(Ω)
(Ω) =
0|λ|k
+
k+1|λ|m
α−m+|λ| (lg )β Dλ upLp(Ω)
1/p
.
We also define the semi-norm
m,p
|u|Wα,β
(Ω) =
|λ|=m
α (lg )β Dλ upLp(Ω)
1/p
,
and for any integer q, we denote by Pq the space of polynomials in N variables of
a degree smaller than or equal to q, with the convention that Pq is reduced to {0}
when q is negative. The weights defined in (1.1) are chosen so that the corresponding
space satisfies two properties:
(1.2)
D(ÊN
+)
m,p
is dense in Wα,β
(ÊN
+ ),
m,p
and the following Poincaré-type inequality holds in Wα,β
(ÊN
+ ).
266
Theorem 1.1. Let α and β be two real numbers and m 1 an integer not
satisfying simultaneously
N
+ α ∈ {1, . . . , m} and (β − 1)p = −1.
p
(1.3)
m,p
m,p
Then the semi-norm | · |Wα,β
defines on Wα,β
(ÊN
( ÊN
+ )/Pq a norm which is equiv+)
alent to the quotient norm,
(1.4)
m,p
m,p
m,p
∀u ∈ Wα,β
(ÊN
c|u|Wα,β
( ÊN
( ÊN
+ ), uWα,β
+ )/Pq
+)
with q = inf(q, m − 1), where q is the highest degree of the polynomials contained
in Wαm,p (ÊN
+ ),
.
First, we construct a linear continuous extension operator such that
m,p
m,p
N
P : Wα,β
(ÊN
+ ) → Wα,β (Ê )
(1.5)
satisfying
m,p
m,p
P uWα,β
.
(ÊN ) uWα,β
( ÊN
+)
(1.6)
Since
(1.6)
m,p
m,p
m,p
∀u ∈ Wα,β
(ÊN ), uWα,β
(ÊN )/Pq c|u|Wα,β
( ÊN )
holds [cf. 1], it automatically implies the statement of our theorem.
Now, we define the space
◦
Ê
·W m,p (
N
N
W m,p
α,β (Ê+ ) = D(Ê+ )
◦
α,β
N)
+
;
−m,p
N
N
the dual space of W m,p
α,β (Ê+ ) is denoted by W−α,−β (Ê+ ), where p is the conjugate
1
1
of p : p + p = 1.
Theorem 1.2.
Under the assumptions of Theorem 1.1, the semi-norm
◦
N
m,p
| · |Wα,β
is a norm on W m,p
( ÊN
α,β (Ê+ ) such that it is equivalent to the full norm
+)
m,p
· Wα,β
.
( ÊN
+)
m,p
(ÊN
We recall now some properties of weighted Sobolev spaces Wα,β
+ ). We have
the algebraic and topological imbeddings
m,p
m−1,p
0,p
N
N
Wα,β
(ÊN
+ ) ⊂ Wα−1,β (Ê+ ) ⊂ . . . ⊂ Wα−m,β (Ê+ )
267
if
N
p
+α∈
/ {1, . . . , m}. When
N
p
+ α = j ∈ {1, . . . , m}, then we have:
m,p
m−j+1,p
m−j,p
0,p
N
N
N
Wα,β
(ÊN
+ ) ⊂ . . . ⊂ Wα−j+1,β (Ê+ ) ⊂ Wα−j,β−1 (Ê+ ) ⊂ . . . ⊂ Wα−m,β−1 (Ê+ ).
m,p
(ÊN
Note that in the first case, the mapping u → γ u is an isomorphism from Wα,β
+)
m,p
N
onto Wα−γ,β (Ê+ ) for any integer m. Moreover, in both cases and for any multi-index
λ ∈ Æ N , the mapping
m,p
λ
(ÊN
u ∈ Wα,β
+ ) → D u ∈ Wα,β
m−|λ|,p
(ÊN
+)
is continuous.
Finally, it can be readily checked that the highest degree q of the polynomials
m,p
contained in Wα,β
(ÊN
+ ) is given by
N

+ α ∈ {1, . . . , m} and (β − 1)p −1

 m − ( N + α) − 1 if p
p
N
q=
p + α ∈ {j ∈ Z; j 0} and βp −1


N
[m − ( p + α)] otherwise,
where [s] denotes the integer part of s.
In the sequel, for any integer q 0, we will use the following polynomial spaces:
— Pq (Pq∆ ) is the space of polynomials (respectively, harmonic polynomials) of
degree q,
— Pq is the subspace of polynomials in Pq depending only on the N − 1 first
variables, x = (x1 , . . . , xN −1 ),
∆
∆
— A∆
q (Nq ) is the subspace of polynomials Pq satisfying the condition p(x , 0) = 0
∂p
(respectively, ∂xN (x , 0) = 0) or equivalently odd with respect to xN (even with
respect to xN ), with the convention that Pq , Pq∆ , Pq , . . . are reduced to {0} when q
is negative.
2. The spaces of traces
m,p
In order to define the traces of functions of Wα,β
(ÊN
+ ), we introduce for any
σ ∈]0, 1[ the space
W0σ,p (ÊN ) = u ∈ D (ÊN ); w−σ u ∈ Lp (ÊN ),
(2.1)
+∞
t−1−σp dt
|u(x + tei ) − u(x)|p dx < ∞ ,
ÊN
0
where
w=
268
if
1/σ
(lg )
if
N
p
N
p
= σ,
= σ,
and e1 , . . . , eN is a canonical basis of
with its natural norm
u
W0σ,p (
ÊN )
ÊN .
It is a reflexive Banach space equipped
N ∞
u p
1/p
−1−σp
= σ
+
t
dt
|u(x + tei ) − u(x)|p dx
w Lp (ÊN ) i=1 0
ÊN
which is equivalent to the norm
u p
1/p
|u(x) − u(y)|p
+
dx dy
.
σ p N
N
+σp
w L (Ê )
ÊN ×ÊN |x − y|
For any s ∈ Ê+ , we set
(2.2)
[s],p
s−[s],p
W0s,p (ÊN ) = u ∈ W[s]−s (ÊN ); ∀|λ| = [s], Dλ u ∈ W0
(ÊN ) .
It is a reflexive Banach space equipped with the norm
uW0s,p (ÊN ) = uW [s],p (ÊN ) +
[s]−s
|λ|=s
Dλ uW s−[s],p(ÊN ) .
0
We notice that this definition and the next one coincide with the definition in the
first section when s = m is a nonnegative integer. For any s ∈ Ê+ and α ∈ Ê, we
then set
(2.3)
[s],p
s−[s],p
Wαs,p (ÊN ) = u ∈ W[s]+α−s (ÊN ), ∀|λ| = [s], α Dλ u ∈ W0
(ÊN ) .
Finally, for any integer m 1, we define the space
(2.4)
N
|λ|−m
X0m,p (ÊN
(lg )−1 Dλ u ∈ Lp (ÊN
+ ) = u ∈ D (Ê+ ); 0 |λ| k, + ),
k + 1 |λ| m, |λ|−m Dλ u ∈ Lp (ÊN
+)
with = (1 + |x |2 )1/2 and lg = ln(2 + |x |2 ). It is a reflexive Banach space. We
can prove that
m,p
(ÊN
D(ÊN
+ ) is dense in X0
+ ).
m,p
(ÊN
We observe that the functions from X0m,p (ÊN
+ ) and W0
+ ) have the same traces
on Γ = ÊN −1 (see below). If u is a function, we denote its traces on Γ = ÊN −1 by
∂j u
x ∈ ÊN −1 , γ0 u(x ) = u(x , 0), . . . , γj u(x ) = ∂x
j (x , 0).
N
As in [3], we can prove the following trace lemma:
269
Lemma 2.1. For any integer m 1 and real number α, the mapping
γ : D(ÊN
+)→
m−1
D(ÊN −1 )
j=0
u → (γ0 u, . . . , γm−1 u)
can be extended by continuity to a linear and continuous mapping still denoted by
m−1
m−j− p1 ,p
γ from Wαm,p (ÊN
Wα
(ÊN −1 ). Moreover, γ is onto and
+ ) to
j=0
◦
N
Ker γ = W m,p
α (Ê+ ).
3. The Laplace equation
The aim of this section is to study the problem (P):
(P)
−∆u = f
in
u=g
in
ÊN+ ,
Γ = ÊN −1 .
Theorem 3.1. Let 0 be an integer and assume that
N
∈
/ {1, . . . , }
p
(3.1)
with the convention that this set is empty if = 0. For any f in W−1,p (ÊN
+ ) and g
1
in Wp
,p
(Γ) satisfying the compatibility condition
(3.2)
∂ϕ ∀ϕ ∈ A∆ [+1− N ] , f, ϕ
W −1,p ×W 1,p = g,
p
−
∂γN Γ
1
,p
−
1
,p
where ·, ·
Γ denotes the duality between Wp (Γ) and W−p (Γ), problem (P) has
a unique solution u ∈ W1,p (ÊN
+ ) and there exists a constant C independent of u, f
and g such that
(3.3)
.
uW 1,p (ÊN ) C(f W −1,p (ÊN ) + g
+
+
1 ,p
Wp
).
First, the kernel of the operator
1
−1,p
p
(ÊN
(−∆, γ0 ) : W1,p (ÊN
+ ) → W
+ ) × W
270
(Γ)
,p
(Γ)
∆
is precisely the space A∆
[+1−N/p ] for any integer and A[+1− N ] is reduced to {0}
p
Ê be the lifting function of g
when 0. Thanks to Lemma 2.1, let ug ∈
such that
ug = g on Γ and ug W 1,p (ÊN ) C1 g 1 ,p .
W1,p ( N
+)
Wp
+
Then problem (P) is equivalent to
−∆v = f + ∆ug
(3.4)
v=0
in
(Γ)
ÊN+ ,
on Γ.
1,p
Set h = f + ∆ug . For any ϕ ∈ W−
(ÊN ) set
ϕ(x , xN ) = ϕ(x , xN ) − ϕ(x , −xN ) if
◦
xN > 0.
−1,p
N
(ÊN )
It is clear that ϕ ∈ W 1,p
− (Ê+ ). Then h can be extended to hπ ∈ W
defined by
1,p
ϕ ∈ W−
(ÊN ), hπ (ϕ) = h, ϕ
W −1,p (ÊN )×W 1,p (ÊN ) .
+
−
+
Moreover,
hπ W −1,p (ÊN ) = hW −1,p (ÊN ) .
Let q be a polynomial in
+
∆
P[+1−N/p
].
We can write it in the form
∆
q = r + s, r ∈ A∆
[+1−N/p ] and s ∈ N[+1−N/p] .
Then,
hπ , q
= f + ∆ug , r
W −1,p (ÊN )×W 1,p (ÊN )
+
+
−
and applying the Green formula we get
∇ug · ∇r dx
∆ug , r
= −
ÊN+
= − g,
(note that ∆r = 0 in
ÊN+
∂r 1 ,p
− 1 ,p
∂xN Wp (Γ)×W−p (Γ)
and r = 0 on Γ). Thus, hπ ∈ W−1,p (ÊN ) and it satisfies
∆
∀q ∈ P[+1−N/p
] , hπ , q
= 0.
Recall that (cf. [1]) since (3.1) holds, the operators
∆
if 1,
∆ : W1,p (ÊN ) → W−1,p ⊥ P[+1−
N
]
p
∆:
Ê
W01,p ( N )/P[1− N ]
p
→
Ê
W0−1,p ( N )
⊥ P[1− N ] if = 0
p
271
are isomorphisms. Hence, there exists ṽ in W1,p (ÊN ) such that −∆ṽ = hπ . Now we
remark that the function w = 12 ṽ belongs to W1,p (ÊN
+ ) and
−∆w = h
in
ÊN+
and w = 0
on Γ,
i.e. w is a solution of (3.4).
.
= 0.
The kernel A∆
[−+1−N/p] is reduced to {0} if 0 and to P[1−N/p] if
With similar arguments, we can prove the following theorem:
Theorem 3.2. Let 1 be an integer and assume that
N
∈
/ {1, . . . , −}.
p
(3.5)
1
,p
−1,p
p
Then for any f in W−
(ÊN
+ ) and g in W− (Γ), problem (P) has a unique solution
1,p
∆
u ∈ W− (ÊN
+ )/A[+1−N/p] and there exists a constant C independent of u, f and g
such that
inf
q∈A∆
[+1− N ]
p
u + qW 1,p (ÊN ) C(f W −1,p (ÊN ) + g
+
−
+
−
1 ,p
p
W−
(Γ)
).
1
Theorem 3.3. Let m be a nonnegative integer, let g belong to Wmp
assume that
−1+m,p
f ∈ Wm
(ÊN
+ ) if
(3.6)
+m,p
(Γ) and
N
= 1 or m = 0,
p
or
(3.7)
−1,p
−1+m,p
(ÊN
(ÊN
f ∈ Wm
+ ) ∩ W0
+ ) if
N
= 1 and m = 0.
p
1+m,p
(ÊN
Then problem (P) has a unique solution u ∈ Wm
+ ) and u satisfies
m+1,p
−1+m,p
uWm
(ÊN ) C(f Wm
(ÊN ) + g
+
+
1 +m,p
p
Wm
(Γ)
) if
N
= 1 or m = 0
p
and
1,p
m+1,p
−1+m,p
uWm
(ÊN ) + f Wm
(ÊN ) C(f W
(ÊN ) + g
+
0
+
+
if
272
1 +m,p
p
Wm
(Γ)
)
N
= 1 and m = 0.
p
.
First, we observe that for any integer m 0 we have the inclusion
−1,p
−1+m,p
Wm
(ÊN
(ÊN
+ ) ⊂ W0
+)
if
N
p
= 1 or m = 0. Thus, under the assumptions (3.6) or (3.7) and thanks to
Theorem 3.1, there exists a unique solution u ∈ W01,p (ÊN
+ ) of problem (P). Let us
prove by induction that
1
g ∈ Wmp
(3.8)
+m,p
m+1,p
(Γ) and f satisfies (3.6) or (3.7) =⇒ u ∈ Wm
(ÊN
+ ).
For m = 0, (3.8) is valid. Assume that (3.8) is valid for 0, 1, . . . , m and suppose that
1
+m+1,p
p
g ∈ Wm+1
m,p
(Γ) and f ∈ Wm+1
(ÊN
+ ) with
N
p
= 1 (a similar argument can be used
m+2,p
(ÊN
for f satisfying (3.7)). Let us prove that u ∈ Wm+1
+ ). We observe first that
1
+m+1,p
m,p
p
m−1,p
(ÊN
(ÊN
Wm+1
+ ) ⊂ Wm
+ ) and Wm+1
1
(Γ) ⊂ Wmp
+m,p
(Γ),
m+1,p
hence u belongs to Wm
(ÊN
+ ) thanks to the induction hypothesis. Now, for i =
1, . . . , N − 1,
2
1
2
+ 3 ∂i u.
∆(∂i u) = ∂i f + r · ∇(∂i u) +
m−1,p
m+1,p
Thus, ∆(∂i u) ∈ Wm
(ÊN
(ÊN −1 ). Applying the induc+ ) and γ0 (∂i u) ∈ Wm
tion hypothesis, we can deduce that
m+1,p
∂i u ∈ Wm+1
(ÊN
+ ) for i = 1, . . . , N − 1.
m+1,p
It remains to prove that v = ∂N u ∈ Wm+1
(ÊN
+ ). This is a consequence of the fact
m,p
N
that v belongs to Wm (Ê+ ) and
m,p
∂i ∂N u = ∂N ∂i u ∈ Wm+1
(ÊN
+ ), i = 1, . . . , N − 1,
∂N (∂N u) = ∆u −
N
−1
m,p
∂i2 u ∈ Wm+1
(ÊN
+ ).
i=1
m+2,p
We can conlude that u ∈ Wm+1
(ÊN
+ ).
Corollary 3.4. Let 1 and m 1 be two integers.
(i) Under the assumption
N
∈
/ {1, . . . , + 1},
p
273
1
+m,p
m−1,p
p
for any f ∈ Wm+
(ÊN
(Γ) satisfying the compatibility condition
+ ) and g ∈ Wm+
m+1,p
(3.2) there exists a unique solution u ∈ Wm+
(ÊN
+ ) of (P) and u satisfies
uW m+1,p (ÊN ) C(f W m−1,p (ÊN ) + g
m+
+
m+
+
1 +m,p
p
Wm+
(Γ)
)
where C = C(m, p, , N ) is a constant independent of u, f and g.
(ii) Under the assumption
m
or
N
∈
/ {1, . . . , − m},
p
1
+m,p
m−1,p
p
for any f ∈ Wm−
(ÊN
(Γ) there exists a unique solution u ∈
+ ) and g ∈ Wm−
m+1,p
N
∆
Wm− (Ê+ )/A[1+−N/p] of (P) and u satisfies
inf
q∈A∆
[1+−N/p]
u + qW m+1,p (ÊN ) C(f W m−1,p (ÊN ) + g
m−
m−
+
+
1 +m,p
p
Wm−
(Γ)
).
References
[1] C. Amrouche, V. Girault, J. Giroire: Weighted Sobolev spaces for Laplace’s equation in
ÊN . J. Math. Pures Appl. 73 (1994), 579–606.
[2] T. Z. Boulmezaoud: Espaces de Sobolev avec poids pour l’équation de Laplace dans le
demi-espace. C. R. Acad. Sci. Paris, Ser. I 328 (1999), 221–226.
[3] B. Hanouzet: Espaces de Sobolev avec poids. Application au problème de Dirichlet dans
un demi-espace. Rend. Sem. Univ. Padova 46 (1971), 227–272.
[4] V. G. Maz’ya, B. A. Plamenevskii, L. I. Stupyalis: The three-dimensional problem of
steady-state motion of a fluid with a free surface. Amer. Math. Soc. Transl. 123 (1984),
171–268.
[5] C. C. Simader, H. Sohr: The Dirichlet Problem for the Laplacian in Bounded and Unbounded Domains. Pitman Research Notes in Matehmatics Series 360, Addison Wesley
Longman, Harlow, Great Britain, 1996.
Authors’ addresses: Cherif Amrouche, Université de Pau et des Pays de L’Adour, Laboratoire de Mathématiques Appliquées, I.P.R.A, Av. de l’Université, 6400 Pau, France, e-mail:
cherif.amrouche@univ-pau.fr; Šárka Nečasová, Mathematical Institute of the Academy
of Sciences, Žitná 25, 115 67 Praha 1, Czech Republic, e-mail: matus@math.cas.cz.
274