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On the limit Sobolev regularity for Dirichlet and Neumann problems on Lipschitz domains
Martin Costabel
To cite this version:
Martin Costabel. On the limit Sobolev regularity for Dirichlet and Neumann problems on Lipschitz domains. Mathematical News / Mathematische Nachrichten, Wiley-VCH Verlag, 2019, 292 (10), pp.2165-2173. �10.1002/mana.201800077�. �hal-01638088v2�
ON THE LIMIT SOBOLEV REGULARITY FOR DIRICHLET AND NEUMANN PROBLEMS ON LIPSCHITZ DOMAINS
MARTIN COSTABEL
ABSTRACT. We construct a boundedC1 domainΩinRn for which theH3/2regularity for the Dirichlet and Neumann problems for the Laplacian cannot be improved, that is, there existsf in C∞(Ω)such that the solution of∆u = f inΩand eitheru = 0on ∂Ωor∂nu = 0on∂Ωis contained inH3/2(Ω)but not in H3/2+ε(Ω) for anyε > 0. An analogous result holds forLp Sobolev spaces withp∈(1,∞).
1. INTRODUCTION
The motivation for this note comes from a question of regularity of the time-harmonic Maxwell equations in Lipschitz domains. In the variational theory of Maxwell’s equations, basis for the analysis of many algorithms of numerical electrodynamics, the following two function spaces are fundamental:
XN =H(div,Ω)∩H0(curl,Ω)
={u∈L2(Ω;C3)|divu∈L2(Ω),curlu∈L2(Ω;C3), u×n = 0on∂Ω} (1.1) XT =H0(div,Ω)∩H(curl,Ω)
={u∈L2(Ω;C3)|divu∈L2(Ω),curlu∈L2(Ω;C3), u·n = 0on∂Ω} (1.2) Herenis the outward unit normal vector field on the boundary of the domainΩ⊂R3.
If Ω is a bounded Lipschitz domain, then it has been known for a long time [15, 11] that XN
andXT are compactly embedded subspaces of L2(Ω;C3), and it has been shown more precisely [5,10] that they are contained in the Sobolev spaceH12(Ω,C3) =W12,2(Ω,C3). For large classes of more regular domains,XN andXT are contained inH1(Ω,C3)(see [3] forC1,1 domains, [7]
for C32+ε domains, [12] for XN on convex domains, [13] for “almost convex” domains). The regularity is diminished by corner singularities, but one also knows [3] that for every Lipschitz polyhedron or, more generally, piecewise smooth domainΩthat is at leastC2-diffeomorphic to a polyhedron, there existsε >0such that
XN ∪XT ⊂H12+ε(Ω;C3). (1.3)
The additional regularity described by ε is of some use in the numerical analysis of Maxwell’s equations (see for example [2, 1]). The parameter εcan become arbitrarily small, depending on the corner angles of∂Ω, but it depends only on these angles, that is, on the local Lipschitz constant of∂Ω. Based on this observation, one could ask the question whether for any Lipschitz domain Ω, there exists such anε >0for which (1.3) holds. This question is the motivation for the present investigation.
Date: 2018-12-04.
1
To the best of the author’s knowledge, the conjecture that such an ε > 0 always exists is not incompatible with the currently available regularity results for Maxwell’s equations on Lipschitz domains, but we shall show that it is not true. As a corollary of our constructions, we obtain a counterexample that is evenC1.
Proposition 1.1. There exists a bounded C1 domain Ω ⊂ R3, an L2(Ω) function g and an L2(Ω;C3)functionhsuch that the solutionsu∈L2(Ω;C3)of the system
divu=g , curlu=h inΩ (1.4)
and either
u×n = 0 on∂Ω (1.5)
or
u·n= 0 on∂Ω (1.6)
do not belong toH12+ε(Ω;C3)for anyε >0.
In the system(1.4), the fieldhcan be chosen to be zero andg can be chosen to be continous onΩ.
As we will see in the following, analogous results are true in dimension2 and in higher dimen- sions, and also for non-Hilbert Sobolev spaces overLp withpdifferent from2.
Non-regular solutions of the div-curl system (1.4) are typically sought as gradients of solutions of the inhomogeneous Laplace (Poisson) equation with either Dirichlet (for (1.5)) or Neumann (for (1.6)) boundary conditions. A non-regularity result for these Laplace boundary value problems is the main result of this paper, see Theorem1.2below. It will be proved in Section3for dimension d= 2and in Section4for higher dimensions.
We use the standard notation Ws,p(Ω) for the Sobolev-Slobodeckij spaces onΩ ⊂ Rd, and we recall that for0< s <1the seminorm
|u|s,p;Ω= Z
Ω
Z
Ω
|u(y)−u(x)|p
|y−x|d+sp dx dy 1p
(1.7) defines the normkukWs,p(Ω)=kukLp(Ω)+|u|s,p;Ω, thatW0,p(Ω) =Lp(Ω), and that for anysthere holds
u∈ Ws+1,p(Ω) ⇐⇒ u∈Ws,p(Ω)and∇u∈Ws,p(Ω;Cd).
As usual, we write W0s,p(Ω) for the closure of C0∞(Ω) in Ws,p(Ω). In view of an interesting property of the domain we are going to construct (see equations (1.10) and (4.1)), we recall that for 1p < s < 1 + 1p the subspace W0s,p(Ω) is characterized by the condition that the boundary trace vanishes, whereas for1 +1p < s <2 + 1p the condition is that both the trace and the normal derivative vanish on∂Ω.
In order to describe known regularity results, we also need the Bessel potential spacesHs,p(Ω), which are different fromWs,p(Ω)ifp 6= 2. For the main properties of these spaces, see [14]. In Triebel’s notationWm,p(Ω) =Fp,2m(Ω)form ∈Nand
Hs,p(Ω) =Fp,2s (Ω), and fors6∈Z: Ws,p(Ω) =Bsp,p(Ω).
Note that the trace space for both Ws,p(Ω) and Hs,p(Ω) on a sufficiently smooth boundary is Ws−1p,p(∂Ω)ifs > 1p.
LIMIT SOBOLEV REGULARITY FOR DIRICHLET AND NEUMANN PROBLEMS ON LIPSCHITZ DOMAINS 3
Comprehensive regularity results in the Hs,p spaces for the Dirichlet problem on Lipschitz do- mains were given by Jerison and Kenig [8]. They had previously studied the homogeneous Laplace equation with inhomogeneous Neumann conditions [9], and corresponding results for the homogeneous Neumann problem of the inhomogeneous Laplace equation were obtained by Fabes, Mendez and Mitrea [6] and Zanger [16]. In particular, there exist precise answers to the question for whichsandpthe conditiong ∈ Hs−2,p(Ω)impliesv ∈ Hs,p(Ω) for the solutionsv of the problems
∆v =g inΩ, v = 0 on∂Ω (1.8)
∆v =g inΩ, ∂v∂n = 0 on∂Ω (1.9)
For the maximal regularity one finds a limit at s = 1 + 1p. We summarize the main results pertaining to the question of maximal regularity (here formulated for the Dirichlet problem, see [8, Thms 1.1–1.3], whereHs,pis writtenLps; the results for the Neumann problem are similar):
For any bounded Lipschitz domainΩ⊂Rd,d≥2, there existsp0 ∈[1,2)such that forp0 < p <
p0
p0−1 and 1p < s <1 + 1p the solutionvof the Dirichlet problem (1.8) withg ∈Hs−2,p(Ω)belongs toHs,p(Ω). The following 4 points indicate the known borders of this result.
1. In general, p0 > 1 and there are counterexamples as soon as p ors are outside of the given bounds, but when Ω is a C1 domain, one can choose p0 = 1, so that the result is true for any p∈(1,∞).
2. Whenp >2, there are Lipschitz counterexamples withg ∈C∞(Ω)andv 6∈W1+1p,p(Ω). There is aC1counterexample forp= 1withg ∈ C∞(Ω)andv 6∈W2,1(Ω).
3. In the optimal regularity-shift result forC1 domains, the condition onscannot be weakened, because for any p > 1 there exists a bounded C1 domain Ω and a g ∈ H−1+1p,p(Ω) such that v 6∈H1+1p,p(Ω).
4. On the other hand, if g is more regular, for example g ∈ H−1+1p+ε,p(Ω) for some ε > 0and p >1, thenv ∈H1+1p,p(Ω)follows. The latter result is obtained by subtracting fromv a solution v0 ∈ H1+p1+ε,p(Ω) of ∆v0 = g without boundary conditions and observing that a harmonic function with trace inW1,p(∂Ω)belongs toH1+1p,p(Ω).
We will prove that one will have v 6∈ H1+1p+ε,p(Ω) for any ε > 0, in general, even for more regularg. Because of the mutual inclusionsHs+ε,p ⊂Ws,p ⊂Hs−ε,pfor any ε >0, the result is equivalently formulated in the scale ofWs,pspaces.
Theorem 1.2. In Rd, d ≥ 2, there exists a bounded C1 domain Ω and for both the Dirichlet problem(1.8)and the Neumann problem(1.9) functionsg ∈ L∞(Ω)such that the solutionsv ∈ H1(Ω)do not belong toW1+p1+ε,p(Ω)for anyp∈[1,∞)and anyε >0.
Remark1.3. It will follow from the proof that in dimensiond= 2, there are functionsg ∈C∞(Ω) that provide examples, even g = 1 is possible for the Dirichlet problem and a second degree polynomialg for the Neumann problem. See also Remark3.3. In dimensiond ≥ 3, there is still an example withg = 1for the Dirichlet problem, and examples withg ∈ Cα(Ω), α >0, for the Neumann problem.
Remark 1.4. Not all of this is new: For p = 1, the counterexample from [8, Theorem 1.2(b)]
shows that the result for the Dirichlet problem holds even with ε = 0. Moreover, forp > 2the result of Theorem1.2 is not interesting in the class of Lipschitz domains, because singularities at
conical points provide a limit of regularity that is strictly belows = 1 + 1p. This follows from the well-known singular asymptotic behaviorO(rα)near a straightd−2dimensional facet of the boundary (corner in dimensiond= 2or “edge” in dimension≥3) of opening angle απ of generic solutions of the Dirichlet and Neumann problems with smooth right hand sides, where r is the distance to the corner or edge. Such functions are inWs,p(Ω)only for s < α+ 2p, hence not in W1+1p,p(Ω)as soon as the opening angle exceeds p−1p π. But forC1 domains the result still seems to be new even forp >2. We provide a proof that works for anyp≥ 1, because there is no extra cost with respect to the proof forp= 2. One just has to be careful to observe that the same domain Ωand the same functiong give an example valid for allpand allε.
Proposition1.1 follows from Theorem 1.2 for p = 2, d = 3 if we take u = ∇v (“electrostatic field”). The Laplace equation for v implies the div-curl system (1.4) for uwith h = 0, and the Dirichlet and Neumann conditions in (1.8) and (1.9) for v imply the vanishing of the tangential component (1.5) or of the normal component (1.6), respectively. Finally, v ∈ W1+1p+ε,p(Ω) is equivalent tou∈Wp1+ε,p(Ω;C3).
The construction of our counterexample uses the ideas of Filonov in the paper [7], where he considers a related question for ε = 12 and constructs aC32 domainΩthat satisfies, among other interesting properties
H2(Ω)∩H01(Ω) =H02(Ω),
that is, the homogeneous Dirichlet condition forH2functions implies the homogeneous Neumann condition, see also [4]. Generalizing this, theC1domainΩthat we will construct satisfies
W1+1p+ε,p(Ω)∩W01,p(Ω) =W1+
1 p+ε,p
0 (Ω) ∀1≤p < ∞,0< ε <1. (1.10)
2. GENERALIZING FILONOV’S SEPARATING FUNCTION
We construct a continuous real-valued functionf onT = R/(2πZ)with the following property:
Ifaandb belong toWε,p(T)for someε >0,p≥1, andaf =b, thena=b= 0.
The construction and proof are modeled after Filonov’s construction of aC12 function that has the above separation property forε = 12 andp = 2. It is in the lineage of Weierstrass’ example of a continuous nowhere differentiable function.
We definef via a lacunary Fourier series f(x) =
∞
X
k=1
aksin(bkx) =
∞
X
k=1
fk(x) (2.1)
where the sequencesak > 0andbk ∈ Nare chosen so that they satisfyP
ak < ∞andbk ≥ 2, bk+1 ≥2bk,k ≥1, and the following properties for a given small constantγ >0to be fixed later
LIMIT SOBOLEV REGULARITY FOR DIRICHLET AND NEUMANN PROBLEMS ON LIPSCHITZ DOMAINS 5
on (see (2.7)):
m−1
X
k=1
akbk ≤γ ambm ∀m≥2 (2.2)
∞
X
k=m+1
ak ≤γ am ∀m≥1 (2.3)
∞
X
m=1
apmbmpε = +∞ ∀ε >0, p≥1. (2.4) We first show that for sufficiently large q ∈ N the sequences ak = q−k, bk = 2qk have the properties (2.2)–(2.4), and we shall keep this choice from now on1 .
For (2.2), letsm = a 1
mbm
Pm−1
k=1 akbk. Noting that forq ≥ 7we have q221−q < 1, we show by induction that thensm < q−11 for allm≥2, which implies (2.2) forqlarge enough. Indeed,
s2 = aa12bb12 =q2(1−q)q < q21−q < 1q < q−11 , and ifsm < q−11 it follows that
sm+1 = (sm+ 1)a ambm
m+1bm+1 = (sm+ 1)q2(1−q)qm <(sm+ 1)q2(1−q) <(q−11 + 1)1q = q−11 . For (2.3), we have
∞
X
k=m+1
ak
am
=
∞
X
k=1
q−k = 1 q−1 which again is less thanγforqlarge enough.
For (2.4) we use that2t ≥tlog 2for allt >0, so thatapmbmpε = (2εqm/qm)p ≥(εlog 2)p for allm.
Lemma 2.1. The functionf defined by(2.1)is continuous onTand satisfies Z 2π
0
|f(y)−f(x)|p
|y−x|1+pε dy= +∞ for allx∈[0,2π], ε >0,1≤p <∞. (2.5) Proof. We first note that we have f(2π−x) = −f(x), so that it is sufficient to prove (2.5) for x∈[0, π]. In this case[x, x+ 1]⊂[0,2π], and therefore with the disjoint intervalsIm = [b1
m,b2
m] we have
Z 2π
0
|f(y)−f(x)|p
|y−x|1+pε dy≥
∞
X
m=1
Z
Im
|f(x+h)−f(x)|p
|h|1+pε dh (2.6)
Now forh∈Im we estimate Z
Im
|f(x+h)−f(x)|p
|h|1+pε dh1p
≥J1−J2 withJ1 =Z
Im
|fm(x+h)−fm(x)|p
|h|1+pε dh1p
andJ2 = X
k6=m
Z
Im
|fk(x+h)−fk(x)|p
|h|1+pε dh1p .
1By plotting approximate values of the integral in (2.7) against the variablezand visual inspection of the graph, one can obtain a rough numerical approximation of γ that indicates that γ ≥ 0.0154. In view of the condition 1/(q−1)< γ, this suggests that a value ofq= 66should be “sufficiently large”.
To estimateJ1, we assume that0< ε <1and make the change of variablest=bmhto obtain J1 =ambεmZ 2
1
|sin(bmx+t)−sin(bmx)|pt−(1+pε)dt1p
≥5γ ambmε , where we defined
γ = 15min
z∈T
Z 2
1
|sin(z+t)−sin(z)|t−2dt >0. (2.7) Here we used H¨older’s inequality,
Z 2
1
|sin(z+t)−sin(z)|
t2 dt≤
Z 2
1
|sin(z+t)−sin(z)|
t1+ε dt
≤Z 2 1
|sin(z+t)−sin(z)|pt−(1+pε)dt1pZ 2 1
dt t
1−1p
. To estimateJ2, we use fork ≤m−1
|fk(x+h)−fk(x)| ≤akbk|h| ≤2akbk 1 bm
and fork ≥m+ 1
|fk(x+h)−fk(x)| ≤2ak
so that we obtain with (2.2)
m−1
X
k=1
Z
Im
|fk(x+h)−fk(x)|p
|h|1+pε dh1p
≤2γam
Z
Im
dh
|h|1+pε 1p
≤2γambmε and with (2.3)
∞
X
k=m+1
Z
Im
|fk(x+h)−fk(x)|p
|h|1+pε dh1p
≤2γam
Z
Im
dh
|h|1+pε 1p
≤2γambmε , henceJ2 ≤4γambmε .
Together, this gives
Z
Im
|f(x+h)−f(x)|p
|h|1+pε dh1p
≥γ ambεm, and finally with (2.6) and (2.4)
Z 2π
0
|f(y)−f(x)|p
|y−x|1+pε dy ≥
∞
X
m=1
γpapmbmpε = +∞.
Proposition 2.2. The functionf defined by(2.1)has the following separation property: Let0 <
ε <1,p≥1anda, b∈Wε,p(0,2π). If af =b, thena =b = 0.
Proof. Write theWε,pseminorm as in (1.7)
|b|ε,p =Z 2π 0
Z 2π
0
|b(y)−b(x)|p
|y−x|1+pε dy dx1p . Using
b(y)−b(x) = (f(y)−f(x))a(x) +f(y)(a(y)−a(x))
LIMIT SOBOLEV REGULARITY FOR DIRICHLET AND NEUMANN PROBLEMS ON LIPSCHITZ DOMAINS 7
and the triangle inequality, we find fora, b∈Wε,p(0,2π) Z 2π
0
Z 2π
0
|a(x)|p|f(y)−f(x)|p
|y−x|1+pε dy dx1p
≤ |b|ε,p+kfkL∞(T)|a|ε,p<∞.
Because of (2.5) from Lemma2.1, this impliesa(x) = 0for almost allx∈Tand thenb =af = 0.
3. 2DDOMAIN WITH LIMITED REGULARITY
LetF(x) = 1 +Rx
0 f(t)dt. ThenF ∈C1(T),F′ =f, and 12 < F(x)< 32. The latter estimate follows easily from
|F(x)−1|=|
∞
X
k=1
ak1−cos(bkx)
bk | ≤2−q
∞
X
k=1
2q−k = 21−qq−11 ≤ 1 2. We define now theC1 domainω ⊂R2using polar coordinates(r, θ)
ω={(r, θ)|r < F(θ)}.
Proposition 3.1. Let p ≥ 1, ε > 0 andu ∈ W1p+ε,p(ω;C2) be such that its normal tracen ·u vanishes on∂ω. Thenu= 0on∂ω. The same conclusion is valid when the tangential tracen×u vanishes on∂ω.
Proof. (Following Filonov [7,§5]) The unit normalnon∂ωhas the Cartesian components n1 = (F2+f2)−12(F cosθ+fsinθ), n2 = (F2+f2)−12(F sinθ−fcosθ). Therefore the conditionn1u1+n2u2 = 0impliesaf =bif we define
a=u2cosθ−u1sinθ , b = (u1cosθ+u2sinθ)F
Now, since the tracesujon∂ω, understood as functionsθ 7→uj(F(θ), θ)onT, belong toWε,p(T), we also havea, b ∈ Wε,p(T). According to Proposition 2.2 we finda = b = 0, which implies u1 =u2 = 0on∂ω. The result using vanishing tangential trace follows by a rotation byπ/2.
Corollary 3.2. (i) There existsg ∈ C∞(ω)such that the solution vD ∈ H01(ω) of the Dirichlet problem
∆vD =g inω; vD = 0 on∂ω does not belong toW1+1p+ε,p(ω)for anyε >0,p≥1.
(i) There existsg ∈C∞(ω)such that any solutionvN ∈H1(ω)of the Neumann problem
∆vN =g inω; ∂nvN = 0 on∂ω does not belong toW1+1p+ε,p(ω)for anyε >0,p≥1.
Proof. ForvD one can take g = 1. Set u = ∇vD. If vD ∈ W1+1p+ε,p(ω), then u satisfies the hypotheses of Proposition 3.1with vanishing tangential trace. Hence also the normal trace of u vanishes, i.e. ∂nvD = 0on∂ω. Then Green’s formula impliesR
ωg = 0, which is not the case.
ForvN ∈ W1+1p+ε,p(ω)one obtains similarly that the tangential derivative on the boundary van- ishes, hence the trace ofvN on∂ω is constant, without loss of generality equal to zero. ThusvN
is also solution of the Dirichlet problem. That there existsg ∈L2(ω)for which this is impossible can be seen as follows:
Letg be a non-zero harmonic polynomial such thatR
ωg = 0, for exampleg(x1, x2) = αx1x2 + β(x21−x22)with suitably chosen coefficientsα, β ∈R. ThenvN exists, and Green’s formula gives the contradiction
0 = Z
∂ω
(∂nvNg−vN∂ng)ds = Z
ω
(∆vNg−vN∆g)dx= Z
ω
g2dx .
Remark 3.3. No eigenfunction of the Laplacian with Dirichlet conditions on ω can belong to W1+1p+ε,p(ω)withε >0, because it would also have vanishing normal derivative. Its extension by zero outsideωwould then be a Dirichlet eigenfunction with the same eigenvalue on any domain containing ω. This contradicts for example the well known behavior of Dirichlet eigenvalues on disks or squares with varying size. It contradicts also the well known interior analyticity of Dirichlet eigenfunctions.
4. EXAMPLE IN HIGHER DIMENSIONS
Fromω ⊂R2one can constructΩ⊂Rdas follows (see [7], forn= 3also [4,§6]). In cylindrical coordinates(r, θ, z),z ∈Rd−2:
Ω = {(r, θ, z)| r2
F(θ)2 +|z|2 <1}
The intersection with the planez = z0 gives for|z0| < 1 the scaled domainp
1− |z0|2ω. One can still prove that for this domainΩand0< ε <1there holds
W1+p1+ε,p(Ω)∩W01,p(Ω) =W1+
1 p+ε,p
0 (Ω), (4.1)
that is, for functions in W1+1p+ε,p(Ω) the vanishing of the boundary trace implies that also the normal derivative is zero on the boundary.
Indeed, suppose that v ∈ W1+1p+ε,p(Ω), v = 0 on ∂Ω and let u = ∇v. Then the tangential components ofuare zero on the boundary, and we have to show that the normal component ofu vanishes, too, on∂Ω. Define
˜
u(r, θ, z) =u(p
1− |z|2r, θ, z). Thenu˜is defined on the product domain
Ω =˜ ω×B1 ={(r, θ, z)|(r, θ)∈ω,|z|<1}.
For anyδ ∈(0,1), letΩ˜δ =ω×Bδ. On this product domain, there holds the inclusion Ws,p( ˜Ωδ)⊂Lp Bδ;Ws,p(ω)
,
as can be seen first for integersdirectly from the definition of the Sobolev space Ws,pand then for alls ≥0by interpolation. Thusu ∈W1p+ε,p(Ω;Cd)implies thatu˜restricted toΩ˜δ belongs to
LIMIT SOBOLEV REGULARITY FOR DIRICHLET AND NEUMANN PROBLEMS ON LIPSCHITZ DOMAINS 9
Lp Bδ;W1p+ε,p(ω;Cd)
,and for almost everyz0 ∈Bδ, the restrictionwz0 ofu˜to the planez =z0
belongs toWp1+ε,p(ω,Cd). The vanishing of the tangential components ofu on∂Ωimplies that the component ofwz0 that is parallel to the planez = 0and tangential to∂ωvanishes on∂ω. Then Proposition3.1tells us that the component ofwz0 that is parallel to the planez = 0and normal to
∂ωvanishes on∂ω, too. This means that at such a point(r, θ, z)∈∂Ωwith(p
1− |z|2r, θ)∈∂ω, z =z0, in addition to the tangential components a component ofuvanishes that is not tangential, and hence all components ofuvanish there. Since this is true for almost allz0 satisfying|z0|< δ and for all0< δ <1, we see that the trace ofuon∂Ωis zero, which proves (4.1).
The non-regularity result of Theorem1.2for the Dirichlet problem inΩthen follows in the same way as in the two-dimensional case. In particular, one can takeg = 1for the counterexample.
For the Neumann problem, a slightly different variant of addingd−2variables works, and this variant could also be used for the Dirichlet problem, giving a counterexample with a somewhat less regular right hand sideg. For this variant, (4.1) still holds. We redefine the domainΩso that it contains a cylindrical part (see also [7, §5.2]). This is done by modifying the function1− |z|2 in the previous example. Choose a decreasingC∞functionµonR+satisfying
µ(t) = 1 fort ≤1 ; µ(t)≤0 fort ≥4 ; µ′(t)<0 fort≥2, and define
Ω ={(r, θ, z)|r2 < µ(|z|2)F(θ)2}. (4.2) It is not hard to see thatΩhas aC1boundary.
We now use the two-dimensional example presented in the previous section and denote byv0 the function found there that satisfies the Neumann problem onω with right hand side g0 ∈ C∞(ω) and that does not belong to anyW1+1p+ε,p(ω)forε > 0, p≥ 1. In addition, we choose a function χ∈C0∞(R+)satisfyingχ(t) = 1fort < 12,χ(t) = 0fort ≥1. Then we define
v(x, z) =v0(x)χ(|z|); g(x, z) =g0(x)χ(|z|) +v0(x)∆zχ(|z|); (x∈ω, |z|<1). Initially,vandgare defined on the cylinderω×B1 ⊂Ω, and we extend them by zero on the rest ofΩ.
One easily verifies thatv satisfies
∆v =g inΩ ; ∂nv = 0 on∂Ω.
Noting that both χ(|z|) and ∆zχ(|z|) define C∞(Ω) functions and using the regularity ofv0 ∈ W1+1p,p(ω)for allp >1, so thatv0is H¨older continuous onω, one finds thatgis H¨older continuous onΩ. Finally the non-regularity ofv0implies clearly that alsov 6∈W1+1p+ε,p(Ω)forε >0,p≥1.
This concludes the proof of Theorem1.2.
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