A Remark on Global $W^{1,p}$ Bounds for Harmonic Functions with Lipschitz Boundary Values

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by  Nikos Katzourakis

A REMARK ON GLOBAL W1,p BOUNDS FOR HARMONIC
FUNCTIONS WITH LIPSCHITZ BOUNDARY VALUES
NIKOSKATZOURAKIS
6
1 Abstract. In this note we show that gradient of Harmonic functions on a
0 smooth domain with Lipschitz boundary values is pointwise bounded by a
2 universalfunctionwhichisinLp forallfinitep≥1.

l
u
J

1
1. Introduction

]
P
Kellogg in [K] pioneered the study of the boundary behaviour of the gradient of
A
Harmonic functions on a bounded domain. Roughly speaking, he established that
. in a domain of R3 near a boundary region which can represented as the graph of
h
t a planar function, the gradient of any Harmonic function is continuous up to the
a
boundaryprovidedthatthegradientoftheboundaryfunctionandoftheHarmonic
m
function are Dini continuous themselves on the boundary. The celebrated theory
[
ofSchauderestimates[GT]establishesstrongrelevantresultsforgeneraluniformly

2 elliptic PDEs, providing interior and global H¨older bounds for solutions and their
v derivatives in terms of the H¨older norms of the boundary values of the solution
0
and the right hand side of the PDE. The Schauder theory has been improved
9
and extended by many authors, but typically for second order elliptic PDEs with
1
0 boundary values of the solutions and right hand sides of the PDEs in the H¨older
0 spaces C2,α or C1,α, in order to obtain uniform estimates for the solutions in the
. respective H¨older spaces.
1
0 In [GH] Gilbarg-H¨ormanderhave extendedSchaudertheory to include hypothe-
6 ses of lower regularity of the boundary values of the solution, of the boundary of
1 the domain and of the coefficients of the equations. Troianiello [T] relaxed further
:
v some conditions of Gilbarg-H¨ormander [GH]. In the paper [HS] Hile-Stanoyevitch,
i extending an older result of Hardy-Littlewood [HL], proved that the gradient of a
X
HarmonicfunctionwithLipschitzcontinuousboundaryvaluesispointwisebounded
r
a uptoaconstantbythelogarithmofamultipleoftheinverseofthedistancetothe
boundary.
However, it appears that in none of these results, even for the special case of
the Laplacian, there is an explicit global bound in Lp for the gradient of Harmonic
functions which have just Lipschitz boundary values and not C1,α. In this note
establish the following consequence of the result of Hile-Stanoyevitch:
Theorem 1. Let n ≥ 2, Ω ⊆ Rn a bounded open set with C2 boundary. Let also
g :∂Ω→R with g ∈Lip(∂Ω), that is g ∈C0(∂Ω) and
|g(x)−g(y)|
Lip(g,∂Ω) := sup < ∞.
|x−y|
x,y∈∂Ω,x(cid:54)=y
1
2 NIKOSKATZOURAKIS
(1) There exists a positive function f : Ω → (0,∞) depending on Ω,n such
Ω,n
that
(cid:92)
(1.1) f ∈ LpΩ)∩C0(Ω)
Ω,n
p∈[1,∞)
and if u∈C2(Ω)∩C0(Ω) is the Harmonic function solving
(cid:26) ∆u = 0, in Ω,
(1.2)
u = g, on ∂Ω,
then we have the estimate
(cid:12) (cid:12)
(1.3) (cid:12)Du(x)(cid:12) ≤ Lip(g,∂Ω)fΩ,n(x), x∈Ω.
(2) Let (gm)∞ ⊆Lip(∂Ω) satisfy for some C >0
1
(1.4) Lip(gm,∂Ω) + max|gm| ≤ C, m∈N.
∂Ω
Let also (um)∞ ∈C2(Ω)∩C0(Ω) be the Harmonic functions solving
1
(cid:26) ∆um = 0, in Ω,
(1.5)
um = gm, on ∂Ω.
Then, (um)∞ is strongly precompact in (cid:84)∞ W1,p(Ω) and if
1 p=1
(1.6) gmk −→g in C0(Ω), as k →∞,
then there is a unique limit point u ∈ C2(Ω)∩C0(Ω) of the subsequence
(umk)∞ such that along perhaps a further subsequence
1
(1.7) umk −→u in W1,p(Ω) ∀p≥1, as k →∞,
and the limit function u solves
(cid:26) ∆u = 0, in Ω,
u = g, on ∂Ω.
The motivation to derive the above integrability result and its consequences
comes from certain recent advances in generalised solutions of nonlinear PDE and
vectorial Calculus of Variations in the space L∞ ([Ka4] and [Ka2, Ka3]). The vec-
torial counterparts of Harmonic functions provide useful energy comparison maps
since they are “stable” in Lp for all 1<p<∞.
2. Proofs
Ournotationiseitherself-explanatoryorotherwisestandardase.g.in[E], [Ka].
ThestartingpointofourproofisthefollowingestimateofHile-Stanoyevitch: under
thehypothesesofTheorem1, thegradientDuofaHarmonicfunctionu∈C2(Ω)∩
C0(Ω) which solves (1.2) with g ∈Lip(∂Ω) satisfies the logarithmic estimate
(cid:18) (cid:19)
(cid:12) (cid:12) diam(Ω)
(2.1) (cid:12)Du(x)(cid:12) ≤ C(Ω,n)Lip(g,∂Ω) ln , x∈Ω.
dist(x,∂Ω)
for some C depending just on Ω (and the dimension). In (2.1), diam(Ω) is the
diameter of the domain and dist(x,∂Ω) the distance of x from the boundary:
(cid:8) (cid:9)
diam(Ω) := sup |x−y| : x,y ∈Ω ,
(cid:8) (cid:9)
dist(x,∂Ω) := inf |x−z| : z ∈∂Ω .
GLOBAL W1,p BOUNDS FOR HARMONIC FUNCTIONS 3
Proof of (1) of Theorem 1. Fix ε > 0 smaller than the diameter of Ω and
consider the inner open ε neighbourhood of Ω:
Ωε := (cid:8)x∈Ω : dist(x,∂Ω)>ε(cid:9).
It is well known that (see e.g. [GT])
dist(·,∂Ω) ∈ W1,∞(Rn)
loc
and
(cid:12) (cid:12)
(2.2) (cid:12)Ddist(·,∂Ω)(cid:12) = 1, a.e. on Ω.
Let p ∈ [1,∞). By the Co-Area formula (see e.g. [[EG], Proposition 3, p. 118])
applied to the function
(cid:18) (cid:18) diam(Ω) (cid:19)(cid:19)p
Rn (cid:51) x(cid:55)−→χ (x) ln ∈R
Ωε dist(x,∂Ω)
(where χ is the characteristic function of Ωε), we have
Ωε
(cid:90) (cid:18) (cid:18) diam(Ω) (cid:19)(cid:19)p
ln dx =
dist(x,∂Ω)
Ωε
 (cid:18) (cid:18) diam(Ω) (cid:19)(cid:19)p 
(2.3) ln
= (cid:90) diam(Ω)(cid:90) (cid:12) dist(z,∂Ω)(cid:12) dHn−1(z)dt
ε  {dist(·,∂Ω)=t} (cid:12)Ddist(z,∂Ω)(cid:12) 
where Hn−1 is the (n−1)-dimensional Hausdorff measure. By using (2.2), (2.3)
simplifies to
(cid:90) (cid:18) (cid:18) diam(Ω) (cid:19)(cid:19)p
ln dx =
dist(x,∂Ω)
Ωε
(cid:90) diam(Ω)(cid:32)(cid:90) (cid:18) (cid:18) diam(Ω) (cid:19)(cid:19)p (cid:33)
= ln dHn−1(z) dt
dist(z,∂Ω)
ε {dist(·,∂Ω)=t}
Further, since
dist(z,∂Ω)=t, for all z ∈{dist(·,∂Ω)=t},
by setting
(cid:90) (cid:18) (cid:18) diam(Ω) (cid:19)(cid:19)p
(2.4) Iε,p := ln dx
dist(x,∂Ω)
Ωε
we obtain
(cid:90) diam(Ω)(cid:32)(cid:90) (cid:18) (cid:18)diam(Ω)(cid:19)(cid:19)p (cid:33)
Iε,p = ln dHn−1(z) dt
t
ε {dist(·,∂Ω)=t}
(2.5)
(cid:90) diam(Ω)(cid:18) (cid:18)diam(Ω)(cid:19)(cid:19)p (cid:16) (cid:17)
= ln Hn−1 {dist(·,∂Ω)=t} dt.
t
ε
As a consequence of the regularity of the boundary, standard results regarding
the equivalence between the Hausdorff measure and the Minkowski content for
rectifiable sets (see e.g. [[AFP], Section 2.13, Theorem 2.106]) imply that there is
a C =C(Ω) such that
(cid:16) (cid:17)
ess sup Hn−1 {dist(·,∂Ω)=t} ≤ C(Ω)
0<t<diam(Ω)
4 NIKOSKATZOURAKIS
and hence the inequality (2.5) gives
(cid:90) diam(Ω)(cid:18) (cid:18)diam(Ω)(cid:19)(cid:19)p
(2.6) Iε,p ≤ C(Ω) ln dt.
t
ε
By the change of variables
diam(Ω)
ω :=
t
we can rewrite the estimate (2.6) as
(cid:90) diam(Ω)/ε (lnω)p
Iε,p ≤ C(Ω)diam(Ω) dω
ω2
1
and by enlarging perhaps the constant C(Ω), we rewrite this as
(cid:90) diam(Ω)/ε(cid:18)lnω(cid:19)p
(2.7) Iε,p ≤ C(Ω) dω.
ω2/p
1
Claim 2. We have that
lim Iε,p ≤ C(Ω,n,p) <∞.
ε→0
First proof of Claim 2(proposedbyoneofthereferees): Byusingthefollowing
known property of the Gamma function
(cid:90) ∞ lnpx
dx = Γ(1+p)
x2
1
we readily conclude. (cid:3)
Second proof of Claim 2: We now give a direct argument without quoting
special functions. Consider now the function
lnω
g(ω) := , g : (1,∞)→(0,∞).
ω2/p
Since
1−(2/p)lnω
g(cid:48)(ω) = ,
ω(2/p)+1
wehavethatgisstrictlyincreasingon(1,e2/p)andstrictlydecreasingon(e2/p,∞).
Further, note that t (cid:55)→ gp(t) also enjoys the exact same monotonicity properties
since s(cid:55)→sp is strictly increasing. Moreover, since
e2/p ≤ 10
for all p ∈ [1,∞) and by using that ε (cid:55)→ Iε,p is decreasing (in view of (2.4)), we
have
(cid:20)(cid:90) 10 (lnω)p (cid:90) ∞ (lnω)p (cid:21)
limIε,p ≤ C(Ω) dt + dt
ε→0 1 ω2 10 ω2
(cid:20) (cid:18) (lnω)p(cid:19) (cid:90) ∞ (lnω)p (cid:21)
≤ C(Ω) 10 sup + dt
ω2 ω2
1<ω<10 10
(cid:20) (cid:18)(lnω)p(cid:19)(cid:12) (cid:90) ∞ (lnω)p (cid:21)
= C(Ω) 10 (cid:12) + dt
ω2 (cid:12)ω=e2/p 10 ω2
GLOBAL W1,p BOUNDS FOR HARMONIC FUNCTIONS 5
and hence
(cid:34) (2/p)p (cid:88)∞ (cid:90) k+1 (lnω)p (cid:35)
limIε,p ≤ C(Ω) 10 + dt
ε→0 e4/p k=10 k ω2
(cid:34) (2/p)p (cid:88)∞ (cid:18) (lnω)p(cid:19)(cid:35)
≤ C(Ω) 10 + sup
e4/p ω2
k<ω<k+1
k=10
which gives
(cid:34) (2/p)p (cid:88)∞ (lnk)p(cid:35)
(2.8) limIε,p ≤ C(Ω) 10 + .
ε→0 e4/p k2
k=10
Now we show that the series
(cid:88)∞ (lnk)p
S :=
k2
k=10
converges.
Method 1: Since the sequence
(lnk)p
, k = 10,11,12,...
k2
is decreasing, by the Cauchy condensation test the series S converges if and only if
(cid:88)∞ (lnk)p
2mA < ∞, A := .
2m k k2
m=10
Since
2m+1A (m+1)p(ln2)p2−m−1 1(cid:18) 1 (cid:19)p 1
2m+1 = = 1+ −→ ,
2mA mp(ln2)p2−m 2 m 2
2m
as m→∞, by the Ratio test we have that
(cid:88)∞ (lnk)p
= C(p) < ∞
k2
k=10
since (cid:80)∞ 2mA converges.
m=10 2m
Method 2 (proposed by one of the referees): By repeated applications of the del
Hospital rule (p∈N), we have
(lnk)p
k2 p!
lim = lim = 0
k→∞ 1 k→∞ k
k3/2
and hence
(lnk)p 1
≤
k2 k3/2
for k ∈N large enough. Since the series
∞
(cid:88) 1
k3/2
k=10
converges and hence so does S by the comparison test.
6 NIKOSKATZOURAKIS
In either case, by (2.4) and (2.8) we have that there is a constant C(Ω,n,p)
depending only on Ω,n,p such that
(cid:90) (cid:18) (cid:18) diam(Ω) (cid:19)(cid:19)p
ln dx = limIε,p
(2.9) Ω dist(x,∂Ω) ε→0
≤ C(Ω,n,p).
By combining (2.9) with (2.1), we see that by setting
(cid:18) (cid:19)
diam(Ω)
f (x) := C(Ω,n)ln , x∈Ω
Ω,n dist(x,∂Ω)
(1) of Theorem 1 is established. (cid:3)
Proof of (2) of Theorem 1. Let um solve (1.5). By standard interior bounds on
the derivatives of Harmonic functions in terms of their boundary values (see e.g.
[GT]) and (1.4), we have that theHessians (D2um)∞ are bounded in C0(Ω,Rn×n),
1
thatisuniformlyoverthecompactsubsetsofΩ. Thesameistrueforthe3rdorder
derivatives as well; thus, for any Ω(cid:48) (cid:98)Ω, there is C(Ω(cid:48)) such that
3
(cid:88) (cid:13)(cid:13)Dkum(cid:13)(cid:13)C0(Ω(cid:48)) ≤ C(Ω(cid:48))(cid:107)um(cid:107)C0(Ω)
k=1
and by the Maximum Principle we have
(cid:107)um(cid:107) ≤ max|gm| ≤ C.
C0(Ω)
∂Ω
As a consequence,
(cid:12) (cid:12) (cid:41)
(cid:12)Dkum(x)−Dkum(y)(cid:12) ≤ C(Ω(cid:48))|x−y|,
x,y ∈Ω(cid:48), k =0,1,2,3, m∈N
(cid:12) (cid:12)
(cid:12)Dkum(x)(cid:12) ≤ C(Ω(cid:48)),
and by the Ascoli-Arzela theorem, the sequence
(cid:16) (cid:17)∞
um,Dum,D2um
m=1
is precompact uniformly over the compact subsets of Ω. Again by (1.4), we have
(cid:12) (cid:12) (cid:41)
(cid:12)gm(x)−gm(y)(cid:12) ≤ C|x−y|,
x,y ∈∂Ω, m∈N
(cid:12) (cid:12)
(cid:12)gm(x)(cid:12) ≤ C,
whichgivesthat(gm)∞ isboundedandequicontinuouson∂Ω. Thus,bytheAscoli-
1
Arzela theorem and by the lower semicontinuity of the Lipschitz seminorm with
respect to uniform convergence, there is a subsequence (gmk)∞ and g ∈ Lip(∂Ω)
1
such that
gmk −→g, as k →∞ in C0(∂Ω).
Along perhaps a further subsequence, by the above bounds on (um)∞ ⊆ C2(Ω)∩
1
C0(Ω), there is u∈C2(Ω) such that
 umk −→u, in C0(Ω),

(2.10) Dumk −→Du, in C0(Ω,Rn),
 D2umk −→D2u, in C0(Ω,Rn×n),
GLOBAL W1,p BOUNDS FOR HARMONIC FUNCTIONS 7
as k → ∞. By passing to the limit in the equation ∆um = 0 we get that ∆u = 0.
Since the measure of Ω is finite, for any p ∈ [1,∞) by H¨older inequality we have
that
(cid:16) (cid:17)
(cid:107)um(cid:107) ≤ |Ω|1/p (cid:107)um(cid:107) ≤ C(Ω,p).
Lp(Ω) C0(Ω)
By item (1) of the theorem and by (1.4), we have that
(cid:107)Dum(cid:107) ≤ C(Ω,n,p).
Lp(Ω)
Hence, we have the bound
(cid:107)um(cid:107) ≤ C(Ω,n,p), p≥1.
W1,p(Ω)
By the Morrey embedding theorem, by choosing p > n we have that (umk)∞ is
1
precompact in C0(Ω) and hence by (2.10) we have that
(2.11) umk −→u, in C0(Ω) as k →∞.
Hence, u = g on ∂Ω and as a consequence u solves the limit Dirichlet problem.
Finally, if E ⊆Ω is a measurable subset, by the H¨older inequality we have that
(cid:90) (cid:18)(cid:90) (cid:19) p
(cid:12)(cid:12)Dum(x)(cid:12)(cid:12)pdx ≤ |E|1−p+p1 (cid:12)(cid:12)Dum(x)(cid:12)(cid:12)p+1dx p+1
E E
= |E|1−p+p1(cid:16)(cid:107)Dum(cid:107)Lp+1(Ω)(cid:17)p
≤ |E|1−p+p1C(Ω,n,p).
Hence, thesequenceofgradients(Dumk)∞ isp-equi-integrableonΩ. By (2.10), we
1
have
Dumk −→Du in measure on Ω, as k →∞.
SinceΩhasfinitemeasure, theVitaliConvergencetheorem(e.g.[FL])impliesthat
Dumk −→Du in Lp(Ω), as k →∞.
Item (2) of Theorem 1 has been established. (cid:3)
Acknowledgement. The author would like to thank the referees of this paper
most warmly for the careful reading of the manuscript and for providing thought-
through alternative simpler proofs of certain of the original arguments.
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Department of Mathematics and Statistics, University of Reading, Whiteknights,
PO Box 220, Reading RG6 6AX, Reading, UK
E-mail address: [email protected]