Sharp Hardy and Hardy--Sobolev inequalities with point singularities on the boundary

0.44 MB·

by  Gerassimos Barbatis

Sharp Hardy and Hardy–Sobolev inequalities with point
singularities on the boundary
G. Barbatis∗ S. Filippas † A. Tertikas †
7
1
January 24, 2017
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Contents

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P
A 1 Introduction and main results 2
.
h
t 2 Distance from the vertex of a cone 9
a
m 2.1 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
[ 2.2 Improved inequalities in half balls. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1
v 3 Hardy inequality in bounded domains 16
6
3
3 4 Improved Hardy and Hardy-Sobolev inequalities for bounded domains 18
6
0
5 Characterizing maximal potentials 22
.
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0
7 6 Logarithmic improvements and maximal potentials 27
1
6.1 Logarithmic improvements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
:
v
6.2 Maximal logarithmic potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
i
X
r
7 Further results 33
a
7.1 Maximal potentials in finite cones. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
7.2 Hardy-Sobolev inequalities involving multiple boundary points . . . . . . . . . . . . . . . . . 35
Abstract
We initially study the Hardy inequality when the singularity is placed on the boundary of a bounded
domain in Rn that satisfies both an interior and exterior ball condition at the singularity. We obtain
the sharp Hardy constant n2/4 in case the exterior ball is large enough and show the necessity of the
large exterior ball condition. We improve Hardy inequality with the best constant by adding a sharp
Sobolev term. We next produce criteria that lead to characterizing maximal potentials that improve
Hardy inequality. Breaking the criteria one produces successive improvements with sharp constants.
Our approach goes through in less regular domains, like cones. In the case of a cone, contrary to the
smooth case, theSobolev constant does dependon theopening of thecone.
∗Department ofMathematics, NationalandKapodistrianUniversityofAthens,15784Athens,Greece
†Department ofMathematics andAppliedMathematics,UniversityofCrete,70013Heraklion,Greece
Emailaddresses: [email protected]; fi[email protected]; [email protected]
1
Keywords: Hardyinequality,Hardyconstant,boundarysingularity,Sobolevinequality,maximalpotential,
best constant, conformality.
2010 Mathematics Subject Classification: 35A23, 35J20,35J75 (46E35, 26D10, 35J60)
1 Introduction and main results
For n 3 Hardy inequality states, that for any u C (Rn) there holds
≥ ∈ c∞
n 2 2 u2
u2dx − dx ,
ZRn|∇ | ≥(cid:16) 2 (cid:17) ZRn |x|2
where (n−2)2 is the best constant. On the other hand Sobolev inequality reads as follows
4
n 2
|∇u|2dx≥Sn |u|n2−n2dx −n
ZΩ (cid:18)ZΩ (cid:19)
2
where S =πn(n 2) Γ(n)/Γ(n) n is the best Sobolev constant.
n − 2
Therearevariousimpr(cid:0)ovedversion(cid:1)sofeitherHardyorSobolevinequalitiesinthecaseofaboundeddomain
Ω containing the origin see e.g [7, 23, 22, 1, 16, 3, 4, 15, 6, 5]. We mention in particular the following sharp
Hardy–Sobolev inequality from [16, 2] that combines both inequalities
n 2
|∇u|2dx≥ n−2 2 2 ux22dx+(n−2)−2(nn−1) Sn X12nn−−22|u|n2−n2dx n−
ZΩ (cid:16) (cid:17) ZΩ | | (cid:18)ZΩ (cid:19)
for all u C (Ω). Here X =X (x/D), with
∈ c∞ 1 1 | |
1
X (t)= , t (0,1), D := sup x .
1
1 lnt ∈ | |
x Ω
− ∈
A natural question is what are the analogues of Hardy and Hardy–Sobolev inequalities in case the origin is
on the boundary of Ω instead of being in the interior. As we shall see, contrary to the previous case, the
geometry of Ω plays an important role. In the simplest case of the half space Rn = (x,x ) : x > 0 ,
+ { ′ n n }
Hardy inequality with best constant reads (cf. [21, 17])
n2 u2
u2dx dx, u C (Rn) . (1)
Rn |∇ | ≥ 4 Rn x2 ∀ ∈ c∞ +
Z + Z + | |
InthemoregeneralcasewherethedomainisaconeC withitsvertexattheoriginthesharpHardyinequality
reads (cf. [21])
(n 2)2 u2
C |∇u|2dx≥ −4 +µ1(Σ) C x2dx, ∀u∈Cc∞(C) , (2)
Z (cid:18) (cid:19)Z | |
where Σ = C Sn 1 and µ (Σ) is the first Dirichlet eigenvalue of the Dirichlet Laplace-Beltrami operator
− 1
∩
on Σ.
If on the other hand, the origin is on the boundary of a smooth near zero domain, then, related types of
problems have been studied in [18, 19, 12, 20]. More precisely the following minimization problem has been
considered for 0<s<2 and n 4,
≥
u2dx
µ (Ω)= inf Ω|∇ | , (3)
s u∈H01(Ω) Ω R|u|2x(nn−s−2s)dx nn−−s2
(cid:18) | | (cid:19)
R
anditwasestablishedthatthegeometryofΩaroundzeroplaysanimportantrole. Inparticularifthemean
curvature at zero is negative then
µ (Ω)<µ (Rn) ,
s s +
2
and there exists a minimizer for (3). In the limit case s = 2 the infimum µ (Ω) is the best Hardy constant
2
and under certain geometric assumptions on Ω has been studied in [8, 9, 10, 11, 14].
In [13] it was realized that the geometry plays no role for the local best Hardy constant. That is, for r >0
small enough if we denote by B the ball of radius r centered at the origin, then for a smooth near zero
r
domain Ω one has
n2 u2
u2dx dx, u C (Ω B ) , (4)
|∇ | ≥ 4 x2 ∈ c∞ ∩ r
ZΩ∩Br ZΩ∩Br | |
which in particular implies the existence of a constant λ 0 such that
≥
n2 u2
λ u2dx+ u2dx dx u C (Ω) .
|∇ | ≥ 4 x2 ∈ c∞
ZΩ ZΩ ZΩ | |
Thefirstquestionweraiseinthisworkistofindamorequantitativeresultthatconnectsthelocalinequality
(4) to the global inequality in the half space (1). To state our results we first introduce some notation. We
denote by B (x ) the ball of radius ρ centered at x and simply by B in case the ball is centered at the
ρ 0 0 ρ
origin. For annuli we use the notation
D(x ;r ,r ):= x Rn : r < x x <r .
0 1 2 1 0 2
{ ∈ | − | }
or simply D(r ,r ) in case x =0. Also, e is the unit vector in the x direction.
1 2 0 n n
In our first result we consider the case of annulus:
Theorem 1 Let n 2. There exists a constant τ >0 which depends only on n such that
n
≥
(i) For all 0<τ τ , all ρ>0 and all u C (D(ρ,ρ(1+τ))) there holds
≤ n ∈ c∞
n2 u2
u2dx dx; (5)
ZD(ρ,ρ(1+τ))|∇ | ≥ 4 ZD(ρ,ρ(1+τ)) |x−ρen|2
(ii) For all τ >τ and for all ρ>0 we have
n
n−2 2 <λ := inf D(ρ,ρ(1+τ))|∇u|2dx < n2.
(cid:16) 2 (cid:17) τ Cc∞(D(ρ,ρ(1+τ))) DR(ρ,ρ(1+τ)) |x−uρ2en|2dx 4
R
Moreover λ is strictly decreasing in (τ ,+ ) and lim λ =(n 2)2/4.
τ n τ + τ
∞ → ∞ −
We note that this result implies in a straightforward way inequality (1) for the half-space by using the
inclusion
R R R(1+τ )
B+ D e ; , n
R ⊂ − τ n τ τ
n n n
(cid:0) (cid:1)
for any R>0.
Our next result is a consequence of Theorem 1 and addresses the case of a general domain Ω.
Theorem 2 Let Ω Rn, n 2, be a domain with 0 ∂Ω having the property that there exists a ball
⊂ ≥ ∈
B ( ρe ) Ω and let τ be the constant appearing in Theorem 1. Then the following hold true:
ρ n n
(a)−(local i⊂neqCuality) For all u C (Ω D( ρe ; ρ,ρ(1+τ ))) there holds
∈ c∞ ∩ − n n
n2 u2
u2dx dx . (6)
ZΩ∩D(−ρen;ρ,ρ(1+τn))|∇ | ≥ 4 ZΩ∩D(−ρen;ρ,ρ(1+τn)) |x|2
In particular for all u C (Ω B ) there holds
∈ c∞ ∩ ρτn
n2 u2
u2dx dx . (7)
|∇ | ≥ 4 x2
ZΩ∩Bρτn ZΩ∩Bρτn | |
3
(b) (global inequality) If Ω D( ρe ; ρ,ρ(1+τ )) then for all u C (Ω) there holds
⊂ − n n ∈ c∞
n2 u2
u2dx dx . (8)
|∇ | ≥ 4 x2
ZΩ ZΩ | |
In particular if Ω is bounded and ρτ D :=sup x then
n ≥ Ω| |
n2 u2
u2dx dx ,
|∇ | ≥ 4 x2
ZΩ ZΩ | |
for all u C (Ω).
∈ c∞
If in addition Ω satisfies an interior ball condition at 0 then the constant n2/4 is sharp in all inequalities.
A domain Ω with Hardy constant n2/4 (cf. Theorem 2(b))
Thus in the case of a smooth (near zero) domain Ω, if the largest exterior ball at zero is large enough
compared to the size of Ω then the Hardy constant is n2/4. If however the exterior ball is not large enough
then the Hardy constantis smaller than n2/4, see the example at the end of Section 3 and the figure below.
A domain Ω with Hardy constant smaller than n2/4
We next improve Hardy inequality by adding a Sobolev term:
4
Theorem 3 Let Ω Rn, n 3, be a domain with 0 ∂Ω and having the property that there exists a ball
⊂ ≥ ∈
B ( ρe ) Ω. There exist σ > 0 and a constant C that both depend only on n such that the following
ρ n n n
− ⊂ C
holds true:
(a) (local inequality) For all r σ ρ there holds
n
≤
n 2
|∇u|2dx≥ n42 ux22dx+Cn X12nn−−22|u|n2−n2dx −n ,
ZΩ∩Br ZΩ∩Br | | (cid:18)ZΩ∩Br (cid:19)
for all u C (Ω B ); here X =X (x/3r).
∈ c∞ ∩ r 1 1 | |
(b) (global inequality) Suppose in addition Ω is bounded and D =sup x.
Ω| |
(1) (small exterior ball) Assume that ρ 1D. There exist constants λ , C > 0 depending only on n
≤ 3 n n
such that
n 2
|∇u|2dx+ λρn2 u2dx≥ n42 xu22dx+Cn X12nn−−22|u|n2−n2dx −n ,
ZΩ ZΩ ZΩ | | (cid:18)ZΩ (cid:19)
for all u C (Ω); here X =X (x/D).
∈ c∞ 1 1 | |
(2) (large exterior ball) Assume that ρ 1 D. There exists C >0 depending only on n such that
≥ σn n
n 2
|∇u|2dx≥ n42 ux22dx+Cn X12nn−−22|u|n2−n2dx −n ,
ZΩ ZΩ | | (cid:18)ZΩ (cid:19)
for all u C (Ω). Here X =X (x/3D).
∈ c∞ 1 1 | |
IfinadditionΩsatisfiesaninteriorballconditionat0thentheconstantn2/4andtheexponent(2n 2)/(n 2)
− −
of X are sharp in all inequalities.
1
In particular, in the case of a smooth (near zero) domain Ω, there always exists a nonnegative constant λ
(that depends on Ω) so that we have
λ u2dx+ |∇u|2dx≥ n42 ux22dx+Cn X12nn−−22|u|n2−n2dx n−n2, u∈Cc∞(Ω) .
ZΩ ZΩ ZΩ | | (cid:16)ZΩ (cid:17)
Using Holder’s inequality one easily concludes the existence of homogeneous improvements of the form
n2 u2
λ u2dx+ u2dx dx+C Vu2dx, u C (Ω), (9)
|∇ | ≥ 4 x2 ∈ c∞
ZΩ ZΩ ZΩ | | ZΩ
where the potential V is nonnegative and the constant C is positive. This is the case for instance if the
potential V belongs in Ln/2(Ω 0 ) and satisfies
loc \{ }
Vn/2X11−ndx<+∞ . (10)
ZΩ
The potential V(x)=Xα/x2, α>2, for example, satisfies (10).
1 | |
In connection with this we make the following definition
Definition 1 A non-negative potential V Ln/2(Ω 0 )is called admissible if there exist λ 0 and C >0
∈ loc \{ } ≥
such that (9) holds. The class of all admissible potentials for the domain Ω is denoted by (Ω).
A
For a given V (Ω) we denote by b(λ) > 0 the best constant C of inequality (9). We next address the
∈ A
question whether inequality (9) with best constant b(λ) can be further improved. That is, whether there
exists potential W (Ω) and a positive constant C such that the following inequality holds true as well
∈A
n2 u2
λ u2dx+ |∇u|2dx≥ 4 x2dx+b(λ) Vu2dx+C Wu2dx, u∈Cc∞(Ω) .
ZΩ ZΩ ZΩ | | ZΩ ZΩ
5
In case there does not exist such a potential W we say that the potential
n2 1
+b(λ)V(x) ,
4 x2
| |
is a maximal potential. Our next goal is to characterize maximal potentials. In this direction for V (Ω)
∈A
and small r>0 we define
λ u2dx+ u2dx n2 u2 dx
Cr(V)=u∈Cinc∞f(Ω) RΩ∩Br RΩ∩ΩB∩rB|r∇Vu|2dx− 4 RΩ∩Br |x|2 .
Since C (V) is a non-increasing function we can define R
r
0(V)= lim C (V),
r
C r 0+
→
whichmayalsobeequalto+ . Thisdefinitiongivestheimpressionthat 0(V)mightdependonthechoice
∞ C
of λ, however we establish in Section 5 that 0(V) is independent of λ. We also define
C
Definition 2 Suppose Ω is a bounded domain. We say that the potential V (Ω) is subcritical if 0(V)=
∈A C
+ .
∞
We now can state our next result
Theorem 4 Let Ω Rn, n 3, be a bounded domain with 0 ∂Ω and satisfying both an interior as well
⊂ ≥ ∈
as an exterior ball condition at zero. Let V be a non-negative potential in (Ω). (a) Let λ 0 and b(λ)>0
A ≥
be such that
n2 u2
λ u2dx+ u2dx dx+b(λ) Vu2dx, u C (Ω), (11)
|∇ | ≥ 4 x2 ∈ c∞
ZΩ ZΩ ZΩ | | ZΩ
where b(λ) is the best constant. If in addition
b(λ)< 0(V) ,
C
then the potential n2/4x2+b(λ)V(x) is a maximal potential, that is inequality (11) cannot be improved by
| |
adding a non-negative potential W in the RHS.
(b) If V is a subcritical potential then there exist λ 0 and a best constant b(λ)>0 such that (11) is true.
≥
Moreover the potential n2/4x2+b(λ)V(x) is a maximal potential.
| |
It turns out (see Lemma 17) that the potentials considered earlier in (10) are subcritical ones and therefore
the corresponding potentials n2/4x2+b(λ)V(x) are maximal.
| |
We consider the case of successively improving Hardy inequality. Clearly one has to consider potentials V
for which b(λ)= 0(V). Before stating our result we first define the iterated logarithms (cf. [16])
C
X (t)=X (X (t)), t (0,1], k=1,2,...
k+1 k 1
∈
One can check that for t ∈ (0,1) the series ∞i=1X1(t)X2(t)...Xi(t) converges and that it is a strictly
increasing function of t. We denote by κ the unique κ>1 for which
P
∞
X (1/κ)...X (1/κ)=1/4.
1 i
i=1
X
We then have
Theorem 5 Let Ω Rn, n 2, be a domain with 0 ∂Ω and having the property that there exists a ball
B ( ρe ) Ω¯. Th⊂ere exist≥σ >0 and C >0 that d∈epend only on n such that the following holds true:
ρ n n n
− ⊂C
(a) (local inequality) For all r σ ρ there holds
n
≤
n2 u2 1 ∞ u2
u2dx dx+ X2...X2dx,
|∇ | ≥ 4 x2 4 x2 1 i
ZΩ∩Br ZΩ∩Br | | Xi=1ZΩ∩Br | |
6
for all u C (Ω B ); here X =X (x/3κr).
∈ c∞ ∩ r i i | |
(b) (global inequality) Suppose in addition Ω is bounded and D =sup x.
Ω| |
(1) (small exterior ball) Assume that ρ 1 D. Then there exists λ >0 depending only on n such that
≤ 3κ n
u2dx+ λn u2dx n2 u2 dx+ 1 ∞ u2 X2...X2dx
|∇ | ρ2 ≥ 4 x2 4 x2 1 i
ZΩ ZΩ ZΩ | | i=1ZΩ | |
X
for all u C (Ω); here X =X (x/D).
∈ c∞ i i | |
(2) (large exterior ball) Assume that ρ 1D. Then
≥ σ
n2 u2 1 ∞ u2
u2dx dx+ X2...X2dx,
|∇ | ≥ 4 x2 4 x2 1 i
ZΩ ZΩ | | i=1ZΩ | |
X
for all u C (Ω); here X =X (σ x/(3κD)).
∈ c∞ i i n| |
If in addition Ω satisfies an interior ball condition at 0 then the constants 1/4 are sharp at each step in all
inequalities. For example for the last inequality we have
inf Ω|∇u|2dx− n42 Ω |ux2|2dx = 1 ,
u∈Cc∞(Ω)R Ω ux22X12dRx 4
| |
R
and for each m=2,3,...
inf Ω|∇u|2dx− n42 Ω |ux2|2dx− 14 mi=−11 Ω |ux|22X12...Xi2dx = 1 .
u∈Cc∞(Ω)R R Ω ux22X12..P.Xm2dxR 4
| |
R
We also have the Hardy-Sobolev analogues:
Theorem 6 Let Ω Rn, n 3 be a domain with 0 ∂Ω and having the property that there exists a ball
B ( ρe ) Ω¯. T⊂here exis≥t σ > 0 and C > 0 t∈hat depend only on n such that for any m N the
ρ n n n
− ⊂ C ∈
following holds true:
(a) (local inequality) For all r σ ρ there holds
n
≤
n2 u2 1 m u2
u2dx dx+ X2...X2dx
|∇ | ≥ 4 x2 4 x2 1 i
ZΩ∩Br ZΩ∩Br | | Xi=1ZΩ∩Br | |
n 2
2n 2 2n −n
+Cn (X1...Xm+1)n−−2 |u|n−2dx ,
(cid:18)ZΩ∩Br (cid:19)
for all u C (Ω B ); here X =X (x/3κr).
∈ c∞ ∩ r i i | |
(b) (global inequality) Suppose in addition Ω is bounded and D =sup x. Then
Ω| |
(1) (small exterior ball) Assume that 3κρ D. Then there exist constants λ , C > 0 depending only
n n
≤
on n such that
λ n2 u2 1 m u2
u2dx+ n u2dx dx+ X2...X2dx
|∇ | ρ2 ≥ 4 x2 4 x2 1 i
ZΩ ZΩ ZΩ | | i=1ZΩ | |
X
n 2
+Cn (X1...Xm+1)2nn−−22|u|n2−n2dx −n ,
(cid:18)ZΩ (cid:19)
for all u C (Ω); here X =X (D).
∈ c∞ i i
7
(2) (large exterior ball) Assume that σ ρ D. Then there exist constants C >0 depending only on n
n n
≥
such that
n2 u2 1 m u2
u2dx dx+ X2...X2dx
|∇ | ≥ 4 x2 4 x2 1 i
ZΩ ZΩ | | i=1ZΩ | |
X
n 2
2n 2 2n n−
+Cn (X1...Xm+1)n−−2 |u|n−2dx ,
(cid:18)ZΩ (cid:19)
for all u C (Ω); here X =X (σ x/(3κD)).
∈ c∞ i i n| |
If in addition Ω satisfies an interior ball condition at 0 then the constants 1/4 are sharp and the exponents
(2n 2)/(n 2) of X are also sharp.
i
− −
Theorems 5 and 6 are applicable in particular in the case of half balls. For instance we have
n2 u2 1 m u2
u2dx dx+ X2...X2dx
ZBR+|∇ | ≥ 4 ZBR+ |x|2 4Xi=1ZBR+ |x|2 1 i
n 2
+Cn (X1...Xm+1)2nn−−22|u|n2−n2dx n− ,
(cid:18)ZBR+ (cid:19)
In fact we have produced a stronger result than (12), see Lemmas 13 and 14. In addition we have produced
similar results for the case of finite cones, see Section 2.
We finally describe maximal potentials in the context of logarithmic improvements.
Definition 3 Suppose Ω is a bounded domain. A non-negative potential V Ln/2(Ω 0 ) is called m-
∈ loc \{ }
admissible if there exist λ 0, D˜ D :=sup x and c>0 such that
≥ ≥ Ω| |
n2 u2 1 m u2
λ u2dx+ u2dx dx+ X2...X2dx+C Vu2dx, u C (Ω) , (12)
|∇ | ≥ 4 x2 4 x2 1 i ∈ c∞
ZΩ ZΩ ZΩ | | i=1ZΩ | | ZΩ
X
where X =X (x/D˜). The class of all m-admissible potentials for the domain Ω is denoted by (Ω).
i i m
| | A
We note that there is a big variety of m-admissible potentials. For example if V satisfies
V n2(X1...Xm+1)1−ndx<+ , (13)
∞
ZΩ
where X =X (x/D), i=1,...,m+1, then V is m-admissible by Theorem 6.
i i
| |
ForagivenV (Ω)wedenotebyb (λ)>0the bestconstantC ofinequality(12). Wenextaddressthe
m m
∈A
question whether inequality (12) with best constant b (λ) can be further improved. That is, whether there
m
exists potential W (Ω) and a positive constant C such that the following inequality holds true as well
m
∈A
n2 u2 m u2
λ u2dx+ u2dx dx+ X2...X2dx+b (λ) Vu2dx+C Wu2dx ,
|∇ | ≥ 4 x2 x2 1 i m
ZΩ ZΩ ZΩ | | i=1ZΩ | | ZΩ ZΩ
X
for u C (Ω). In case there does not exist such a potential W, we say that the potential
∈ c∞
n2 1 m X2...X2
+ 1 i +b (λ)V(x) ,
4 x2 x2 m
| | i=1 | |
X
is an m–maximal potential. Our next goal is to characterize m–maximal potentials. In this direction for
V (Ω) and small r >0 we define
m
∈A
λ u2dx+ u2dx n2 u2 dx+ 1 m u2 X2...X2dx
Cm,r(V)=u∈Cc∞in(Ωf∩Br) RΩ∩Br RΩ∩Br|∇ | − 4ΩR∩ΩB∩rBVru|x2|d2x 4Pi=1RΩ∩Br |x|2 1 i ,
R
8
where X =X (x/D˜) and D˜ D. We also define
i i
| | ≥
0(V)= lim C (V),
Cm r 0+ m,r
→
which may also equal + .
∞
Definition 4 Suppose Ω is a bounded domain. We say that the potential V (Ω) is m-subcritical if
m
∈ A
0 =+ .
Cm ∞
It turns out for example (see Lemma 24) that potentials that satisfy (13) are m-subcritical. We then have
Theorem 7 Let Ω Rn, n 3, be a bounded domain with 0 ∂Ω and satisfying both an interior and an
exterior ball conditio⊂n at zero≥. Let V be a non-negative potenti∈al in (Ω) and D˜ D. (a) Let λ 0 and
m
A ≥ ≥
b (λ)>0 be such that
m
n2 u2 1 m u2
λ u2dx+ |∇u|2dx≥ 4 x2dx+ 4 x2X12...Xk2dx+bm(λ) Vu2dx, u∈Cc∞(Ω), (14)
ZΩ ZΩ ZΩ | | k=1ZΩ | | ZΩ
X
where X =X (x/D˜) and b (λ) is the best constant. If in addition
i i m
| |
b (λ)< 0 ,
m Cm
then the potential n2/4x2+1/4x2 m X2...X2+b (λ)V(x) is a maximal potential, that is inequality
| | | | k=1 1 k m
(14) cannot be improved by adding a non-negative potential W in the RHS.
P
(b) If V is an m-subcritical potential then there exist λ 0 and a best constant b (λ)>0 such that (14) is
m
true. Moreover the potential n2/4x2+1/4x2 m X≥2...X2+b (λ)V(x) is a maximal potential.
| | | | k=1 1 k m
P
Analogues to Theorems 3–7 hold true if the domain Ω is a cone with vertex at zero, see Sections 2 and
7. What is interesting in this case is that the Sobolev constant does depend on the cone. In particular if
C :=C B , we have
1 1
∩
Theorem 8 Let n 3. There exists a positive constant C that depends only on Σ such that
≥
n 2
C |∇u|2dx≥ (n−42)2 +µ1(Σ) C xu22dx+C C X12nn−−22|u|n2−n2dx −n , (15)
Z 1 (cid:16) (cid:17)Z 1 | | (cid:18)Z 1 (cid:19)
for all u C (C ); here X =X (x). The exponent (2n 2)/(n 2) of X is the best possible. Moreover
∈ c∞ 1 1 1 | | − − 1
the best constant C for inequality (15) satisfies the estimate
C Cn Σ n2 (16)
≤ | |
for some positive constant C that depends only on n. In particular the best constant C of inequality (15)
n
cannot be taken to be independent of Σ.
Finally, asimilaranalysisgoesthroughif one haspotentials withmultiple singularitiesonthe boundary,see
Theorem 34 for one such result.
2 Distance from the vertex of a cone
Inthissectionweconsiderthecaseofthefinite coneandweobtainbothhomogeneousandnonhomogeneous
improvements of the Hardy inequality (2). We pay particular attention to the special case where the cone
is the half ball B+. In this case the estimates we obtain are stronger than in the case of a general cone and
R
play a crucial role in our subsequent analysis.
9
Let Σ Sn 1 be a domain in Sn 1 (that is a set that is open and connected in the relative topology) with
− −
⊂
Lipschitz boundary. Let µ =µ (Σ) be the kth Dirichlet eigenvalue of the Laplace-Beltrami operator on Σ
k k
and let φ be a corresponding eigenfunction that is,
k
∆ φ (ω)=µ φ (ω), ω Σ ,
Sn 1 k k k
− − ∈
( φk ∂Σ =0.
(cid:12)
Wemayassumethat φk isacomplete(cid:12)orthonormalsysteminL2(Σ). Wenotethatµ1 isasimpleeigenvalue
{ }
and we take φ to be positive.
1
We define
x
C = x Rnr 0 : Σ .
{ ∈ { } x ∈ }
| |
and
x
C =C B = x Rnr 0 : Σ, x <1 .
1 1
∩ { ∈ { } x ∈ | | }
| |
2.1 The general case
We begin by recalling the following well-known result; see [21].
Theorem 9 For n 2 we have
≥
n 2 2 u2
C |∇u|2dx≥ −2 +µ1(Σ) C x2dx, ∀u∈Cc∞(C) .
Z (cid:18)(cid:16) (cid:17) (cid:19)Z | |
Moreover the constant (n 2)2+µ (Σ) is sharp.
−2 1
Proof. Using polar decomposition we calculate
ZC |∇u|2dx = Z0∞rn−1ZΣ(cid:16)u2r + r12|∇ωu|2(cid:17)dSdr
≥ n−2 2 2 ∞rn−1 ur22dSdr+µ1(Σ) ∞rn−1 ur22dSdr
(cid:16) (cid:17) Z0 ZΣ Z0 ZΣ
n 2 2 u2
= − +µ (Σ) dx .
2 1 C x2
(cid:18)(cid:16) (cid:17) (cid:19)Z | |
The optimality of the constant is proved by considering the functions
u (x)=v (r)φ (ω)
ǫ ǫ 1
where v (r) are radial functions with compact support in (0,1) satisfying
ǫ
1
rn−1vε2,rdr n 2 2
Z0 − , as ǫ 0
1 v2 −→ 2 →
rn−1 εdr (cid:16) (cid:17)
r2
Z0
and φ is the first Dirichlet eigenfunction of the Laplace-Beltrami operator on Σ. (cid:3)
1
We next give the proof of Theorem 8.
Proof of Theorem 8. Let u C (C ) be given and let
∈ c∞ 1
∞
u(x)= u (r)φ (ω)
k k
k=1
X
10