Extremal Functions for Singular Moser-Trudinger Embeddings

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by  Stefano Iula

Extremal Functions for Singular Moser-Trudinger Embeddings
Stefano Iula Gabriele Mancini
∗ ∗ †
6 Universit¨at Basel Universit¨at Basel
1 [email protected] [email protected]
0
2

n
a
J Abstract

1
WestudyMoser-Trudingertypefunctionalsinthepresenceofsingularpotentials. Inpar-
2
ticular we propose a proof of a singular Carleson-Changtype estimate by means of Onofri’s

] inequality for the unit disk in R2. Moreover we extend the analysis of [1] and [8] consid-
P
ering Adimurthi-Druet type functionals on compact surfaces with conical singularities and
A
discussing the existence of extremals for such functionals.
.
h
t
a 1 Introduction
m
[ Let Ω R2 be a bounded domain, from the well known Sobolev’s inequality

⊆
1
v 1,p
u S u p (1,2), u W (Ω), (1)
6 k kL22−pp(Ω) ≤ pk∇ kLp(Ω) ∈ ∈ 0
6
6 one can deduce that the Sobolev space H1(Ω) := W1,2(Ω) is embedded into Lq(Ω) q 1. A
5 0 0 ∀ ≥
much more precise result was proved in 1967 by Trudinger [27]: on bounded subsets of H1(Ω)
0 0
. one has uniform exponential-type integrability. Specifically, there exists β > 0 such that
1
0
6 sup eβu2dx < + . (2)
1 ∞
: u∈H01(Ω), Ω|∇u|2dx≤1ZΩ
v R
i This inequality was later improved by Moser in [20], who proved that the sharp exponent in (2)
X
is β = 4π, that is
r
a sup e4πu2dx < + , (3)
∞
u∈H01(Ω), Ω|∇u|2dx≤1ZΩ
R
and
sup eβu2dx = + (4)
∞
u∈H01(Ω), Ω|∇u|2dx≤1ZΩ
R
for β > 4π. An interesting question consists in studying the existence of extremal functions
for (3). Indeed, while there is no function realizing equality in (1), one can prove that the
supremum in (3) is always attained. This was proved in [4] by Carleson and Chang for the unit
disk D R2, and by Flucher ([9]) for arbitrary bounded domains (see also [23] and [15]). The
⊆
∗The authors are supported by theSwiss National ScienceFoundation project nr. PP00P2-144669.
†The author is supported by SISSA and the PRIN project Variational and perturbative aspects of nonlinear
differential problems.
1
proof of these results is based on a concentration-compactness alternative stated by P. L. Lions
([16]): for a sequence u H1(Ω) such that u = 1 one has, up to subsequences, either
n ∈ 0 k∇ nkL2(Ω)
e4πu2ndx e4πu2dx
→
ZΩ ZΩ
where u is the weak limit of u , or u concentrates in a point x Ω, that is
n n
∈
u2dx ⇀ δ and u ⇀ 0. (5)
x n
|∇ |
The key step in [4] consists in proving that if a sequence of radially symmetric functions u
n
∈
H1(D) concentrates at 0, then
0
limsup e4πu2ndx π(1+e). (6)
n→∞ ZD ≤
Since for the unit disk the supremum in (3) is strictly greater than π(1+e), one can exclude
concentrationformaximizingsequencesbymeansof (6)andthereforeproveexistenceofextremal
functions for (3). In [9] Flucher observed that concentration at arbitrary points of a general
domain Ω can always be reduced, through properly defined rearrangements, to concentration
of radially symmetric functions on the unit disk. In particular he proved that if u H1(Ω)
n ∈ 0
satisfies u = 1 and (5), then
n 2
k∇ k
limsup e4πu2ndx πe1+4πAΩ(x)+ Ω . (7)
n→∞ ZΩ ≤ | |
where A (x) is the Robin function of Ω, that is the trace of the regular part of the Green
Ω
function of Ω. He also proved
sup e4πu2dx > πe1+4πmaxΩAΩ + Ω , (8)
| |
u∈H01(Ω), Ω|∇u|2dx≤1ZΩ
R
which implies the existence of extremals for (3) on Ω. Similar results hold if Ω is replaced by a
smooth closed surface (Σ,g). Let us denote
:= u H1(Σ) : u2dv 1, u dv = 0 .
g g
H ∈ |∇ | ≤
(cid:26) ZΣ ZΣ (cid:27)
Fontana [10] proved that
sup e4πu2dv < + (9)
g
∞
u∈HZΣ
and
sup eβu2dv =+ (10)
g
∞
u∈HZΣ
β > 4π. Existence of extremal functions for (9) was proved in [13] (see also [12], [14]), again
∀
by excluding concentration for maximizing sequences.
In this paper we are interested in Moser-Trudinger type inequalities in the presence of singular
potentials. Thesimplestexampleis given bythesingularmetric x 2α dx 2 onaboundeddomain
| | | |
Ω R2 containing the origin. In [2] Adimurthi and Sandeep observed that α ( 1,0],
⊂ ∀ ∈ −
sup x 2αe4π(1+α)u2dx < + , (11)
| | ∞
u∈H01(Ω), Ω|∇u|2dx≤1ZΩ
R
2
and
sup x 2αeβu2dx = + , (12)
| | ∞
u∈H01(Ω), Ω|∇u|2dx≤1ZΩ
R
for any β > 4π(1 + α). Existence of extremals for (11) has recently been proved in [7] and
[8]. The strategy is similar to the one used for the case α = 0. One can exclude concentration
for maximizing sequences using the following estimate, which can be obtained from (6) using a
simple change of variables (see [2], [8]).
Theorem 1.1. Let u H1(D) be such that u 2dx 1 and u ⇀ 0 in H1(D), then
n ∈ 0 D|∇ n| ≤ n 0
α ( 1,0] we have
∀ ∈ − R
π(1+e)
limsup x 2αe4π(1+α)u2ndx . (13)
n→∞ ZD| | ≤ 1+α
In the first part of this work we will give a simplified version of the argument in [4] and show
that (6) (and therefore (13)) can be deduced from Onofri’s inequality for the unit disk.
Proposition 1.1 (See [21], [3]). For any u H1(D) we have
∈ 0
1 1
log eudx u2dx+1. (14)
π ≤ 16π |∇ |
(cid:18) ZD (cid:19) ZD
Theorem 1.1 can be used to prove existence of extremals for several generalized versions of (3).
In [1] Adimurthi and Druet proved that
4πu2(1+λ u 2 )
sup e k kL2(Ω) dx < + (15)
∞
u∈H01(Ω), Ω|∇u|2dx≤1ZΩ
R
for any λ < λ(Ω), where λ(Ω) is the first eigenvalue of ∆ with respect to Dirichlet boundary
−
conditions. This bound on λ is sharp, that is
4πu2(1+λ(Ω) u 2 )
sup e k kL2(Ω) dx = . (16)
∞
u∈H01(Ω), Ω|∇u|2dx≤1ZΩ
R
Existence of extremal functions for sufficiently small λ for this improved inequality has been
proved in [17] and [28]. Similar results hold for compact surfaces on the space . We refer to
H
[25], [29] and references therein for further improved inequalities.
InthisworkwewillfocusonAdimurthi-Druettypeinequalities oncompactsurfaceswithconical
singularities. Given a smooth, closed Riemannian surface (Σ,g), and a finite number of points
p ,...,p Σ we will consider functionals of the form
1 m
∈
EΣβ,,λh,q(u) := heβu2(1+λkuk2Lq(Σ,g))dvg (17)
ZΣ
where λ,β 0, q > 1 and h C1(Σ p ,...,p ) is a positive function satisfying
1 m
≥ ∈ \{ }
h(x) d(x,p )2αi with α > 1 near p i = 1,...,m. (18)
i i i
≈ −
Oneofthemainmotivations forthechoiceofthesesingularweights comesindeedfromthestudy
of surfaces with conical singularities. We recall thatasmooth metric g on Σ p ,...,p is said
1 m
\{ }
3
to have conical singularities of order α ,...,α in p ,...,p if g = hg with g smooth metric on
1 m 1 m
Σ and 0 < h C (Σ p ,...,p ) satisfying (18). Thus the functional (17) naturally appears
∞ 1 m
∈ \{ }
in the analysis of Moser-Trudinger embeddings for the singular surface (Σ,g) (see [26]).
β,λ,q
If m = 0 and h 1, E corresponds to the functional studied in [17]. In particular, one has
≡ Σ,1
4π,λ,q
supE < + λ < λ (Σ,g), (19)
Σ,1 ∞ ⇐⇒ q
u
∈H
where
u2dv
λ (Σ,g) := inf Σ|∇ | g.
q u u 2
∈HRk kLq(Σ,g)
As it happens for (11), if h has singularities the critical exponent becomes smaller. More
precisely, in [26] Troyanov (see also [5]) proved that if h is a positive function satisfying (18),
then
β,0,q
supE < + β 4π(1+α) (20)
Σ,h ∞ ⇐⇒ ≤
u
∈H
where α= min 0, min α . In this paper we combine (19) and (20) obtaining the following
i
1 i m
(cid:26) ≤≤ (cid:27)
singular version of (19).
Theorem 1.2. Let (Σ,g) be a smooth, closed, surface. If h C1(Σ p ,...,p ) is a positive
1 m
∈ \{ }
function satisfying (18), then β [0,4π(1+α)] and λ [0,λ (Σ,g)) we have
q
∀ ∈ ∈
β,λ,q
supE (u) <+ , (21)
Σ,h ∞
u
∈H
and the supremum is attained if β < 4π(1+α) or if β = 4π(1+α) and λ is sufficiently small.
Moreover
β,λ,q
supE (u) = +
Σ,h ∞
u
∈H
for β > 4π(1+α), or β = 4π(1+α) and λ > λ (Σ,g).
q
Notethatwedonottreatthecaseβ = 4π(1+α)andλ = λ (Σ,g)(seeRemark5.1). InTheorem
q
1.2, it is possible to replace , λ (Σ,g) and , with , λ (Σ,g ) and
k·kLq(Σ,g) q H k·kLq(Σ,gh) q h
:= u H1(Σ) : u2dv 1, u dv = 0 ,
Hh ∈ 0 |∇gh | gh ≤ gh
(cid:26) ZΣ ZΣ (cid:27)
whereg := hg. InparticularwecanextendtheAdimurthi-Druetinequality tocompactsurfaces
h
with conical singularities.
Theorem 1.3. Let (Σ,g) be a closed surface with conical singularities of order α ,...,α > 1
1 m
−
in p ,...,p Σ. Then for any 0 λ < λ (Σ,g) we have
1 m q
∈ ≤
sup e4π(1+α)u2(1+λkuk2Lq(Σ,g))dvg < + ,
∞
u∈HZΣ
and the supremum is attained for β < 4π(1+α) or for β = 4π(1+α) and sufficiently small λ.
Moreover
sup eβu2(1+λkuk2Lq(Σ,g))dvg = + ,
∞
u∈HZΣ
if β > 4π(1+α) or β = 4π(1+α) and λ > λ (Σ,g).
q
4
As in [13], [29] and [17], our techniques can be adapted to treat the case of compact surfaces
with boundary.
Theorem1.4. Let(Σ,g)beasmooth compact Riemanniansurfacewithboundary. Ifp ,...,p
1 m
∈
Σ ∂Σ and h C1(Σ p ,...,p ) satisfies (18), then β [0,4π(1+α)] and λ [0,λ (Σ,g))
1 m q
\ ∈ \{ } ∀ ∈ ∈
we have
β,λ,q
sup E (u) < +
Σ,h ∞
u∈H01(Σ), Σ|∇u|2dvg≤1
R
and the supremum is attained if β < 4π(1+α) or if β = 4π(1+α) and λ is sufficiently small.
Furthermore if β > 4π(1+α), or β = 4π(1+α) and λ λ (Σ,g), we have
q
≥
β,λ,q
sup E (u) = + .
Σ,h ∞
u∈u∈H01(Σ), Σ|∇u|2dvg≤1
R
Inparticular, ifΣ = Ωistheclosureof aboundeddomaininR2, Theorem1.4gives thefollowing
generalization of the results in [9], [1], [8].
Corollary 1.1. Let Ω R2 be a bounded domain. For any choice of V C1(Ω), V > 0,
⊆ ∈
α ,...,α > 1, x ,...,x Ω, q > 1 and λ [0,λ (Ω)), the supremum
1 m 1 m q
− ∈ ∈
m
sup V(x) x xi 2αie4π(1+α)u2(cid:16)1+λkuk2Lq(Ω)(cid:17)dx
| − |
u∈H01(Ω), Ω|∇u|2dx≤1ZΩ Yi=1
R
is finite. Moreover it is attained if λ is sufficiently small.
This paper is organized as follows. Section 2 contains a simple proof of Theorem 1.1. Theorem
1.2 will be proved in the remaining three sections. In section 3 we will state some useful lemmas
β,λ,q
and prove existence of extremals for E with β < 4π(1+α). In Section 4 we will deal with
Σ,h
the blow-up analysis for maximizing sequences for the critical case β = 4π(1+α) and we will
prove an estimate similar to (7), which implies the finiteness of the supremum in (21). Finally,
in Section 5 we test the functionals with a properly defined family of functions and complete the
proof Theorem 1.2. In theAppendixwe will discuss some Onofri-type inequalities. In particular
we will show how to deduce (14) from the standard Onofri inquality on S2 and discuss its
extensions to singular disks. The proof of Theorems 1.3 and 1.4 is very similar to the one of
Theorem 1.2, hence it will not be discussed in this work.
2 A Carleson-Chang type estimate.
In this section we will prove Theorem 1.1 by means of (14). We will consider the space
H := u H1(D) : u2dx 1
∈ 0 |∇ | ≤
(cid:26) ZD (cid:27)
and, for any α ( 1,0], the functional
∈ −
E (u) := x 2αe4π(1+α)u2dx.
α
| |
ZD
By (11) we have sup E < + . For any δ > 0, we will denote with D the disk with radius δ
H α δ
∞
centered at 0.
5
Remark 2.1. With a trivial change of variables, one immediately gets that if δ > 0 and u
∈
H1(D ) are such that u 2dx 1, then
0 δ Dδ|∇ n| ≤
R
x 2αe4π(1+α)u2dx δ2(1+α)supE .
α
| | ≤
ZDδ H
In order to control the values of the Moser-Trudinger functional on a small scale, we will need
the following scaled version of (14) (cfr. Lemma 1 in [4]).
Corollary 2.1. For any δ,τ > 0 and c R we have
∈
ecudx πe1+1c26πτδ2
≤
ZDδ
for any u H1(D ) such that u2dx τ.
∈ 0 δ Dδ|∇ | ≤
As in the original proof in [4],Rwe will first assume α = 0 and work with radially symmetric
functions. For this reason we introduce the spaces
H1 (D) := u H1(D) : u is radially symmetric and decreasing .
0,rad ∈ 0
(cid:8) (cid:9)
and
H := H H1 (D).
rad ∩ 0,rad
Functions in H satisfy the following useful decay estimate.
rad
Lemma 2.1. For any u H we have
rad
∈
1
u(x)2 1 u2dy log x x D 0 .
≤ −2π − |∇ | | | ∀ ∈ \{ }
ZD|x| !
Proof. We bound
1
1 1 2
u(x) u′(t)dt tu′(t)2dt ( log x )12
| | ≤ | | ≤ − | |
Z|x| Z|x| !
1
1 u2dy 2 ( log x )12
≤ √2π ZD\D|x||∇ | ! − | |
1
1 1 u2dy 2 ( log x )12 .
≤ √2π −ZD|x||∇ | ! − | |
On a sufficiently small scale, it is possible to control E using only Corollary 2.1 Lemma 2.1 and
0
Remark 2.1.
6
Lemma 2.2. If u H and δ 0 satisfy
n rad n
∈ −→
u 2dx 0, (22)
n
|∇ | −→
ZDδn
then
limsup e4πu2ndx πe.
n→∞ ZDδn ≤
Proof. Take v := u u (δ ) H1(D ) and set τ := u 2dx. If τ = 0, then
n n − n n ∈ 0 δn n Dδn |∇ n| n
u u (δ ) in D and, using Lemma 2.1, we find
n ≡ n n δn R
e4πu2ndx =πδ2e4πun(δn)2 π < πe.
n
≤
ZDδn
Thus, w.l.o.g. we can assume τ > 0 for every n N. By Holder’s inequality and Remark 2.1
n
∈
we have
e4πu2ndx = e4πun(δn)2 e4πvn2+8πun(δn)vndx
ZDδn ZDδn
e4πun(δn)2 e4πvτnn2dx τn e8πu1n−(δτnn)vndx 1−τn (23)
≤
ZDδn ! ZDδn !
e4πun(δn)2 δn2supE0 τn e8πu1n−(δτnn)vndx 1−τn.
≤
(cid:18) H (cid:19) ZDδn !
Applying Corollary 2.1 with τ = τ ,δ = δ and c = 8πun(δn) we find
n n 1 τn
−
e8πu1n−(δτnn)vndx δn2πe1+4π(1u−nτ(nδn)2)2τn
≤
ZDδn
thus from (23) it follows
e4πu2ndx δn2 supE0 τn(πe)1−τne4πu2n(δn)+4πu(n1−(δτnn))2τn
≤
ZDδn (cid:18) H (cid:19)
= δn2 supE0 τn(πe)1−τne4π1u−nτ(nδ)2.
(cid:18) H (cid:19)
Lemma 2.1 yields
δn2e4πun1−(δτnn)2 1,
≤
therefore
τn
e4πu2ndx supE0 (πe)1−τn.
≤
ZDδn (cid:18) H (cid:19)
Since τ 0, we obtain the conclusion by taking the limsup as n on both sides.
n
−→ → ∞
7
In order to prove Theorem 1.1 on H for α = 0, it is sufficient to show that, if u ⇀ 0, there
rad n
exists a sequence δ satisfying the hypotheses of Lemma 2.2 and such that
n
e4πu2n 1 dx 0. (24)
− −→
ZDδn (cid:16) (cid:17)
Note that, by dominated convergence theorem, (24) holds if there exists f L1(D) such that
∈
e4πu2n f (25)
≤
in D D . In the next lemma we will chose a function f L1(D) with critical growth near 0
\ δn ∈
(i.e. f(x) 1 ) and define δ so that (25) is satisfied.
≈ x2log2 x n
| | | |
Lemma 2.3. Take u H such that
n rad
∈
sup u 0 r (0,1). (26)
n
−→ ∀ ∈
D Dr
\
Then there exists a sequence δ (0,1) such that
n
∈
1. δ 0.
n
−→
2. τ := u 2dx 0.
n Dδn |∇ n| −→
3. Re4πu2ndx π.
D\Dδn −→
R
Proof. We consider the function
1 x e 1
x2log2 x | | ≤ −
f(x):= | | | | (27)
( e2 x (e 1,1].
−
| | ∈
Note that f L1(D) and
∈
inf f = e2. (28)
(0,1)
Let us fix γ (0, 1) such that u 2dx 1. We define
n ∈ n Dγn |∇ n| ≤ n
R
δn := inf r (0,1) : e4πu2n(x) f(x) for r x 1 [0,1).
∈ ≤ ≤ | |≤ ∈
n o
and e
δ if δ > 0
n n
δ :=
n
( γn if δn = 0.
e e
By definition we have
e
e4πu2n ≤ f in D\Dδn,
thus 3 follows by dominated convergence Theorem. To conclude the proof it suffices to prove
that if n + is chosen so that δ = δ k, then
k ր ∞ nk nk ∀
klim δenk = klim τnk = 0. (29)
→∞ →∞
8
For such n one has
k
e4πunk(δnk)2 = f(δnk). (30)
In particular using (28) we obtain
e4πunk(δnk)2 = f(δnk) ≥ e2 > 1
k
which, by (26), yields δnk −→→∞0. Finally, Lemma 2.1 and (30) imply
1 ≥ δn2(k1−τnk)e4πunk(δnk)2 = lδon−gk22τδnk
nk
k
so that τnk −→→∞0 (otherwise the limit of the RHS would be +∞).
CombiningLemma2.2 withLemma2.3weimmediately get(6)forradially symmetricfunctions:
Proposition 2.1. Let u H and α ( 1,+ ]. If for any r (0,1)
n rad
∈ ∈ − ∞ ∈
sup u 0,
n
−→
D Dr
\
then
π(1+e)
limsupE (u ) .
α n
n ≤ (1+α)
→∞
Proof. If α= 0 theprooffollows directly applyingLemma2.3 and Lemma2.2. If α= 0 consider
6
1 1
vn(x) = (1+α)2un(x 1+α).
| |
We have
v 2dx = u 2dx
n n
|∇ | |∇ |
ZD ZD
and hence v H . Moreover we compute
n rad
∈
1
x 2αe(1+α)u2ndx = e4πvn2 dx
| | 1+α
ZD ZD
and the claim follows at once from the case α = 0.
To pass from Proposition 2.1 to Theorem 1.1 we will use symmetric rearrangements. We recall
thatgivenameasurablefunctionu: R2 [0,+ ),thesymmetricdecreasingrearrangementof
−→ ∞
u is the uniqueright-continuous radially symmetric and decreasing function u : R2 [0,+ )
∗
−→ ∞
such that
u > t = u > t t > 0.
∗
|{ }| |{ }| ∀
Among the properties of u we recall that
∗
1. If u Lp(R2), then u Lp(R2) and u = u .
∗ p p
∈ ∈ k ∗k k k
9
2. If u H1(D), then u H1 (D) and
∈ 0 ∗ ∈ 0,rad
u 2dx u2dx. (31)
∗
|∇ | ≤ |∇ |
ZD ZD
3. If u,v : R2 [0,+ ), then
−→ ∞
u (x)v (x)dx u(x)v(x)dx. (32)
∗ ∗
R2 ≥ R2
Z Z
In particular if u H1(D) and α 0,
∈ 0 ≤
x 2αeu∗dx x 2αeudx. (33)
| | ≥ | |
ZD ZD
Notethat(33)doesnotholdifα > 0. Wereferthereaderto[11]foramoredetailedintroduction
to symmetric rearrangements.
Proof of Theorem 1.1. Take u H such that u ⇀ 0 and let u be the symmetric decreasing
n n ∗n
∈
rearrangement of u . Then u H and since u = u 0 we have sup u 0
n ∗n ∈ rad k ∗nk2 k nk2 −→ D\Dr ∗n −→
r > 0. Thus from (33) and Proposition 2.1 we get
∀
π(1+e)
limsupE (u ) limsupE (u ) .
n α n ≤ n α ∗n ≤ 1+α
→∞ →∞
In the next section we will need the following local version of Theorem 1.1.
Corollary 2.2. Fix δ > 0, α ( 1,0] and take u H1(D ) such that u 2dx 1 and
∈ − n ∈ 0 δ Dδ|∇ n| ≤
u ⇀ 0 in H1(D ). For any choice of sequences δ 0, x Ω such that D (x ) D we
n 0 δ n → n ∈ R δn n ⊂ δ
have
πe
limsup x 2αe4π(1+α)u2ndx δ2(1+α).
n→∞ ZDδn(xn)| | ≤ 1+α
Proof. Letus defineu (x) := u (δx). Note thatu H and satisfies thehypotheses of Theorem
n n n
∈
1.1, thus
e e
limsup x 2α(e4πu2n 1)dx
n→∞ ZDδ| | −
= δ2(1+α)limsup x 2α(e4πu2n 1)dx
nπ→e∞ ZD| | e −
δ2(1+α) .
≤ 1+α
Thus we get
limsup x 2αe4π(1+α)u2ndx
n→∞ ZDδn(xn)| |
= limsup x 2α e4π(1+α)u2n 1 dx
n→∞ ZDδn(xn)| | (cid:16) − (cid:17)
x 2α(e4πu2n 1)dx
≤ | | −
ZDδ
πe
δ2(1+α) .
≤ 1+α
10