Table Of ContentAN OPTIMAL MASS TRANSPORT APPROACH FOR
LIMITS OF EIGENVALUE PROBLEMS FOR THE
FRACTIONAL p-LAPLACIAN.
5
1
0
L. M. DEL PEZZO, J. D. ROSSI,N.SAINTIER,ANDA. SALORT
2
n
a Abstract. Wefindinterpretationusingoptimalmasstransporttheory
J
foreigenvalueproblemsobtainedaslimitsoftheeigenvalueproblemsfor
8 the fractional p−Laplacian operators as p → +∞. We deal both with
2 Dirichlet and Neumann boundaryconditions.
]
P
A
.
h 1. Introduction
t
a
Our main goal in this paper is to use tools from mass transport theory
m
to study eigenvalue problems that are obtained taking limits as p → +∞
[
in eigenvalue problems that involve fractional spaces Ws,p (with 0 < s < 1
1 and 1 < p ≤ +∞). We deal both with Dirichlet and Neumann boundary
v
conditions.
5
5 AlongthispaperwetakeU asmoothboundeddomaininRn,1 < p < +∞
1 and0 < s< 1. Wealsofixd(·,·)adistanceinRn equivalenttotheEuclidean
7
one.
0
.
1 Let λD be the first eigenvalue of the fractional p-Laplacian of order s in
0 s,p
U with Dirichlet boundary conditions, that is, let us consider
5
1
: λD := inf [u]p : u∈ Ws,p(U), |u|pdx = 1
v s,p s,p
i (cid:26) ZU (cid:27)
X
where f
r |u(x)−u(y)|p
a [u]p := dxdy
s,p Rn Rn |x−y|n+sp
Z Z
is the semi-norm of Ws,p(Rn) and
Ws,p(U) := {u ∈Ws,p(Rn) : u =0 in Rn\U}.
For this problem Lindgren and Lindqvist in [18] proved that
f
ΛD := lim λD 1/p = 1
s,∞ p→+∞ s,p Rs
(cid:0) (cid:1)
Date: January 29, 2015.
Key words and phrases. Fractional p−Laplacian, eigenvalues, mass transport.
2010 Mathematics Subject Classification: 35J92, 35P30, 45C05.
1
2 L.M.DELPEZZO,J.D.ROSSI,N.SAINTIER,ANDA.SALORT
where
R := maxdist(x,∂U) = max min |x−y|.
x∈U x∈U y∈∂U
Moreover, via a subsequence, the eigenfunctions u suitable normalized
p
(a minimizer for λD ) converge uniformly to a minimizer for ΛD .
s,p s,∞
Our first purpose in this work is to relate ΛD to an optimal mass trans-
s,∞
port problem with cost function c(x,y) = |x−y|s. We prove the following
result:
Theorem 1.1. There holds that
1
= sup W (µ,P(∂U)),
ΛD s
s,∞ µ∈P(U)
where
W (µ,ν) := inf |x−y|sdπ(x,y).
s
π
ZU ZU
Here P(A) is the set of probability measures on A and π ∈ P(U ×U) is a
measure with marginals µ and ν.
Note that W (µ,ν) is the total cost when we have to transport the mea-
s
sure µ onto ν using as cost for transporting one unit of mass from position
x to position y the Euclidean distance to the power s, that is |x−y|s. We
referto [20] andto Section 2for precisedefinitions, notations and properties
of optimal mass transporttheory. Hence, our resultsays that theeigenvalue
ΛD is related to the problem of finding a probability measure supported
s,∞
inside U, µ, that is far (in terms of the transport cost) from the set of prob-
ability measures supported on the boundary, ∂U. One easy solution to this
problem is the following: take B (x ) a ball with maximum radius R inside
R 0
U and let y ∈ ∂U∩∂B (x ) (there exists such y due to the maximality of
0 R 0 0
R). Then, µ = δ (with ν = δ ) solves sup W (µ,P(∂U)). Observe
x0 y0 µ∈P(U) s
that from Theorem 1.1 we can recover that ΛDs,∞ = 1/Rs.
Now, let usturnourattention tothecase of thefirstnontrivialeigenvalue
for Neumann boundary conditions, i.e., let us consider
λN := inf JuKp : u∈ C ,
s,p s,p
where (cid:8) (cid:9)
|u(x)−u(y)|p
(1.1) JuKp := dxdy
s,p d(x,y)n+sp
ZU ZU
and
C := u∈ Ws,p(U): kuk = 1, |u|p−2udx = 0 .
Lp(U)
(cid:26) ZU (cid:27)
For this problem, in the case d(x,y) = |x−y|, Del Pezzo and Salort in [8]
proved that
ΛN := lim λN 1/p = 2
s,∞ p→+∞ s,p (diam(U))s
(cid:0) (cid:1)
OPTIMAL MASS TRANSPORT FOR EIGENVALUE PROBLEMS. 3
where diam(U) is the extrinsic diameter, i.e.
diam(U) := max |x−y|.
x,y∈U
p
Their proof actually extends to the case in which we consider JuK with
s,p
d(x,y) any distance as above (for instance, for the geodesic distance in U).
In this case, it holds that
(1.2) ΛN = lim λN 1/p = 2
s,∞ p→+∞ s,p (diamd(U))s
(cid:0) (cid:1)
where diam (U) is the diameter of U according to d, that is
d
diam (U) = max d(x,y).
d
x,y∈U
Moreover, as happens for the Dirichlet problem, via a subsequence the nor-
malized eigenfunctions u (a minimizer for λN ) converge uniformly to a
p s,p
minimizer for ΛN .
s,∞
In order to introduce the mass transport interpretation we need the fol-
lowing notations. We denote by M(U) the space of finite Borel measures
over U. Given σ ∈ M(U), we denote its positive and negative part by σ+
and σ− so that σ = σ+−σ−, and |σ| = σ++σ−. Then we have,
Theorem 1.2. There holds
2
(1.3) = max W (σ+,σ−): σ ∈ M(U), σ+(U) = σ−(U) = 1
ΛN s
s,∞
(cid:8) (cid:9)
where W is as in Theorem 1.1.
s
Here we relate ΛN to the problem of finding two probability measures,
s,∞
σ+ and σ−, supported in U, such that the cost of transporting one into the
other is maximized. To obtain a solution to this problem one can argue as
follows: take two points x and y in U that realize the diameter, that is, we
0 0
have d(x ,y ) = diam (U). Then take σ+ = δ and σ− = δ as a solution
0 0 d x0 y0
to max W (σ+,σ−): σ ∈M(U), σ+(U) = σ−(U) = 1 . Note that we can
s
recover (1.2) from Theorem 1.2.
(cid:8) (cid:9)
A different concept of Neumann boundary condition for fractional opera-
tors was recently introduced in [9]. More precisely, for (−∆)s the fractional
p
p−Laplacian given by
|u(x)−u(y)|p−2(u(x)−u(y))
(−∆)su(x) = P.V. dy
p Rn d(x,y)n+sp
Z
(the symbol P.V. stands for the principal value of the integral), we consider
the following non-local non-linear fractional normal derivative
|u(x)−u(y)|p−2(u(x)−u(y))
N u(x) = dy x ∈ Rn\U.
s,p d(x,y)n+sp
ZU
4 L.M.DELPEZZO,J.D.ROSSI,N.SAINTIER,ANDA.SALORT
Associatedwiththisoperator,weconsiderthefollowingeigenvalueproblems
(−∆)su= λ|u|p−2u in U,
(1.4) p
(Ns,pu= 0 in Rn\U.
Before stating our main result concerning these problems, we need to
introduce some notations. Let Ws,p(U) be the set of measurable functions
with finite
kukp := kukp +H (u),
Ws,p(U) Lp(U) s,p
where
|u(x)−u(y)|p
H (u) := dxdy,
s,p ZZR2n\(Uc)2 d(x,y)n+ps
and (Uc)2 = Uc×Uc. Let us also introduce
|u(x)−u(y)|
H (u) :=sup : (x,y) ∈ R2n\(Uc)2 .
s,∞ d(x,y)s
(cid:26) (cid:27)
Then, for (1.4) we have the following result.
Theorem 1.3. The first non-zero eigenvalue of (1.4) is given by
H (v)
λ = inf s,p : v ∈ Ws,p(U)\{0}, |v|p−2vdx = 0 .
s,p p
2kvk
( Lp(U) ZU )
Concerning the limit as p → +∞ of these eigenvalues we have
2 H (u)
lim (λ )1/p = = Λ := inf s,∞ : u ∈ A ,
p→+∞ s,p (diamd(U))s s,∞ (cid:26)kukL∞(U) (cid:27)
where
A:= v ∈ Ws,∞(U)\{0}: supu(x)+ inf u(x) = 0 .
(cid:26) x∈U x∈U (cid:27)
Moreover, if u is a minimizer of λ normalized by ku k = 1, then, up
p s,p p Lp(U)
to a subsequence, u converges in C(U) to some minimizer u ∈ Ws,∞(U)
p ∞
of ΛN .
s,∞
Note that, since the limit of (λ )1/p, Λ , coincides with ΛN (given in
s,p s,∞ s,∞
(1.2)), we get the same interpretation in terms of optimal mass transporta-
tion given in Theorem 1.2.
To end this introduction, let us briefly comment on previous results. The
limit as p → +∞ of the first eigenvalue λD of the usual local p-Laplacian
p
with Dirichlet boundary condition was studied in [15, 16], (see also [3] for
an anisotropic version). In those papers the authors prove that
λD := lim λD 1/p = inf k∇vkL∞(U): v ∈W1,∞(Ω) = 1,
∞ p→+∞ p (cid:26) kvkL∞(U) 0 (cid:27) R
(cid:0) (cid:1)
whereR is, as before, the largest possible radius of a ball contained in U. In
addition, the authors show the existence of extremals, i.e. functions where
OPTIMAL MASS TRANSPORT FOR EIGENVALUE PROBLEMS. 5
the above infimum is attained. These extremals can be constructed taking
the limit as p → +∞ in the eigenfunctions of the p−Laplacian eigenvalue
problems (see [15]) and are viscosity solutions of the following eigenvalue
problem (called the infinity eigenvalue problem in the literature):
min |Du|−λDu, ∆ u = 0 in U,
∞ ∞
(u =(cid:8)0 (cid:9) on ∂U.
The limit operator ∆ that appears here is the ∞-Laplacian given by
∞
∆ u = −hD2uDu,Dui. Remark that solutions to ∆ v = 0 with a Dirich-
∞ p p
let data v = f on ∂U converge as p → +∞ to the viscosity solution to
p
∆ v = 0 with v = f on ∂U, see [2, 4, 6]. This operator appears naturally
∞
when one considers absolutely minimizing Lipschitz extensions in U of a
boundary data f, see [1, 2, 13].
Recently in [5], the authors relate λD with the Monge-Kantorovich dis-
∞
tance W . Recall that the Monge-Kantorovich distance W (µ,ν) between
1 1
two probability measures µ and ν over U is defined by
(1.5) W1(µ,ν):= max v(dµ−dν): v ∈ W1,∞(U), k∇vkL∞(U) ≤ 1 .
(cid:26)ZU (cid:27)
It was proved in [5] that
1
= sup W (µ,P(∂U)).
λD 1
∞ µ∈P(U)
Notice that this result is the analogous to Theorem 1.1 in the local case.
For the Neumann problem for the local p−Laplacian we refer to [10, 19]
where the authors prove the local analogous to Theorem 1.2. In this local
case the distance that appears in the limit is the geodesic distance inside
U. This is, in contrast with the non-local case studied here, where we can
consider any distance d equivalent to the Euclidean one, see (1.1).
For limits as p → +∞ in non-local p−Laplacian problems and its relation
with optimal mass transport we refer to [14]. Eigenvalue problems were not
considered there.
The case of a Steklov boundary condition has also been investigated re-
cently. Indeed, the authors in [12] (see also [17] for a slightly different prob-
lem)studiedthebehaviorasp → +∞oftheso-called variationaleigenvalues
λS , k ≥ 1, of the p-Laplacian with a Steklov boundary condition. In par-
k,p
ticular they proved that
lim λS 1/p = 1 and λS := lim λS 1/p = 2 ,
p→+∞ 1,p 2,∞ p→+∞ 2,p diam(U)
(cid:0) (cid:1) (cid:0) (cid:1)
and also identify the limit variational problem defining λS .
2,∞
Thepaperisorganizedasfollows: inSection2wecollectsomepreliminary
results concerning optimal mass transport with cost d(x,y)s, in particular,
weprovideastatement oftheKantorovich duality resultthatwillbeusedin
6 L.M.DELPEZZO,J.D.ROSSI,N.SAINTIER,ANDA.SALORT
theproofsofourresults; inSection 3wedealwiththeDirichlet problemand
prove Theorem 1.1; in Section 4 we study the Neumann case (Theorem 1.2).
Finally, in Section 5 we deal with problem (1.4) and we prove Theorem 1.3.
2. Kantorovich duality for the cost c(x,y) = d(x,y)s
In this section we follow [20]. We first recall the definition of c-concavity
and c-transform.
Definition 2.1 ([20, Definitions 5.2 and 5.7]). Let X,Y be two sets and
c : X × Y → R ∪ {+∞}. A function ψ : X → R ∪ {+∞} is said to be
c-convex if ψ 6≡ +∞ and there exists φ: Y → R∪{±∞} such that
(2.1) ψ(x) = supφ(y)−c(x,y) for all x ∈ X.
y∈Y
Its c-transform is the function ψc defined by
ψc(y) = inf ψ(x)+c(x,y) for all y ∈ Y.
x∈X
A function φ: Y → R∪{−∞} is c-concave if φ 6≡ −∞ and φ= ψc for some
function ψ : X → R∪{±∞}. Then its c-transform φc is
φc(x) = supφ(y)−c(x,y) for all x ∈X.
y∈Y
There holds:
Proposition 2.2 ([20, Proposition 5.8]). For any ψ : X → R ∪ {+∞},
ψc = ψccc and ψ is c-convex iff ψ = ψcc.
In the case where the cost function is c(x,y) = d(x,y)s, we have the
following characterization of c-convex function.
Lemma 2.3. Let c(x,y) = d(x,y)s and X =Y = U. Any c-convex function
ψ satisfies ψc = ψ and
(2.2) |ψ(x)−ψ(x˜)| ≤ d(x,x˜)s for all x,x˜ ∈ U.
Proof. Notice that
ψc(y)= inf ψ(x)+d(x,y)s ≤ ψ(y)
x∈U
and that the opposite inequality holds if (2.2) holds. We now verify (2.2).
Let φ : U → R∪{±∞} such that ψ = φc as in (2.1). Since s ∈ (0,1), we
have d(x,y)s ≤ d(x,x˜)s +d(y,x˜)s for any x,x˜,y ∈U. It follows that
ψ(x) = φc(x) = supφ(y)−d(x,y)s
y∈U
≥ supφ(y)−d(y,x˜)s −d(x,x˜)s = φ(x˜)−d(x,x˜)s,
y∈U
i.e. φ(x˜)−φ(x) ≤ d(x,x˜)s. The opposite inequality holds as well by switch-
ing x and x˜. Thus (2.2) holds. (cid:3)
OPTIMAL MASS TRANSPORT FOR EIGENVALUE PROBLEMS. 7
We recall the following result, see [20, Theorem 5.9].
Theorem 2.4. Let (X,µ) and (Y,ν) be two Polish probability spaces (i.e.
metric complete separable) and let c : X × Y → R ∪ {+∞} be a lower
semicontinuous function such that
c(x,y) ≥ a(x)+b(y) for all (x,y) ∈ X ×Y
for some real-valued upper semicontinuous functions a ∈ L1(µ) and b ∈
L1(ν). Then, letting J(φ,ψ) := φdν − ψdµ,
Y X
R R
W (µ,ν) := min c(x,y)dπ(x,y) = sup J(φ,ψ)
c
π∈Π(µ,ν)ZX×Y (ψ,φ)∈L1(µ)×L1(ν),φ−ψ≤c
= sup J(ψc,ψ)
ψ∈L1(µ)
and in the above sup, one might as well impose ψ to be c-convex. Moreover
if c is real-valued, W (µ,ν) < ∞ and
c
c(x,y) ≤ c (x)+c (y) for all (x,y) ∈ X ×Y
X Y
for some c ∈ L1(ν) and c ∈ L1(µ), then the above sup is a max and one
X Y
might as well impose ψ to be c-convex.
In the particular case c(x,y) = d(x,y)s, X =Y = U with U bounded, we
obtain in view of Lemma 2.3 the following result.
Theorem 2.5. For any µ,ν ∈ P(U),
min d(x,y)sdπ(x,y) = max ψdν − ψdµ.
π∈Π(µ,ν)ZU×U |ψ(x)−ψ(y)|≤d(x,y)sZU ZU
Proof. In view of Lemma 2.3 and the previous theorem, we can write that
W (µ,ν) = max J(ψc,ψ)
c
ψ∈L1(µ)c-convex
≤ max J(ψ,ψ)
|ψ(x)−ψ(y)|≤d(x,y)s
≤ sup J(φ,ψ)
(ψ,φ)∈L1(µ)×L1(ν),φ−ψ≤d(x,y)s
= W (µ,ν)
c
form which we deduce the result. (cid:3)
3. The Dirichlet case
In this section, we borrow ideas from [5]. Let us consider
G ,G : C(U)×M(U) → R∪{+∞}
p ∞
8 L.M.DELPEZZO,J.D.ROSSI,N.SAINTIER,ANDA.SALORT
the functionals given by
− vσdx, if σ ∈ Lp′(U), kσk ′ ≤ 1,
Lp (U)
Gp(v,σ) = ZU and v ∈ Ws,p(U), [v] ≤ (λD )1/p,
s,p s,p
+∞ otherwise,
f
and
− vdσ, if σ ∈M(U), |σ|(U) ≤1,
G∞(v,σ) = ZU and v ∈ Ws,∞(U), |v(x)−v(y)| ≤ ΛD |x−y|s,
s,∞
+∞ otherwise.
f
In the space M(U), we consider the weak convergence of measures, and
in the space C(U) the uniform convergence.
First, we have that G is the limit as p → +∞ of G in the Γ−limit
∞ p
sense (we refer to [7] for the definition of Γ−convergence).
Lemma 3.1. The functionals G Γ−converge as p → +∞ to G .
p ∞
Proof. It follows as in [5]. (cid:3)
Now, we let f : Rn → R defined as
p
f (x) := (u (x))p−1,
p p
where u is a nonnegative eigenfunction associated to λD (U) such that
p s,p
ku k = 1. When we consider f as an element of M(U) together with
p Lp(U) p
u we obtain a minimizer for G . The proof of this fact is immediate.
p p
Lemma 3.2. The pair (f ,u ) minimizes G in C(U)×M(U) with
p p p
G (f ,u ) = −1.
p p p
Now, let us show that we can extract a subsequence p → +∞ such that
n
f and u converge.
p p
Lemma 3.3. There exists a sequence p → +∞ such that
n
u → u
pn ∞
uniformly in Rn. This limit u verifies
∞
|u (x)−u (y)| ≤ ΛD |x−y|s, x,y ∈ Rn.
∞ ∞ s,∞
Moreover, we have
∗
f ⇀ f
pn ∞
weakly-* in M(U) and f is a nonnegative measure that verifies f (U) ≤ 1.
∞ ∞
OPTIMAL MASS TRANSPORT FOR EIGENVALUE PROBLEMS. 9
Proof. The convergence of u , via a subsequence, is contained in [18]. Con-
p
cerning f the conclusion follows from the inequality
pn
p−1
(3.1) f dx ≤ (u )pdx p |U|1/p = |U|1/p,
p p
ZU (cid:18)ZU (cid:19)
thatimpliesthatf isboundedinM(U)andhencewecanextractasequence
p
∗
p → +∞ such that f ⇀ f weakly-* in M(U). That the limit f is a
n pn ∞ ∞
nonnegative measure that verifies f (U) ≤ 1 also follows from (3.1). (cid:3)
∞
From the main property of Γ−convergence we obtain the following corol-
lary.
Corollary 3.4. The pair (f ,u ) minimizes G with
∞ ∞ ∞
G (f ,u ) = −1.
∞ ∞ ∞
Now we are ready to prove Theorem 1.1.
Proof of Theorem 1.1. As (f ,u ) minimizes G we obtain that
∞ ∞ ∞
u
∞
f ,
∞ ΛD
∞,s
(cid:16) (cid:17)
minimizes
− vdσ,
ZU
with (v,σ) belonging to
A := (v,σ) ∈ Ws,∞(U)×M(U): |σ|(U) ≤ 1,|v(x)−v(y)| ≤ d(x,y)s .
n o
Then
f
1 1
= u df
ΛD ΛD ∞ ∞
s,∞ s,∞ ZU
= max vdσ
(v,σ)∈AZU
= max max wdµ
µ∈P(U)|w(x)−w(y)|≤d(x,y)sZU
= max W (µ,P(∂U)),
s
µ∈P(U)
as we wanted to show. (cid:3)
4. The Neumann case
Again, we follow ideas from [5], see also [19]. Let u be an extremal for
p
λN (that is, a minimizer for (1.1)) normalized by ku k = 1. Then
p,s p Lp(U)
f := |u |p−2u ∈ Lp′(U) (where p′ = p ) satisfies
p p p p−1
(4.1) kfpkLp′(U) = 1 and fpdx = 0.
ZU
10 L.M.DELPEZZO,J.D.ROSSI,N.SAINTIER,ANDA.SALORT
The first step consists in extracting from {f } a subsequence converging
p p>1
weakly to some measure f ∈ M(U), the weak convergence meaning here
∞
that
lim φf dx = φdf
p ∞
p→+∞
ZU ZU
for any φ ∈ C(U).
Lemma 4.1. Up to a subsequence, the measures f converge weakly in mea-
p
sure in U to some measure f supported in U satisfying
∞
(4.2) f (U)= 0 and |f |(U)= 1.
∞ ∞
Proof. We claim that
(4.3) lim |f |dx =1.
p
p→+∞
ZU
First, in view of (4.1), we have that
|fp|dx ≤ kfpkLp′(U)|U|1−1/p′ = |U|1−1/p′ → 1 as p → +∞
ZU
and then, recalling that up → u in C(U) with kukL∞(U) = 1,
1 = upfpdx ≤ kupkL∞(U)kfpkL1(U) = (1+o(1))kfpkL1(U).
ZU
It follows in particular that the measures |f | are bounded in M(U) inde-
p
pendently of p. Since U is compact, we can then extract from this sequence
a subsequence converging weakly to some measure f ∈ M(U). Passing to
∞
the limit in (4.1) and (4.3) gives (4.2). (cid:3)
Consider the functionals G ,G : C(U)×M(U) → R∪{+∞} defined by
p ∞
− vσdx if σ ∈ Lp′(U), kσk ′ ≤ 1, σdx = 0,
Lp (U) U
Gp(v,σ) = ZU and v ∈Ws,p(U), JvK ≤ (λNR )1/p,
s,p p,s
+∞ otherwise,
and
− vdσ, if σ ∈ M(U), |σ|(U) ≤ 1, σ(U)= 0,
G∞(v,σ) = ZU and v ∈ Ws,∞(U), |v(x)−v(y)| ≤ ΛN d(x,y)s,
∞,s
+∞ otherwise.
Remark that these functionals are similar to the ones considered for the
Dirichlet case but the spaces involved change. In fact, here we consider
Ws,p(U) instead of Ws,p(U) (that encodes the fact that we are considering
functions that vanish outside U when dealing with the Dirichlet problem).
As for the Dirichlfet case, we can prove as in [5, 19] that G is the limit
∞
of the G in the sense of Γ-convergence:
p
