BASIS PROPERTIES OF EIGENFUNCTIONS OF THE
p-LAPLACIAN
Paul Binding1 , Lyonell Boulton2
Department of Mathematics and Statistics, University of Calgary
Calgary, Alberta, Canada T2N 1N4
Jan Čepička3 , Pavel Drábek3 , Petr Girg3
Department of Mathematics, University of West Bohemia
Pilsen, Czech Republic.
Abstract. For p > 12
11 , the eigenfunctions of the non-linear eigenvalue problem for the p-Laplacian on the interval (0, 1) are shown
to form a Riesz basis of L2 (0, 1) and a Schauder basis of Lq (0, 1)
whenever 1 < q < ∞.
Date: 1st May 2004.
2000 Mathematics Subject Classification. Primary: 34L30; Secondary: 34L10,
42A65.
Key words and phrases. p-Laplacian eigenvalues, eigenfunction completeness.
1
Research supported by I. W. Killam Foundation and NSERC of Canada.
2
PIMS Postdoctoral Fellow.
3
Research supported by GAČR, no. 201/03/0671.
1
2
1. Introduction
Generalized sine functions Sp , 1 < p < ∞, were studied by Elbert [6]
as Dirichlet eigenfunctions for the one-dimensional p-Laplacian equation
(1)
−([y 0 (t)]p−1 )0 = (p − 1)λ[y(t)]p−1
on 0 6 t 6 πp . Here, by definition, [z]p−1 = z|z|p−2 , and
2π
(2)
πp :=
p sin(π/p)
is the first zero of Sp , which is the eigenfunction (normalized by Sp0 (0) =
1) corresponding to the minimal Dirichlet eigenvalue λ = 1 of (1).
Elbert also deduced the relation
(3)
|Sp |p + |Sp0 |p = 1
and considered the higher order eigenfunctions, which are given by
Sp (nt) with corresponding eigenvalues λ = np , n = 2, 3, . . .. When
p = 2, (1) corresponds to Fourier’s equation, S2 (t) = sin(t), π2 = π
and (3) is the Pythagorean relation.
In [15], Ôtani examined analogous functions sinp with the factor
(p − 1) on the right side of (1) replaced by 1. Lindqvist [12], [13]
has studied the sinp functions in some detail, noting that hyperbolic
versions were introduced as early as 1879. We remark that the Sp
and sinp functions have become standard tools in the analysis of more
complicated equations with various applications, cf. e.g. [1], [2], [3],
[4], [5], [6], [7], [8], [14] and [16].
Despite this activity, it seems that analogues for 1 < p 6= 2 of the
standard completeness and expansion theorems for sine functions have
not been discussed previously. While the Sp and sinp functions can
easily be transformed into each other, it turns out that (3) must be
modified when Sp is replaced by sinp . Since relation (3) simplifies our
arguments below, we shall work with Sp normalized to the interval
[0, 1]. To this end, we define
fn (t) := Sp (nπp t),
n ∈ N,
which comprise eigenfunctions of the nonlinear eigenvalue problem (1)
on (0, 1) with y(0) = y(1) = 0, corresponding to eigenvalues λn =
(nπp )p , n = 1, 2, . . . . They admit characterizations, e.g., via a modification of Prüfer’s transformation [6] and via Ljusternik-Schnirelmann
variational characterization [2].
The functions fn depend on p, and in the case p = 2 they become
en (t) := sin(nπt),
n ∈ N,
3
which are proportional to a standard orthonormal basis of the Hilbert
space L2 (0, 1) and form a Schauder basis of the Banach space Lq (0, 1)
for all q > 1 (see the proof of Theorem 6 below).
Recall that {gn } is a Schauder basis [17, p. 152] of Lq (0, 1), 1 < q <
∞, if for any g ∈ Lq (0, 1), there exist unique coefficients cn , depending
continuously on g, so that
N
X
cn gn − g → 0
q
n=1
as N → ∞. For q = 2, such a basis is a Riesz basis [10, p. 74] if the `2
norm of the above coefficients cn generates a norm equivalent to that
in L2 (0, 1).
The aim of this note is to establish the following
Theorem 1. For 12
6 p < ∞, the family {fn }∞
n=1 forms a Riesz basis
11
of L2 (0, 1) and a Schauder basis of Lq (0, 1) whenever 1 < q < ∞.
Our main device is a linear mapping T of the space Lq (0, 1), satisfying T en = fn , n ∈ N, and decomposing into a linear combination of
certain isometries. In Section 3 we show that T is a bounded operator
for 1 < p < ∞. In Section 4 we prove boundedness of the inverse for
the range 12
6 p < ∞.
11
We should point out that the bound 12
for p is certainly not opti11
mal, although it is convenient for explicit calculations. We shall make
some remarks at the end about this bound, but we note here that our
methods will not reduce it to 1.
2. Preliminaries
In this section we describe some elementary properties of πp and Sp ,
which will be employed in the sequel. We start with
Lemma 2. The number πp in (2) is strictly decreasing in p > 1.
Indeed, a simple computation gives
dπp
<0
dp
whenever p > 1. Notice that πp % ∞ as p & 1 and πp & 2 as p % ∞.
We shall also need the specific value
√
11π 2
.
(4)
π12/11 = √
3( 3 − 1)
4
Let x be an arbitrary point in the interval (0, πp /2). By virtue of
(3), 0 6 Sp (x) 6 1 and
Sp0 (x) = (1 − Sp (x)p )1/p > 0.
(5)
Furthermore, by construction, Sp (0) = 0 and Sp (πp /2) = 1.
Lemma 3. Let p > 1. Then Sp is strictly concave on (0, πp /2), and,
when p ≤ 2,
|Sp00 (x)| 6 1.
(6)
Proof. By differentiating (5), one readily sees that
Sp00 (x) = −h(Sp (x))
(7)
where h(y) := y p−1 (1 − y p )
that |h(y)| ≤ 1.
2−p
p
> 0 for y ∈ (0, 1). When p ≤ 2 it is clear
Remark 4. The above results apply to f1 : t 7→ Sp (πp t) on [0, 12 ]. We
extend f1 symmetrically across 12 by f1 (1 − t) = f1 (t). When p ≤ 2,
it is easily seen that Lemma 3 extends to this situation, and so f100 is
continuous on (0, 1). When p > 2, f100 ( 12 ) does not exist, but
Z
(8)
0
1/2
|f100 (t)| dt
=
−πp2
Z
1/2
h
i1/2
Sp00 (πp t) dt = −πp S 0 (πp t)
= πp ,
t=0
0
and so f100 ∈ L1 (0, 1).
For any function f on [0, 1], we define its successive antiperiodic
extension f ∗ over R+ by f ∗ = f on [0, 1], and
(9)
f ∗ (t) = −f ∗ (2k − t) if k ≤ t ≤ k + 1,
k ∈ N.
Then we extend f1 over R+ via f1∗ , in agreement with [6].
Finally we need a monotonicity property of this function.
Lemma 5. For each t ∈ (0, 1), f1 (t) = Sp (πp t) is strictly decreasing in
p > 1.
Ry
Proof. From (3), f1−1 (y) = (πp )−1 0 (1 − τ p )−1/p dτ , for 0 6 y 6 1.
Then the proof follows from the fact that (πp )−1 and (1 − τ p )−1/p are
both positive and strictly increasing in p > 1 for fixed τ ∈ (0, 1).
For the reader’s convenience, we depict typical profiles of Sp (πp t),
together with the limiting cases p = 1 and p = ∞, in Fig. 1.
5
Sp (πp t)
S1 (t)
0
p → 1+
1
0
p<2
Sp (πp t)
1
0
S∞ (t)
1
p>2
0
1
p→∞
Figure 1
3. The operator T
We start by justifying the route we shall take in proving Theorem 1.
Theorem 6. If there is a linear homeomorphism T of Lq (0, 1) satisfying T en = fn , n ∈ N for 12
6 p < ∞ and 1 < q < ∞, then Theorem 1
11
holds.
Proof. By [17, Example 11.1, pp.342-345], any function in Lq (−1, 1)
has a unique sine-cosine series representation. For any f ∈ Lq (0, 1),
we can thus represent its odd extension to Lq (−1, 1) uniquely in a sine
series, so the en form a basis of Lq (0, 1). According to [10, p. 75] the
same is true for the fn , and so by [17, Theorem 3.1, p.20] they form a
Schauder basis of Lq (0, 1). The argument for a Riesz basis when q = 2
is similar and follows, e.g., [9, Section VI.2].
In the remainder of this section we define T as a linear combination
of certain isometries of Lq (0, 1). Then we show that T is a bounded
operator satisfying T en = fn , n ∈ N for all p, q > 1.
As in the above proof, the fn (which belong to Lq (0, 1)) have Fourier
sine series expansions
Z 1
∞
X
b
b
fn (t) =
fn (k) sin(kπt), where fn (k) := 2
fn (t) sin(kπt) dt.
0
k=1
An argument involving symmetry with respect to the midpoint x =
easily shows that fb1 (k) = 0 whenever k is even. Furthermore
Z 1
b
fn (k) = 2
f1 (nt) sin(kπt) dt
0
Z 1
X
=2
fb1 (k)
sin(mπnt) sin(kπt) dt
(10)
m−odd
=
0
fb1 (m) if mn = k for some m odd,
0
otherwise.
1
2
6
In what follows we will often denote fb1 (j) by τj . We first find a bound
on |τj | which will be crucial in the definition of T below. Integration
by parts ensures that
Z 1
Z 1/2
τj = 2
f1 (t) sin(jπt) dt = 4
f1 (t) sin(jπt) dt
0
0
(11)
Z 1/2
Z 1/2
4
4
0
=−
f1 (t) cos(jπt) dt = − 2 2
f100 (t) sin(jπt) dt,
jπ 0
j π 0
where the integrals exist because f100 ∈ L1 (0, 1) by Remark 4. In fact
(8) shows that
Z 1/2
Z 1/2
00
|f1 (t) sin(jπt)| dt <
|f100 (t)| dt = πp ,
0
0
so
|τj | <
(12)
4πp
,
j 2π2
for all p > 1, j ∈ N.
In order to construct the linear operator T , we next define isometries
Mm of the Banach space Lq (0, 1) by Mm g(t) := g ∗ (mt), m ∈ N, in the
notation of (9). Notice that Mm en = emn .
Lemma 7. The maps Mm are isometric linear transformations of
Lq (0, 1) for all m ∈ N and q > 1.
Proof.
Z
1
Z 1
|Mm g(t)| dt = |g ∗ (mt)|q dt
q
0
0
m Z
1 X k ∗
|g (u)| du =
|g (u)|q du
m k=1 k−1
0
Z
Z 1
m
1 X 1
q
=
|g(u)| du =
|g(t)|q dt.
m k=1 0
0
1
=
m
Z
m
∗
q
We now define T : Lq (0, 1) −→ Lq (0, 1) by
(13)
T g(t) =
∞
X
τm Mm g(t).
m=1
The above lemma, the triangle inequality and (12) ensure that T is
a bounded everywhere defined operator with kT k(Lq →Lq ) 6 π2p .
7
We conclude this section by showing that T en = fn . Indeed, by
virtue of (10),
T en =
=
∞
X
m=1
∞
X
X
τm emn =
fb1 (m)emn
m−odd
fbn (k)ek = fn .
k=1
4. Bounded invertibility of T
In this section we complete the proof of our main result by showing
12
and 1 < q < ∞.
that T has a bounded inverse for all p > 11
Observe that (13), Lemma 7 and the triangle inequality give
kT − τ1 M1 k(Lq →Lq ) ≤
∞
X
|τj |.
j=3
Since M1 = I, [11, p. 196] shows that it suffices to prove
∞
X
(14)
|τj | < |τ1 |.
j=3
We will carry out the verification of (14) separately for p > 2 and
p < 2.
Case 1 (p > 2). By virtue of (12) and Lemma 2,
∞
X
X
8
4π2 π 2
− 1 < 2,
|τj | =
|τj | < 2
π
8
π
j=3
36j−odd
On the other hand, Lemma 3 ensures that
Z 1
Z 1/2
Sp (πp t) sin πt dt
Sp (πp t) sin πt dt = 4
τ1 = 2
0
Z
0
1/2
>4
2t sin πt dt =
0
8
,
π2
This guarantees (14).
Case 2 ( 12
6 p < 2). By virtue of Lemma 5,
11
Z 1
Z 1
τ1 = 2
Sp (πp t) sin πt dt > 2
(sin πt)2 dt = 1.
0
0
8
On the other hand, (11), Lemma 2 and the Cauchy-Schwartz inequality
yield
4π Z 1/2
p
|τj | = 2 2
Sp00 (πp t) sin jπt dt
j π 0
!1/2 Z
!1/2
Z 1/2
1/2
4π12/11
6 2 2
(Sp00 (πp t))2 dt
(sin jπt)2 dt
.
j π
0
0
Then (4) and (6) give
∞
X
π2
22
√
− 1 < 1.
|τj | 6
3( 3 − 1)π 8
j=3
This completes the proof of Theorem 1.
Remark 8.
A more precise but less explicit statement of Theorem 1 can be given
as follows:
12
There exists p0 ∈ (1, 11
) such that for p0 6 p < ∞, the family
∞
{fn }n=1 forms a Riesz basis of L2 (0, 1) and a Schauder basis of Lq (0, 1)
for all 1 < q < ∞.
12
As already mentioned at the end of Section 1 the value p = 11
is by
no means the optimal one. For example, our proof already shows that
12
inequality (14) holds also for p ∈ 11
− ε, 12
with ε > 0 small enough.
11
Numerical results show that (14) holds when p = 1.05, but not when
p = 1.04, so the method here will not suffice for all p > 1.
Remark 9. It is obvious that {fn }∞
n=1 is not an orthogonal basis in
L2 (0, 1) for p 6= 2. Indeed, the sign of
Z 1
Ii,j =
fi (t)fj (t) dt, i, j = 1, 2, . . .
0
depends on p and can change as p varies from 1 to ∞. Fig. 2 illustrates
the dependence of I1,3 , I1,5 and I3,7 on p.
I1,3 (p)
0
1
I1,5 (p)
2
p
0
1
I3,7 (p)
2
p
0
1
2
p
9
Figure 2
More values of Ii,j , and further numerical experiments with the bases
{fn }, can be found at http://drift.cam.zcu.cz.
5. The limit as p → ∞
Although (1) does not represent a differential equation when p =
∞, we can still discuss thepointwise
limits fn∞ of the fn as p → ∞,
2
constructed as follows. On 0, n we set
1

2nt
, t ∈ 0, 2n
,



1
3
2 − 2nt , t ∈ 2n
, 2n
,
fn∞ (t) :=


 2nt − 4 , t ∈ 3 , 2 ,
2n n
h
2(k+1)
and extend it periodically to the intervals 2k
, k ∈ Z. The
,
n
n
restriction to [0, 1] is a piecewice linear function whose graph is depicted
in Fig. 3.
fn∞ (t)
0
1
2n
1
n
1
2
n
Figure 3
∞
∞
Similarly to Section 3 we can define τj∞ := fc
1 (j). As before, τj = 0
when j is even. For odd j, integration by parts yields
τj∞
Z
1
=2
4
=
jπ
Z
f1∞ (t) sin(jπt) dt
Z
1/2
=4
0
1/2
0
2 cos(jπt) dt = (−1)j
0
8
j 2π2
so
|τj∞ | =
8
.
j 2π2
f1∞ (t) sin(jπt) dt =
10
Hence, as in Section 4, we have
∞
X
∞
τj < 8
π2
j=3
and so
∞
X
|τj∞ | < |τ1∞ |.
j=3
The analogue of Theorem 1 for p = ∞ follows as for p < ∞, the
analogue T ∞ : Lq (0, 1) → Lq (0, 1) of T being defined via
∞
X
∞
∞
Mm g(t).
T g(t) =
τm
m=1
Remark 10. The results of this paper appear to be useful in numerical approaches to boundary value problems for quasilinear equations
involving the p-Laplacian, e.g., of the type considered in [3] and [4].
We hope to report on this elsewhere.
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