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# Basisness of Fucik eigenfunctions for the Dirichlet Laplacian

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• Institute of Mathematics, Ufa Federal Research Centre, Russian Academy of Sciences
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## Abstract and Figures

We provide improved sufficient assumptions on sequences of Fucik eigenvalues of the one-dimensional Dirichlet Laplacian which guarantee that the corresponding Fucik eigenfunctions form a Riesz basis in $L^2(0,\pi)$. For that purpose, we introduce a criterion for a sequence in a Hilbert space to be a Riesz basis.
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BASISNESS OF FUČÍK EIGENFUNCTIONS FOR THE DIRICHLET
LAPLACIAN
Abstract. We provide improved suﬃcient assumptions on sequences of Fučík eigenvalues
of the one-dimensional Dirichlet Laplacian which guarantee that the corresponding Fučík
eigenfunctions form a Riesz basis in 𝐿2(0, 𝜋). For that purpose, we introduce a criterion for
a sequence in a Hilbert space to be a Riesz basis.
1. Introduction
We study basis properties of sequences of eigenfunctions of the Fučík eigenvalue problem
for the one-dimensional Dirichlet Laplacian
𝑢′′(𝑥) = 𝛼𝑢+(𝑥)𝛽𝑢(𝑥), 𝑥 (0, 𝜋),
𝑢(0) = 𝑢(𝜋)=0,(1.1)
where 𝑢+= max(𝑢, 0) and 𝑢= max(𝑢, 0). The Fučík spectrum is the set Σ(0, 𝜋)of pairs
(𝛼, 𝛽)R2for which (1.1) possesses a non-zero classical solution. Any (𝛼, 𝛽)Σ(0, 𝜋)
is called Fučík eigenvalue and any corresponding non-zero classical solution of (1.1) is called
Fučík eigenfunction. The Fučík eigenvalue problem (1.1) was introduced in [4] and [6] to study
elliptic equations with “jumping” nonlinearities, and afterwards it has been widely investigated
in various aspects and for diﬀerent operators, see, e.g., the surveys [3], [8, Chapter 9.4], and
references therein. To the best of our knowledge, basisness of sequences of Fučík eigenfunc-
tions was considered for the ﬁrst time in [2]. In that article, we provided several suﬃcient
assumptions on sequences of Fučík eigenvalues to obtain Riesz bases of 𝐿2(0, 𝜋)consisting of
Fučík eigenfunctions. Let us recall that a sequence is a Riesz basis in a Hilbert space if it
is the image of an orthonormal basis of that space under a linear homeomorphism, see, e.g.,
[9]. The aim of the present note is to use more general techniques to signiﬁcantly improve the
results of [2].
Let us describe the structure of the Fučík spectrum Σ(0, 𝜋). It is not hard to see that
the lines {1} × Rand R× {1}are subsets of Σ(0, 𝜋), since they correspond to sign-constant
solutions of (1.1) which are constant multiples of sin 𝑥, the ﬁrst eigenfunction of the Dirichlet
2010 Mathematics Subject Classiﬁcation. 34L10, 34B25, 34B08, 47A70.
Key words and phrases. Fucik spectrum, Fucik eigenfunctions, Riesz basis, Paley-Wiener stability.
The main part of the research was done during a stay of F. Baustian at the Ufa Federal Research Centre.
The stay was ﬁnanced by the German-Russian Interdisciplinary Science Center (G-RISC), grant no. F-2021b-
8_d. V. Bobkov was supported in the framework of implementation of the development program of Volga
Region Mathematical Center (agreement no. 075-02-2021-1393).
1
arXiv:2111.08329v1 [math.CA] 16 Nov 2021
2 F. BAUSTIAN AND V. BOBKOV
Laplacian in (0, 𝜋). The remaining part of Σ(0, 𝜋)is exhausted by the hyperbola-type curves
Γ𝑛=(𝛼, 𝛽)R2:𝑛
2
𝜋
𝛼+𝑛
2
𝜋
𝛽=𝜋
for even 𝑛N, and
Γ𝑛=(𝛼, 𝛽)R2:𝑛+ 1
2
𝜋
𝛼+𝑛1
2
𝜋
𝛽=𝜋,
Γ𝑛=(𝛼, 𝛽)R2:𝑛1
2
𝜋
𝛼+𝑛+ 1
2
𝜋
𝛽=𝜋
for odd 𝑛3, see, e.g., [6, Lemma 2.8]. Evidently, (𝛼, 𝛽)Γ𝑛for odd 𝑛3implies
(𝛽, 𝛼)
Γ𝑛. If 𝑢is a Fučík eigenfunction for some (𝛼, 𝛽), then so is 𝑡𝑢 for any 𝑡 > 0, while
𝑡𝑢 is a Fučík eigenfunction for (𝛽, 𝛼). Hence, we neglect the curve
Γ𝑛from our investiga-
tion of the basis properties of Fučík eigenfunctions. Each sign-changing Fučík eigenfunction
consists of alternating positive and negative bumps, where positive bumps are described by
𝐶1sin(𝛼(𝑥𝑥1)), while negative bumps are described by 𝐶2sin(𝛽(𝑥𝑥2)), for proper
constants 𝐶1, 𝐶2, 𝑥1, 𝑥2R.
We want to uniquely specify a Fučík eigenfunction for each point of Σ(0, 𝜋). In slight
contrast to [2], we normalize Fučík eigenfunctions in such a way that they are “close” to the
functions
𝜙𝑘(𝑥) = 2
𝜋sin(𝑘𝑥), 𝑘 N,
which form a complete orthonormal system in 𝐿2(0, 𝜋). This choice will be helpful in the
proof of our main result, Theorem 1.3, below.
Deﬁnition 1.1. Let 𝑛2and (𝛼, 𝛽 )Γ𝑛. The normalized Fučík eigenfunction 𝑔𝑛
𝛼,𝛽 is the
𝐶2-solution of the boundary value problem (1.1) with (𝑔𝑛
𝛼,𝛽)(0) >0and which is normalized
by
𝑔𝑛
𝛼,𝛽= sup
𝑥[0,𝜋]|𝑔𝑛
𝛼,𝛽(𝑥)|=2
𝜋.
For 𝑛= 1, we set 𝑔1
𝛼,𝛽 =𝜙1for every (𝛼, 𝛽)({1} × R)(R× {1}).
Piecewise deﬁnitions of the Fučík eigenfunctions 𝑓𝑛
𝛼,𝛽 =𝜋/2𝑔𝑛
𝛼,𝛽 can be found in the
equations (1.2) and (1.3) in [2]. In accordance to [2], we study the basisness of sequences of
Fučík eigenfunctions described by the following deﬁnition.
Deﬁnition 1.2. We deﬁne the Fučík system 𝐺𝛼,𝛽 ={𝑔𝑛
𝛼(𝑛),𝛽(𝑛)}as a sequence of normal-
ized Fučík eigenfunctions with mappings 𝛼, 𝛽 :NRsatisfying 𝛼(1) = 𝛽(1) = 1 and
(𝛼(𝑛), 𝛽(𝑛)) Γ𝑛for every 𝑛2.
We can now formulate our main result on the basisness of Fučík systems which presents a
non-trivial generalization of [2, Theorems 1.4 and 1.9].
Theorem 1.3. Let 𝐺𝛼,𝛽 be a Fučík system. Let 𝑁be a subset of the even natural numbers
and 𝑁*=N𝑁. Assume that
𝑛𝑁*1𝑔𝑛
𝛼,𝛽, 𝜙𝑛2
𝑔𝑛
𝛼,𝛽2+𝐸2sup
𝑛𝑁4 max(𝛼(𝑛), 𝛽(𝑛))
𝑛2<1,(1.2)
BASISNESS OF FUČÍK EIGENFUNCTIONS FOR THE DIRICHLET LAPLACIAN 3
with sup𝑛𝑁4 max(𝛼(𝑛), 𝛽(𝑛))/𝑛2[4,9). Here, 𝐸: [4,9) Ris a strictly increasing
function deﬁned as
𝐸(𝛾) = 22
𝜋
𝛾2
𝛾1
(𝛾2) sin 𝜋
𝛾
(𝛾1)(2𝛾1) +((3 + 𝜋2)𝛾+ (9 2𝜋2)𝛾6)(𝛾2)
3(𝛾1)(𝛾+ 2)(3𝛾2)
+4
3𝜋
𝛾2
𝛾1
(𝛾2) sin 3𝜋
𝛾
(9 𝛾)(2𝛾3)(4𝛾3) +2
𝜋
𝛾2
𝛾1
(𝛾2)
(16 𝛾)(3𝛾4)(5𝛾4)
+6
5
2
𝜋
𝛾2(𝛾2)
𝛾1
𝑘=5
1
(𝑘2𝛾)((𝑘1)𝛾𝑘)((𝑘+ 1)𝛾𝑘).(1.3)
Then 𝐺𝛼,𝛽 is a Riesz basis in 𝐿2(0, 𝜋).
The proof of this theorem is given in Section 3and it is based on a general basisness criterion
provided in Section 2. We visualize special cases of domains on the (𝛼, 𝛽)-plane described in
Theorem 1.3 in Figures 1and 2below.
Notice that, thanks to the orthonormality of {𝜙𝑛}, the terms in the ﬁrst sum in (1.2) satisfy
01𝑔𝑛
𝛼,𝛽, 𝜙𝑛2
𝑔𝑛
𝛼,𝛽2=𝑔𝑛
𝛼,𝛽 𝜙𝑛2(𝑔𝑛
𝛼,𝛽2− ⟨𝑔𝑛
𝛼,𝛽, 𝜙𝑛)2
𝑔𝑛
𝛼,𝛽2≤ ‖𝑔𝑛
𝛼,𝛽 𝜙𝑛2,(1.4)
and we have the following explicit bounds:
𝑔𝑛
𝛼,𝛽 𝜙𝑛2
8(3 + 𝜋2)
9
(max(𝛼, 𝛽)𝑛)2
𝑛2for even 𝑛,
8𝑛2(𝑛2+ 1)
(𝑛1)4
(𝛼𝑛)2
𝑛2for odd 𝑛3with 𝛼𝑛2,
10𝑛2(𝑛2+ 1)
(𝑛+ 1)4
(𝛽𝑛)2
𝑛2for odd 𝑛3with 𝛽 > 𝑛2,
(1.5)
see the estimates (3.2), (3.4), (3.5), (3.6) in [2, Section 3]. In view of (1.4), if we chose 𝑁=,
then Theorem 1.3 is an improvement of [2, Theorem 1.4].
Let us summarize a few properties of the function 𝐸deﬁned in Theorem 1.3, see the end
of Section 3for discussion.
Lemma 1.4. The function 𝐸has the following properties:
(i) 𝐸is continuous in [4,9).
(ii) Each summand in the deﬁnition (1.3)of 𝐸is strictly increasing in [4,9).
(iii) We have 𝐸(4) = 0 and 𝐸(6.49278 . . .)=1.
(iv) The inﬁnite sum in the deﬁnition (1.3)of 𝐸in (4,9) can be expressed as follows:
6
5
2
𝜋
𝛾2(𝛾2)
𝛾1
𝑘=5
1
(𝑘2𝛾)((𝑘1)𝛾𝑘)((𝑘+ 1)𝛾𝑘)
=6
5
2
𝜋
𝛾
𝛾1
𝑘=5 1
𝑘2𝛾1
𝑘2𝛾
(𝛾1)2
4 F. BAUSTIAN AND V. BOBKOV
=6
5
1
𝜋(𝛾1) 𝜋(𝛾1) cot 𝜋𝛾
𝛾1𝜋cot(𝜋𝛾)(𝛾2)
6
5
2
𝜋
𝛾2(𝛾2)
𝛾1
4
𝑘=1
1
(𝑘2𝛾)((𝑘1)𝛾𝑘)((𝑘+ 1)𝛾𝑘).
The interval [4,9) appears naturally in the proof of Theorem 1.3. In fact, Lemma 1.4 (iii)
indicates that the highest possible value of sup𝑛𝑁4 max(𝛼(𝑛), 𝛽(𝑛))/𝑛2to satisfy the
assumption (1.2) is even smaller than 9.
We obtain the following practical corollary of Theorem 1.3 by applying the upper bounds
(1.5) for the case that 𝑁is the set of all even natural numbers, see Figure 1.
Corollary 1.5. Let 𝐺𝛼,𝛽 be a Fučík system, and 𝜀 > 0. Assume that
sup
𝑛Neven 4 max(𝛼(𝑛), 𝛽(𝑛))
𝑛2<6.49278 . . .
and
max(𝛼(𝑛), 𝛽(𝑛)) 𝑛+𝑐𝑛𝑛(1𝜀)/22for all odd 𝑛3,
where
0𝑐𝑛<
1𝐸2sup
𝑛Neven 4 max(𝛼(𝑛), 𝛽(𝑛))
𝑛2
45 11
21+𝜀𝜁(1 + 𝜀)1
with the Riemann zeta function 𝜁. Then 𝐺𝛼,𝛽 is a Riesz basis in 𝐿2(0, 𝜋).
(a) (b)
Figure 1. The assumptions of Corollary 1.5 are satisﬁed for (𝛼(𝑛), 𝛽(𝑛)) be-
longing to bold lines inside the shaded regions. We have 𝜀= 0.5for both panels
and sup𝑛Neven 4 max(𝛼(𝑛),𝛽(𝑛))
𝑛2= 5,6in panel (A), (B), respectively.
If we assume that the ﬁrst sum of (1.2) in Theorem 1.3 is vanishing, which corresponds to
𝑐𝑛= 0 for all odd 𝑛3in the previous corollary, we obtain the following result.
BASISNESS OF FUČÍK EIGENFUNCTIONS FOR THE DIRICHLET LAPLACIAN 5
Corollary 1.6. Let 𝐺𝛼,𝛽 be a Fučík system such that 𝑔𝑛
𝛼,𝛽 =𝜙𝑛for any odd 𝑛. Assume that
sup
𝑛Neven 4 max(𝛼(𝑛), 𝛽(𝑛))
𝑛2<6.49278 . . . (1.6)
Then 𝐺𝛼,𝛽 is a Riesz basis in 𝐿2(0, 𝜋).
Figure 2. The assumption (1.6) is satisﬁed for (𝛼(𝑛), 𝛽(𝑛)) belonging to bold
We remark that Corollaries 1.5 and 1.6 are signiﬁcant improvements of [2, Theorem 1.9]
since each point (𝛼(𝑛), 𝛽(𝑛)) Γ𝑛for even 𝑛2is free to belong to the whole angular sector
in between the line
𝛽=sup
𝑛Neven 4 max(𝛼(𝑛), 𝛽(𝑛))
𝑛212
𝛼
and its reﬂection with respect to the main diagonal 𝛼=𝛽, and the angle of that sector is
allowed to be larger than the one provided by [2, Theorem 1.9]. We refer to Figure 2for
the domain on the (𝛼, 𝛽)-plane given by Corollary 1.6. Moreover, Corollary 1.5 improves [2,
Theorem 1.9] in the sense that 𝑔𝑛
𝛼,𝛽 for odd 𝑛3might diﬀer from 𝜙𝑛, see Figure 1.
2. Basisness criterion
In this section, we formulate a useful generalization of the separation of variables approach
of [5] in a real Hilbert space 𝑋. The provided criterion will be applied to the space 𝐿2(0, 𝜋)
to prove our main result, Theorem 1.3, in the subsequent section.
Theorem 2.1. Let 𝑀N. Let 𝑁*, 𝑁𝑚N,1𝑚𝑀, be pairwise disjoint sets which
form a decomposition of the natural numbers, i.e.,
𝑁*
𝑀
𝑚=1
𝑁𝑚=N.
6 F. BAUSTIAN AND V. BOBKOV
Let {𝜑𝑛}be a complete orthonormal sequence in 𝑋and {𝑓𝑛} ⊂ 𝑋be a sequence that can be
represented as
𝑓𝑛=𝜑𝑛+
𝑘=1
𝐶𝑚
𝑛,𝑘𝑇𝑚
𝑘𝜑𝑛for every 𝑛𝑁𝑚,1𝑚𝑀, (2.1)
and satisﬁes
Λ*:=
𝑛𝑁*1𝑓𝑛, 𝜑𝑛2
𝑓𝑛21
2
<.
In the representation formula (2.1),{𝑇𝑚
𝑘}is a family of bounded linear mappings from 𝑋to
itself with bounds 𝑇𝑚
𝑘*𝑡𝑚
𝑘on the operator norm and {𝐶𝑚
𝑛,𝑘}is a family of constants with
uniform bounds |𝐶𝑚
𝑛,𝑘| ≤ 𝑐𝑚
𝑘that satisfy
Λ𝑚:=
𝑘=1
𝑐𝑚
𝑘𝑡𝑚
𝑘<.(2.2)
Then {𝑓𝑛}is a basis in 𝑋provided that
Λ2
*+
𝑀
𝑚=1
Λ2
𝑚<1.(2.3)
If, in addition, the subsequence {𝑓𝑛}𝑛𝑁*is bounded, then {𝑓𝑛}is a Riesz basis in 𝑋.
Proof. Denote
𝑓𝑛=𝜌𝑛𝑓𝑛, where 𝜌𝑛= 1 for 𝑛N𝑁*, and the values of 𝜌𝑛for 𝑛𝑁*will
be speciﬁed later. Let {𝑎𝑛}𝑛
𝑁be an arbitrary ﬁnite sequence of constants with a ﬁnite index
set
𝑁N. Setting
𝑁*=𝑁*
𝑁and
𝑁𝑚=𝑁𝑚˜
𝑁for every 1𝑚𝑀, we obtain
𝑛
𝑁
𝑎𝑛(
𝑓𝑛𝜑𝑛)
𝑀
𝑚=1
𝑛
𝑁𝑚
𝑎𝑛(𝑓𝑛𝜑𝑛)
+
𝑛
𝑁*
𝑎𝑛(𝜌𝑛𝑓𝑛𝜑𝑛)
.(2.4)
For the ﬁrst sum on the right-hand side of (2.4), we apply the representation (2.1) and get
𝑀
𝑚=1
𝑛
𝑁𝑚
𝑎𝑛(𝑓𝑛𝜑𝑛)
=
𝑀
𝑚=1
𝑛
𝑁𝑚
𝑎𝑛
𝑘=1
𝐶𝑚
𝑛,𝑘𝑇𝑚
𝑘𝜑𝑛
=
𝑀
𝑚=1
𝑘=1
𝑇𝑚
𝑘
𝑛
𝑁𝑚
𝐶𝑚
𝑛,𝑘𝑎𝑛𝜑𝑛
𝑀
𝑚=1
𝑘=1
𝑇𝑚
𝑘
𝑛
𝑁𝑚
𝐶𝑚
𝑛,𝑘𝑎𝑛𝜑𝑛
𝑀
𝑚=1
𝑘=1
𝑡𝑚
𝑘
𝑛
𝑁𝑚
𝐶𝑚
𝑛,𝑘𝑎𝑛𝜑𝑛
𝑀
𝑚=1
𝑘=1
𝑡𝑚
𝑘𝑐𝑚
𝑘
𝑛
𝑁𝑚
𝑎𝑛𝜑𝑛
=
𝑀
𝑚=1
Λ𝑚
𝑛
𝑁𝑚
𝑎𝑛𝜑𝑛
,
while for the second sum we obtain
𝑛
𝑁*
𝑎𝑛(𝜌𝑛𝑓𝑛𝜑𝑛)
𝑛
𝑁*
𝜌𝑛𝑓𝑛𝜑𝑛21
2
𝑛
𝑁*
|𝑎𝑛|21
2.
Let us choose 𝜌𝑛to be a minimizer of the distance 𝜌𝑓𝑛𝜑𝑛2with respect to 𝜌. Since
𝜌𝑓𝑛𝜑𝑛2=𝜌2𝑓𝑛22𝜌𝑓𝑛, 𝜑𝑛+ 1,
BASISNESS OF FUČÍK EIGENFUNCTIONS FOR THE DIRICHLET LAPLACIAN 7
𝜌𝑛𝑓𝑛𝜑𝑛2= min
𝜌R𝜌𝑓𝑛𝜑𝑛2= 1 𝑓𝑛, 𝜑𝑛2
𝑓𝑛2=𝑓𝑛𝜑𝑛2(𝑓𝑛2− ⟨𝑓𝑛, 𝜑𝑛)2
𝑓𝑛2
with 𝜌𝑛=𝑓𝑛, 𝜑𝑛/𝑓𝑛2. Evidently, we have |𝜌𝑛| ≤ 1. We remark that in case of 𝜌𝑛= 0,
we get Λ*1which violates the assumption (2.3). Applying now the Cauchy inequality, we
deduce from (2.4) that
𝑛
𝑁
𝑎𝑛(
𝑓𝑛𝜑𝑛)
𝑀
𝑚=1
Λ𝑚
𝑛
𝑁𝑚
𝑎𝑛𝜑𝑛
+ Λ*
𝑛
𝑁*
|𝑎𝑛|21
2
𝑀
𝑚=1
Λ2
𝑚+ Λ2
*1
2
𝑛
𝑁
𝑎𝑛𝜑𝑛
.
We conclude from the assumption (2.3) that the sequence {
𝑓𝑛}is Paley-Wiener near to the
complete orthonormal sequence {𝜑𝑛}and, thus, it is a Riesz basis in 𝑋, see, e.g., [9, Chapter 1,
Theorem 10]. Clearly, {𝑓𝑛}={𝜌1
𝑛
𝑓𝑛}is a basis in 𝑋. Assume that the subsequence {𝑓𝑛}𝑛𝑁*
is bounded. Then there exists 0< 𝑐 < 1such that |𝜌𝑛| ≥ 𝑐for all 𝑛
𝑁*. This is evident for
ﬁnite 𝑁*since 𝜌𝑛̸= 0. In the case of inﬁnte 𝑁*, if we suppose that 𝜌𝑛goes to zero up to a
subsequence, then the sum
Λ*=
𝑛𝑁*1𝑓𝑛, 𝜑𝑛2
𝑓𝑛21
2
=
𝑛𝑁*1𝜌2
𝑛𝑓𝑛21
2
does not converge. Recalling 𝜌𝑛= 1 for every 𝑛N𝑁*, we obtain 1≤ |𝜌1
𝑛| ≤ 𝑐1for all
𝑛Nwhich implies that {𝑓𝑛}is a Riesz basis in 𝑋, see, e.g., [9, Chapter 1, Theorem 9].
In the case 𝑁1=N, Theorem 2.1 simpliﬁes to Theorem D from [5] and for 𝑁*=Nwe get
the result of Theorem V-2.21 and Corollary V-2.22 i) from [7] which were discussed in [2].
Remark 2.2. It can be seen from the proof of Theorem 2.1 that if we weaken the deﬁnition
of Λ*to
Λ*:=
𝑛𝑁*
𝑓𝑛𝜑𝑛21
2
Λ*,
then we can formulate the following result under the assumptions of Theorem 2.1: the sequence
{𝑓𝑛}is a Riesz basis in 𝑋provided that
Λ2
*+
𝑀
𝑚=1
Λ2
𝑚<1.
The boundedness of the subsequence {𝑓𝑛}𝑛𝑁*is not required under this modiﬁed assumption.
3. Proof of Theorem 1.3
We prove Theorem 1.3 by applying the general basisness criterion introduced in the previous
section. To determine the bounds on the family of constants {𝐶𝑚
𝑛,𝑘}in Theorem 2.1 we will
make use of the Fourier coeﬃcients of Fučík eigenfunctions corresponding to Fučík eigenvalues
8 F. BAUSTIAN AND V. BOBKOV
on the ﬁrst nontrivial curve Γ2. Namely, we provide estimates for the Fourier coeﬃcients of
the odd Fourier expansion of the function
𝑔2
𝛾,𝛾/(𝛾1)2=
𝑘=1
𝐴𝑘(𝛾)𝜙𝑘(𝑥)
for 𝛾 > 4which are given by
𝐴𝑘(𝛾) = 𝜋
0
𝑔2
𝛾,𝛾/(𝛾1)2(𝑥)𝜙𝑘(𝑥) d𝑥=2
𝜋
𝛾2
𝛾1
(2 𝛾) sin 𝑘𝜋
𝛾
(𝑘2𝛾)(𝑘2(𝛾1)2𝛾),
and of the function
𝑔2
𝛿/(𝛿1)2,𝛿 =
𝑘=1
𝐴𝑘(𝛿)𝜙𝑘(𝑥)
for 𝛿 > 4which are given by
𝐴𝑘(𝛿) = 𝜋
0
𝑔2
𝛿/(𝛿1)2,𝛿 (𝑥)𝜙𝑘(𝑥) d𝑥= (1)𝑘𝐴𝑘(𝛿).
In the case 𝛾=𝛿= 4, we have 𝐴2= 1 and 𝐴𝑘= 0 for any other 𝑘N.
Obviously, we have
|𝐴1(𝛾)|=𝐵1(𝛾) := 2
𝜋
𝛾2
𝛾1
(𝛾2) sin 𝜋
𝛾
(𝛾1)(2𝛾1) (3.1)
and it was shown in [2, Section 5] that
|𝐴2(𝛾)1| ≤ 𝐵2(𝛾) := ((3 + 𝜋2)𝛾+ (9 2𝜋2)𝛾6)(𝛾2)
3(𝛾1)(𝛾+ 2)(3𝛾2) .(3.2)
For 𝛾[4,9), we clearly have
|𝐴3(𝛾)|=𝐵3(𝛾) := 2
𝜋
𝛾2
𝛾1
(𝛾2) sin 3𝜋
𝛾
(9 𝛾)(2𝛾3)(4𝛾3) (3.3)
and for 𝑘4we use the simple estimate
|𝐴𝑘(𝛾)| ≤ 𝐵𝑘(𝛾) := 2
𝜋
𝛾2
𝛾1
(𝛾2)
(𝑘2𝛾)((𝑘1)𝛾𝑘)((𝑘+ 1)𝛾𝑘).(3.4)
Evidently, the same bounds hold for
𝐴𝑘. Numerical calculations with the exact coeﬃcients
show that the used estimates in (3.2) and (3.4) do not inﬂuence the results in a signiﬁcant
way.
Lemma 3.1. Let 𝛾[4,9) and 𝑘N. Then 𝐵𝑘is strictly increasing.
Proof. For simplicity, we introduce the change of variables 𝑥=𝛾[2,3). The ﬁrst derivative
of 𝐵𝑘(𝑥2)with 𝑘N∖ {1,3}is a rational function with a positive denominator and we can
easily check that the numerator is positive, as well. Hence, 𝐵𝑘(𝛾)with 𝑘N{1,3}is strictly
increasing for 𝛾[4,9). The ﬁrst derivative of 𝐵1(𝑥2)takes the form
2𝑥2(𝑥1) cos 𝜋
𝑥𝑥(2𝑥44𝑥3𝑥2+ 15𝑥8) tan 𝜋
𝑥𝜋(2𝑥45𝑥3+ 5𝑥2)
𝜋(𝑥1)2(𝑥21)2(2𝑥1)2.
BASISNESS OF FUČÍK EIGENFUNCTIONS FOR THE DIRICHLET LAPLACIAN 9
Noting that 𝑥(2𝑥44𝑥3𝑥2+ 15𝑥8) >0for 𝑥[2,3), we can use the simple lower bound
tan 𝜋
𝑥3to show that the expression in square brackets is positive. Since all other terms
in the derivative are also positive, we conclude that 𝐵1(𝛾)is strictly increasing for 𝛾[4,9).
Finally, the numerator of the ﬁrst derivative of 𝐵3(𝑥2)is given by
2𝑥2𝑥(10𝑥5+ 90𝑥4765𝑥3+ 1872𝑥21863𝑥+ 648) sin 3𝜋
𝑥
+ 3𝜋(8𝑥642𝑥5+ 7𝑥4+ 315𝑥3693𝑥2+ 567𝑥162) cos 3𝜋
𝑥,(3.5)
whereas the denominator is a positive polynomial. We have sin 3𝜋
𝑥<0and cos 3𝜋
𝑥<0for
𝑥[2,3), and taking into account that
𝑥(10𝑥5+ 90𝑥4765𝑥3+ 1872𝑥21863𝑥+ 648) <0,
3𝜋(8𝑥642𝑥5+ 7𝑥4+ 315𝑥3693𝑥2+ 567𝑥162) >0,
we employ the estimates
sin 3𝜋
𝑥<3𝜋
𝑥𝜋+1
63𝜋
𝑥𝜋3
and cos 3𝜋
𝑥>1.
As a result, the expression (3.5) is estimated from below by a polynomial which is positive for
𝑥[2,3). Thus, 𝐵3(𝛾)is strictly increasing for 𝛾[4,9).
Now we are ready to prove our main result.
Proof of Theorem 1.3.We apply Theorem 2.1, where we consider 𝑋=𝐿2(0, 𝜋), the sequence
{𝑓𝑛}is the Fučík system, which is bounded by deﬁnition, and the complete orthonormal set
{𝜑𝑛}is given by {𝜙𝑛}. We set 𝑀= 1 and 𝑁1=𝑁and choose 𝑁*=N𝑁as assumed in
Theorem 1.3. We deﬁne the linear operators 𝑇1
𝑘:𝐿2(0, 𝜋)𝐿2(0, 𝜋)as
𝑇1
𝑘𝑔(𝑥) = 𝑔*𝑘𝑥
2,
where
𝑔*(𝑥)=(1)𝜅𝑔(𝑥𝜋𝜅)for 𝜋𝜅 𝑥𝜋(𝜅+ 1), 𝜅 N∪ {0},
is the 2𝜋-antiperiodic extension for arbitrary functions 𝑔𝐿2(0, 𝜋). In particular, we have
𝑇1
𝑘sin(𝑛𝑥) = sin 𝑘𝑛𝑥
2for every even 𝑛. It was proven in [2, Appendix B] that 𝑇1
𝑘*= 1 for
even 𝑘and 𝑇1
𝑘*=1+1/𝑘 for odd 𝑘.
Let 𝑛𝑁be ﬁxed and recall that 𝑛is even. To begin with, we assume that 𝛼(𝑛)> 𝑛2.
The Fučík eigenfunction 𝑔𝑛
𝛼,𝛽 has the dilated structure
𝑔𝑛
𝛼,𝛽(𝑥) = 𝑔2
𝛾𝑛,𝛾𝑛/(𝛾𝑛1)2𝑛𝑥
2with 𝛾𝑛=4𝛼(𝑛)
𝑛2
and, thus, has the odd Fourier expansion
𝑔𝑛
𝛼,𝛽(𝑥) = 𝑔2
𝛾𝑛,𝛾𝑛/(𝛾𝑛1)2𝑛𝑥
2=
𝑘=1
𝐴𝑘(𝛾𝑛)𝜙𝑘𝑛𝑥
2=
𝑘=1
𝐴𝑘(𝛾𝑛)𝑇1
𝑘𝜙𝑛(𝑥).
From this, we directly see that the representation (2.1) of 𝑔𝑛
𝛼,𝛽 in terms of {𝜙𝑛}holds with
the constants 𝐶1
𝑛,𝑘 =𝐴𝑘(𝛾𝑛)for 𝑘̸= 2 and 𝐶1
𝑛,2= 1 𝐴2(𝛾𝑛). The bounds for the constants
10 F. BAUSTIAN AND V. BOBKOV
|𝐶1
𝑛,𝑘|are given by the functions 𝐵𝑘(𝛾𝑛)deﬁned in (3.1), (3.2), (3.3), and (3.4), which are
strictly increasing in the interval [4,9) by Lemma 3.1. For the case 𝛽(𝑛)> 𝑛2, the Fučík
eigenfunction has the form
𝑔𝑛
𝛼,𝛽(𝑥) = 𝑔2
𝛿𝑛/(𝛿𝑛1)2,𝛿𝑛𝑛𝑥
2with 𝛿𝑛=4𝛽(𝑛)
𝑛2,
and by analogous arguments we get the bounds |𝐶1
𝑛,𝑘| ≤ 𝐵𝑘(𝛿𝑛). If 𝛼(𝑛) = 𝑛2, and hence
𝛽(𝑛) = 𝑛2, then we set 𝐶1
𝑛,𝑘 = 0 for every 𝑘N.
In view of the monotonicity, we have
|𝐶1
𝑛,𝑘| ≤ 𝐵𝑘sup
𝑛𝑁
max(𝛾𝑛, 𝛿𝑛).
Therefore, we can provide the following upper estimate on the constant Λ1deﬁned in (2.2):
Λ12𝐵1sup
𝑛𝑁
max(𝛾𝑛, 𝛿𝑛)+𝐵2sup
𝑛𝑁
max(𝛾𝑛, 𝛿𝑛)
+4
3𝐵3sup
𝑛𝑁
max(𝛾𝑛, 𝛿𝑛)+𝐵4sup
𝑛𝑁
max(𝛾𝑛, 𝛿𝑛)
+6
5
𝑘=5
𝐵𝑘sup
𝑛𝑁
max(𝛾𝑛, 𝛿𝑛)
=𝐸sup
𝑛𝑁
max(𝛾𝑛, 𝛿𝑛)=𝐸sup
𝑛𝑁4 max(𝛼(𝑛), 𝛽(𝑛))
𝑛2,
with the function 𝐸introduced in Theorem 1.3, and 𝐸is strictly increasing in [4,9). Noticing
that we have
Λ*=
𝑛𝑁*1𝑔𝑛
𝛼,𝛽, 𝜙𝑛2
𝑔𝑛
𝛼,𝛽21
2
,
the assumption (1.2) yields the assumption Λ2
*+ Λ2
1<1in Theorem 2.1. This completes the
proof of Theorem 1.3.
We conclude this note by discussing Lemma 1.4. The monotonicity statement (ii) directly
follows from Lemma 3.1, and to obtain the alternative representation (iv), we make use of the
identity
𝑘=1
1
𝑘2𝑎2=1
2𝑎2𝜋cot(𝜋𝑎)
2𝑎, 𝑎 ̸∈ N,
see, e.g., [1, (6.3.13)]. The representation (iv) shows that the function 𝐸is continuous in
[4,9). The combination of the continuity and monotonicity of 𝐸allows us to compute values
of 𝐸with an arbitrary precision. In particular, we have 𝐸(6.49278 . . .)=1.
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(F. Baustian)
Institute of Mathematics, University of Rostock,
Ulmenstraße 69, 18057 Rostock, Germany