Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 12, pp. 1–25.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2025.12

NODAL SETS AND CONTINUITY OF EIGENFUNCTIONS OF

KREĬN-FELLER OPERATORS

SZE-MAN NGAI, MENG-KE ZHANG, WEN-QUAN ZHAO

Abstract. Let µ be a compactly supported positive finite Borel measure on

Rd. Let 0 < λ1 ≤ λ2 ≤ · · · be eigenvalues of the Krĕın-Feller operator
∆µ. We prove that, on a bounded domain, the nodal set of a continuous

λn-eigenfunction of a Krĕın-Feller operator divides the domain into at least 2

and at most n+ rn − 1 subdomains, where rn is the multiplicity of λn. This
work generalizes the nodal set theorem of the classical Laplace operator to

Krĕın-Feller operators on bounded domains. We also prove that on bounded

domains on which the classical Green function exists, the eigenfunctions of a
Krĕın-Feller operator are continuous.

1. Introduction

For a bounded domain (i.e., an open and connected set) Ω ⊆ Rd, consider the
Dirichlet problem

−∆u = λu, in Ω,

u = 0, in ∂Ω,
(1.1)

with eigenvalue λ and eigenfunction u. The nodal set of u is defined as

Z(u) := {x ∈ Ω : u(x) = 0}.
It is known that the eigenvalues can be ordered as

0 < λ1 ≤ λ2 ≤ · · ·
with limn→∞ λn = ∞. Properties of nodal set of the eigenfunctions have been
studied extensively (see [2, 3, 10, 22, 31, 32, 36, 44, 45, 48, 50] and references
therein). Let Ω ⊆ Rd be a bounded domain. By a λ-eigenfunction we mean
an eigenfunction corresponding to the eigenvalue λ. The Courant nodal domain
theorem says that the nodal set of a λn-eigenfunction of (1.1) divides Ω into at
most n subdomains (see e.g. [10]). Gladwell and Zhu [22] studied the nodal sets of
eigenfunctions of the following Helmholtz equation on a bounded domain Ω ⊆ Rd:

∆u+ λρu = 0,

where ρ(x) is positive and bounded. They proved that the nodal set of a λn-
eigenfunction divides Ω into at most n + rn − 1 subdomains, where rn is the

2020 Mathematics Subject Classification. 35J05, 35B05, 34L10, 28A80, 35J08.

Key words and phrases. Krĕın-Feller operators; nodal set; continuous eigenfunctions.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted April 17, 2024. Published February 5, 2025.

1



2 S.-M. NGAI, M.-K. ZHANG, W.-Q. ZHAO EJDE-2025/12

multiplicity of λn. One goal of this paper is to generalize the above theorem to
Laplace operators defined by measures (see definition in Section 2) on a domain
Ω ⊆ Rd. Such operators are also called Krĕın-Feller operators and are introduced
in [17, 27, 30]. These operators are used to describe physical phenomena, such
as wave propagation or heat conduction, in media with an inhomogeneous mass
distribution modeled by a measure µ, such as a fractal measure.

Krĕın-Feller operators have been studied extensively. McKean and Ray [33]
studied spectral asymptotics of Krĕın-Feller operator defined by the Cantor mea-
sure. Freiberg [18] studied analytic properties of the operators defined on the line.
Hu et al [25] studied spectral properties of Krĕın-Feller operators defined on a
bounded domain of Rd. Deng and Ngai [13], Pinasco and Scarola [46] studied
the eigenvalue estimates of such operators. Kesseböhmer and Niemann [28, 29]
studied the relation between the Lq-spectrum and spectral dimension of such an
operator. For additional work associated with these operators, including eigen-
values, eigenfunctions, spectral asymptotics, spectral gaps, spectral dimension,
wave equation, heat equation and heat kernel estimates, the reader is referred to
[7, 8, 9, 13, 14, 18, 19, 20, 24, 25, 28, 29, 37, 38, 39, 40, 41, 42, 46, 47, 52] and
references therein.

We will summarize the definition of a Krĕın-Feller operator in Section 2. We
denote by ∆µ the Krĕın-Feller operator defined by a measure µ (see Section 2). In
this article, we consider eigenvalues and eigenfunctions associated with the Dirichlet
problem

−∆µu = λu, in Ω,

u = 0, in ∂Ω.
(1.2)

Let dim∞(µ) be defined as in (2.1). It is shown in [25, Theorem 1.2] that, under
the assumption dim∞(µ) > d − 2, there exists an orthonormal basis {ϕn}∞n=1 of
L2(Ω, µ) consisting of (Dirichlet) eigenfunctions of ∆µ. The eigenvalues {λn}∞n=1

satisfy 0 < λ1 ≤ λ2 ≤ · · · and limn→∞ λn = ∞. We let

Zµ(u) := {x ∈ Ω : u(x) = 0} (1.3)

be the nodal set of an eigenfunction u of ∆µ. Under the assumption of the continuity
of eigenfunctions, we have the following theorem.

Theorem 1.1. Let Ω ⊆ Rd (d ≥ 1) be a bounded domain and µ be a positive finite
Borel measure on Rd with supp(µ) ⊆ Ω and µ(Ω) > 0. Assume dim∞(µ) > d− 2.
Let the eigenvalues {λn}∞n=1 of (1.2) be arranged in an increasing order and let un
be a λn-eigenfunction. Suppose un ∈ C(Ω). Then

(a) u1 is nonzero on Ω.
(b) For n ≥ 2, if λn has multiplicity rn ≥ 1, then Zµ(un) divides Ω into at

least 2 and at most n+ rn − 1 subdomains.

To prove this result, we need the maximum principle of continuous µ-subharmonic
functions (Definition 3.1) which we will prove in Section 3.

Note that the definition of a nodal set Zµ(u) in (1.3) makes sense only if u is
defined everywhere and not just almost everywhere. As the domain of ∆µ consists
of Sobolev functions, we need to study the continuity of the eigenfunctions of ∆µ.
It is known that for the classical Laplacian, the eigenfunctions are continuous and
differentiable. If the measure µ is absolutely continuous with respect to Lebesgue
measure, then the eigenfunctions of ∆µ are continuous (see e.g. [16]). On R, the



EJDE-2025/12 NODAL SETS AND CONTINUITY OF EIGENFUNCTIONS 3

eigenfunctions of ∆µ are continuous, since Sobolev functions are continuous. To
the best of our knowledge, the continuity of eigenfunctions in higher dimensions is
unknown in general. Motivated by the requirement of the continuity of eigenfunc-
tions in the definition of Zµ(u) and in Theorem 1.1, we prove the following main
theorem. The definition of the regularity of the boundary of a bounded domain is
given in Definition 5.2.

Theorem 1.2. Let Ω be a bounded domain in Rd on which the classical Green
function G(x,y) exists and let µ be a finite positive Borel measure with supp(µ) ⊆ Ω
and µ(Ω) > 0. Assume dim∞(µ) > d − 2. Then the eigenfunctions of ∆µ are
continuous on Ω. Moreover, if the boundary of Ω is regular, then the eigenfunctions
of ∆µ are continuous on Ω.

To prove Theorem 1.2, we apply the inverse operator of −∆µ, called the Green
operator (see Section 5), which is defined by the classical Green function. By
expressing the eigenfunctions of ∆µ in terms of the Green operator and using the
continuity of the Green function, we prove Theorem 1.2; details are given in Section
5. This theorem shows that the Dirichlet problem (1.2) has continuous solutions
on a bounded domain Ω ⊆ Rd on which the Green function exists.

This article is organized as follows. In Section 2, we summarize the definition of
∆µ in [25]. In Section 3, we study the properties of ∆µ and prove the maximum
principle of continuous µ-subharmonic functions. Sections 4 and 5 are devoted
to the proofs of Theorems 1.1 and 1.2, respectively. In Section 6, we construct
examples of continuous eigenfunctions corresponding to singular measures on R2.

2. Preliminaries

In this section, we summarize the definition of Krĕın-Feller operators; details can
be found in [25, 14]. Let Ω ⊆ Rd be a bounded domain and µ be a finite positive
Borel measure with supp(µ) ⊆ Ω and µ(Ω) > 0, where Ω is the closure of Ω. Let
∂Ω := Ω\Ω be the boundary of Ω. Let dx be the Lebesgue measure on Rd, and let
H1(Ω) be the Sobolev space equipped with the inner product

⟨u, v⟩H1(Ω) :=

∫
Ω

uv dx+

∫
Ω

∇u · ∇v dx.

Let H1
0 (Ω) be the completion of C∞

c (Ω) under the above inner product, where
C∞

c (Ω) is the space of all smooth functions with compact support in Ω. Let L2(Ω, µ)
be the space of all square integrable functions with respect to µ. The norm in
L2(Ω, µ) is defined by

∥u∥L2(Ω,µ) :=
(∫

Ω

|u|2 dµ
)1/2

.

Throughout this paper, we write L2(Ω) := L2(Ω, dx). It follows from the Poincaré
inequality (see e.g., [25]), that is, there exists a constant C > 0 such that

∥u∥L2(Ω) ≤ C∥∇u∥L2(Ω) .

The space H1
0 (Ω) admits the equivalent inner product defined by

⟨u, v⟩H1
0 (Ω) :=

∫
Ω

∇u · ∇v dx.



4 S.-M. NGAI, M.-K. ZHANG, W.-Q. ZHAO EJDE-2025/12

The lower L∞-dimension of µ is defined as

dim∞(µ) := lim inf
δ→0+

ln(supx µ(Bδ(x)))

ln δ
, (2.1)

where Bδ(x) is the ball with center x and radius δ, and the supremum is taken over
all x ∈ supp(µ) (see [51] for details). To define Krĕın-Feller operators, we need the
following assumption, which is called the Poincaré inequality for measures (MPI):
there exists a constant C > 0 such that∫

Ω

|u|2 dµ ≤ C

∫
Ω

|∇u|2 dx, for all u ∈ C∞
c (Ω).

We know that if dim∞(µ) > d− 2, then µ satisfies (MPI) (see [25, Theorem 1.1]).
(MPI) implies that each equivalence class u ∈ H1

0 (Ω) contains a unique (in L2(Ω, µ)
sense) member û ∈ L2(Ω, µ) that satisfies the following two conditions:

(a) there exists a sequence {un} in C∞
c (Ω) such that un → û in H1

0 (Ω) and
un → û in L2(Ω, µ);

(b) û satisfies the (MPI).

We call û the L2(Ω, µ)-representative of u. Under the assumption (MPI), we define
a map I: H1

0 (Ω) → L2(Ω, µ) by

I(u) = û. (2.2)

The mapping I is in general not injective. We define the following closed subset of
H1

0 (Ω):

N :=
{
u ∈ H1

0 (Ω) : ∥I(u)∥L2(Ω,µ) = 0
}
.

Let N⊥ be the orthogonal complement of N in H1
0 (Ω). Then I : N⊥ → L2(Ω, µ)

is injective. We denote û simply by u if there is no confusion possible.
Consider a nonnegative bilinear form E(·, ·) in L2(Ω, µ) defined as

E(u, v) :=
∫
Ω

∇u · ∇v dx, (2.3)

with dom(E) = N⊥. (MPI) implies that (E ,dom(E)) is a closed quadratic form on
L2(Ω, µ) (see [25, Proposition 2.1]). Hence, there exists a nonnegative self-adjoint
operator −∆µ such that

dom(E) = dom((−∆µ)
1/2),

E(u, v) =
〈
(−∆µ)

1/2u, (−∆µ)
1/2v

〉
L2(Ω,µ)

, for all u, v ∈ dom(E)

(see [12]), where the ⟨·, ·⟩L2(Ω,µ) is the inner product on L
2(Ω, µ). We call the above

∆µ the (Dirichlet) Laplacian with respect to µ or the Krĕın-Feller operator defined
by µ. It follows from [25, Proposition 2.2] that u ∈ dom(∆µ) and −∆µu = f if and
only if −∆u = f dµ in the sense of distribution, i.e.,∫

Ω

∇u · ∇φdx =

∫
Ω

fφ dµ, for all φ ∈ C∞
c (Ω).

3. Maximum principle

In this section, we prove the maximum principle for a continuous µ-subharmonic
function. We first define µ-subharmonic functions.

Definition 3.1. We call u ∈ dom(∆µ) a µ-subharmonic function if ∆µu ≥ 0
(µ-a.e.). Call u ∈ dom(∆µ) a µ-superharmonic function if ∆µu ≤ 0 (µ-a.e.).



EJDE-2025/12 NODAL SETS AND CONTINUITY OF EIGENFUNCTIONS 5

Note that the class of µ-subharmonic (resp. µ-superharmonic) functions and
the class of classical subharmonic (resp. superharmonic) functions are in general
not equal. In fact, the function u in Example 6.4 is µ-superharmonic but not
superharmonic. Nevertheless, we will prove that the maximum principle still holds
for continuous µ-subharmonic (resp. µ-superharmonic) functions. To prove this,
we use mollifiers.

Let Ω ⊆ Rd be a bounded domain and let u ∈ H1
0 (Ω). Let ũ be the zero-extension

of u, i.e.,

ũ(x) =

{
u(x) if x ∈ Ω,

0 if x ∈ Rn \ Ω.

It is known that ũ ∈ H1(Rd) (see [1, Lemma 3.27]). Let ϵ > 0. Define

ũϵ := ηϵ ∗ ũ, (3.1)

where ηϵ ≥ 0 are mollifiers. It is known that for each ϵ > 0, ηϵ is smooth and
satisfies

∫
Rd ηϵ(x) dx = 1. Moreover, supp(ηϵ(x)) ⊆ Bϵ(0) and

supp(ηϵ(x− ·)) ⊆ Bϵ(x) (3.2)

(see [16, 1] for details). The following proposition follows from [1, Theorem 2.29].

Proposition 3.2. Let ũϵ be defined in (3.1). Then

(a) ũϵ ∈ C∞(Rd).
(b) If u ∈ C(Ω), then ũϵ → u uniformly on Ω.
(c) ũϵ → u in H1(Ω) as ϵ→ 0.

Proposition 3.3. Let Ω ⊆ Rd (d ≥ 1) be a bounded domain and µ be a positive
finite Borel measure on Rd with supp(µ) ⊆ Ω and µ(Ω) > 0. Assume that (MPI)
holds. Let u ∈ dom(∆µ) be a µ-subharmonic function and ũϵ be defined as in (3.1).
Let z ∈ Ω. Then for any 0 < r < dist(z, ∂Ω),

lim
ϵ→0

∫
Br(z)

∆(ũϵ|Ω) dx =

∫
Br(z)

∆µu dµ.

Proof. Let z ∈ Ω and ϵ > 0 be sufficiently small so that ϵ < dist(z, ∂Ω)/4. Let
r ∈ (ϵ,dist(z, ∂Ω)− 3ϵ). Then by Proposition 3.2(a), we have∫

Br(z)

∆(ũϵ|Ω) dx =

∫
Br(z)

∆
(
(ηϵ ∗ ũ)|Ω

)
dx

=

∫
Br(z)

(∆ηϵ ∗ ũ)|Ω dx

=

∫
Br(z)

∫
Bϵ(x)

∆ηϵ(x− y)ũ(y) dydx. (by (3.2))

Using (3.2) and the fact that ηϵ(x− y) = 0 on ∂Bϵ(x) (see [1, 16]), we have∫
Br(z)

∆(ũϵ|Ω) dx = −
∫
Br(z)

∫
Bϵ(x)

∇ηϵ(x− y) · ∇ũ(y) dydx

= −
∫
Br(z)

∫
Bϵ(x)

∇ηϵ(x− y) · ∇u(y) dydx

=

∫
Br(z)

∫
Bϵ(x)

ηϵ(x− y)∆µu(y) dµ(y)dx,



6 S.-M. NGAI, M.-K. ZHANG, W.-Q. ZHAO EJDE-2025/12

where the second inequality holds as ũ = u in Bϵ(x), and the last equality follows
from [25, Proposition 2.2] and (3.2). Therefore, if we let χBϵ(x) be the characteristic
function on Bϵ(x), then∫

Br(z)

∆(ũϵ|Ω) dx

=

∫
Br(z)

∫
Br+2ϵ(z)

χBϵ(x)(y)ηϵ(x− y)∆µu(y) dµ(y)dx

=

∫
Br+2ϵ(z)

∫
Br(z)

χBϵ(x)(y)ηϵ(x− y)∆µu(y) dxdµ(y) (Fubini)

=

∫
Br+2ϵ(z)

∫
Br(z)∩Bϵ(y)

χBϵ(y)(x)ηϵ(x− y)∆µu(y) dxdµ(y)

≤
∫
Br+2ϵ(z)

(∫
Bϵ(y)

ηϵ(x− y) dx
)
∆µu(y) dµ(y)

=

∫
Br+2ϵ(z)

∆µu dµ

(3.3)

(see Figure 2(a)). On the other hand, since ηϵ ≥ 0 and ∆µu ≥ 0 µ-a.e., by (3.3) we
have ∫

Br(z)

∆(ũϵ|Ω) dx

=

∫
Br+2ϵ(z)

∫
Br(z)∩Bϵ(y)

χBϵ(y)(x)ηϵ(x− y)∆µu(y) dxdµ(y)

≥
∫
Br−ϵ(z)

∫
Br(z)∩Bϵ(y)

χBϵ(y)(x)ηϵ(x− y)∆µu(y) dxdµ(y)

=

∫
Br−ϵ(z)

(∫
Bϵ(y)

ηϵ(x− y) dx
)
∆µu(y) dµ(y)

=

∫
Br−ϵ(z)

∆µu dµ

(3.4)

(see Figure 2(b)).
Combining (3.3) and (3.4), we have∫

Br−ϵ(z)

∆µu dµ ≤
∫
Br(z)

∆(ũϵ|Ω) dx ≤
∫
Br+2ϵ(z)

∆µu dµ.

Letting ϵ→ 0 completes the proof. □

Remark 3.4. We know that ∆µu ∈ L2(Ω, µ) ⊆ L1(Ω, µ) for u ∈ dom(∆µ) (see
[25, Proposition 2.2]). Therefore, if u is a µ-subharmonic function, then for any
Br(z) ⊆ Ω, the limit

lim
ϵ→0

∫
Br(z)

∆(ũϵ|Ω) dx =

∫
Br(z)

∆µu dµ

is nonnegative and finite.

In [52, Theorem 3.3], it is shown that if u ∈ C2(Ω) ∩ C(Ω) is a µ-subharmonic
function, then the maximum principle holds. The following theorem generalizes
this theorem to u ∈ C(Ω).



EJDE-2025/12 NODAL SETS AND CONTINUITY OF EIGENFUNCTIONS 7

z

r

r+2ϵ

y
ϵ

x

Ω

z

r

r-ϵ

y
ϵ
x

Ω

(a) (b)

Figure 1. Sets Br(z), Br+2ϵ(z), Br−ϵ(z), Bϵ(y) and the positions
of the points x, y, z.

Theorem 3.5. Let Ω ⊆ Rd (d ≥ 1) be a bounded domain and µ be a positive finite
Borel measure on Rd with supp(µ) ⊆ Ω and µ(Ω) > 0. Assume that (MPI) holds.
If u ∈ C(Ω) is a nonconstant µ-subharmonic function, then u cannot attain its
maximum value in Ω.

Proof. Some basic derivations are similar to the proof of the mean-value formula
(see e.g., [16, §2.2.2 Theorem 2]); we will omit some details. Let u be a nonconstant
continuous µ-subharmonic function. For any fixed x ∈ Ω, let ũϵ be defined as in
(3.1) and let r > 0 be sufficiently small such that Br(x) ⊆ Ω. Define

φϵ(r) := –

∫
∂Br(x)

ũϵ|Ω(y) dS(y) = –

∫
∂B1(0)

ũϵ|Ω(x+ rz) dS(z),

where

–

∫
∂Br(x)

f dS :=
1

nα(n)rn−1

∫
∂Br(x)

f dS (3.5)

is the average of f over the sphere ∂Br(x) and α(n) = πn/2
/
Γ(n/2 + 1) is the

volume of the unit ball B1(0) in Rn. Let

–

∫
Br(x)

f dy :=
1

α(n)rn

∫
Br(x)

f dy (3.6)

be the average of f over Br(x) (see [16, Appendix A]). By the calculation as in the
proof of [16, §2.2.2 Theorem 2], we have

φ′
ϵ(r) = –

∫
∂B1(0)

Dũϵ|Ω(x+ rz) · z dS(z)

=
1

nα(n)rn−1

∫
Br(x)

∆(ũϵ|Ω)(y) dy.

By Remark 3.4, limϵ→0 φ
′
ϵ(r) ≥ 0. It follows that for each t > 0, there exists δt > 0

such that for all ϵ ∈ (0, δt), φ
′
ϵ(r) + t ≥ 0. This implies that for each ϵ ∈ (0, δt),

φϵ(r) + tr is an increasing function of r. For each t > 0, we choose ϵt > 0 so that



8 S.-M. NGAI, M.-K. ZHANG, W.-Q. ZHAO EJDE-2025/12

the function ϵt is decreasing and tends to 0 as t→ 0+. Moreover, φϵt(r) + tr is an
increasing function of r. Letting t→ 0+ and using Proposition 3.2(b), we have

lim
t→0+

(φϵt(r) + tr) = lim
ϵt→0

–

∫
∂Br(x)

ũϵt |Ω(y) dS(y) = –

∫
∂Br(x)

u(y) dS(y) =: φ(r).

Observe that φ(r) is an increasing function of r. Hence, by using this and the
continuity of u, we have that for all s > 0,

φ(s) ≥ lim
ξ→0

φ(ξ) = lim
ξ→0

–

∫
∂Bξ(x)

u(y) dS(y) = u(x). (3.7)

Using (3.6) and the equation∫
Br(x)

u(y) dy =

∫ r

0

(∫
∂Bs(x)

u(y) dS(y)
)
ds,

which can be derived by using [16, §C3]. Then by (3.5) and (3.7) we have

–

∫
Br(x)

u(y) dy =
1

α(n)rn

∫ r

0

(∫
∂Bs(x)

u(y) dS(y)
)
ds

=
1

α(n)rn

∫ r

0

nα(n)sn−1
(
–

∫
∂Bs(x)

u(y) dS(y)
)
ds

≥ 1

α(n)rn

∫ r

0

u(x)nα(n)sn−1

=
u(x)

rn

∫ r

0

nsn−1 ds

= u(x).

(3.8)

Suppose there exists a point x0 ∈ Ω such that u(x0) =M := maxΩ u(x). Then for
0 < r < dist(x0, ∂Ω), it follows from (3.8) that

M = u(x0) ≤ –

∫
Br(x0)

u dy ≤M.

Hence,

–

∫
Br(x0)

u dy =M.

Thus u(y) =M for all y ∈ Br(x0). Since Ω is a domain, and hence is connected, it
follows that u(x) =M for all x ∈ Ω. □

Remark 3.6. (a) By replacing u in the proof of Theorem 3.5 with −u, one can
prove that a nonconstant continuous µ-superharmonic function attains its minimum
only on ∂Ω.

(b) From the proof of Theorem 3.5, one can see that the reason for introducing
the function ũϵ|Ω is that ∆u may exist as a distribution but need not exist as a
function (see Example 6.4).

Let Ω be a bounded domain. For a continuous µ-harmonic function u, i.e.,
∆µu = 0, we have the following proposition.

Proposition 3.7. Let Ω be a bounded domain. If u ∈ C(Ω) is a µ-harmonic
function on Ω and vanishes on ∂Ω, then u ≡ 0.



EJDE-2025/12 NODAL SETS AND CONTINUITY OF EIGENFUNCTIONS 9

To prove Proposition 3.7, we need the following Weyl’s Lemma (see [26, Corollary
2.2.1] or [53, 6, 21] and references therein).

Lemma 3.8 (Weyl’s Lemma). Let u : Ω → R be measurable and locally integrable
on Ω. Suppose that for any v ∈ C∞

c (Ω),∫
Ω

u(x)∆v(x) dx = 0.

Then u is harmonic and, in particular, smooth.

Proof of Proposition 3.7. Let u be µ-harmonic, i.e., ∆µu = 0. Then for all v ∈
C∞

c (Ω),

0 =

∫
Ω

v∆µu dµ = −
∫
Ω

∇v · ∇u dx =

∫
Ω

u∆v dx.

Moreover, u is locally integrable as u ∈ dom(∆µ) ⊂ H1
0 (Ω). It follows from Lemma

3.8 that u is harmonic and smooth on Ω. Therefore, by the classical maximum
principle (see [16, §6.4]) and the fact that u vanish on ∂Ω, we have u ≡ 0. □

4. Courant nodal domain theorem

Let Ω ⊆ Rd be a bounded domain and µ be a positive finite Borel measure on Rd

with supp(µ) ⊆ Ω and µ(Ω) > 0. Let E(u, u) be defined as in (2.3). We define the
Rayleigh quotient of µ, an important and useful quantity in studying eigenvalues.
For related applications, see [3, 22, 50, 14, 9].

Definition 4.1. Use the above assumption and notation. For any u ∈ dom(E), the
Rayleigh quotient associated with µ is defined as

Rµ(u) :=
E(u, u)

(u, u)L2(Ω,µ)
=

∫
Ω
|∇u|2 dx∫

Ω
|u|2 dµ

.

The following lemma is inspired by [14, 43]. It can be proved by using similar
methods as in the proofs of [14, Theorem 1.3] and [43, Corollary 4.3]; we omit the
proof.

Lemma 4.2. Let λ1, . . . , λn be eigenvalues of (1.2) and let u1(x), . . . , un(x) be
corresponding eigenfunctions. Then

(a) λ1 = min
{
Rµ(u) : u ∈ dom(E)

}
and Rµ(u1) = λ1. Moreover, if there exists

some u ∈ dom(E) such that Rµ(u) = λ1, then u is a λ1-eigenfunction.
(b) For n ≥ 2, λn = min

{
Rµ(u) : u ∈ dom(E), (u, ui)L2(Ω,µ) = 0, i =

1, . . . , n− 1
}
and Rµ(un) = λn. Moreover, if there exists some u ∈ dom(E)

such that (u, ui)L2(Ω,µ) = 0 for i = 1, . . . , n− 1, and Rµ(u) = λn, then u is
a λn-eigenfunction.

For each f ∈ H1
0 (Ω), under the assumption dim∞(µ) > d − 2, it is known that

there exists a unique L2(Ω, µ)-representative f̂ of f . Let

Hµ(Ω) := {f ∈ H1
0 (Ω) : ∥f̂∥L2(Ω,µ) > 0}.

For each f ∈ Hµ(Ω), we define

R̃µ(f) :=

∫
Ω
|∇f |2 dx∫

Ω
|f̂ |2 dµ

.



10 S.-M. NGAI, M.-K. ZHANG, W.-Q. ZHAO EJDE-2025/12

Lemma 4.3. Assume the hypotheses of Lemma 4.2. For n ≥ 1, let

Hn
µ(Ω) :=

{
Hµ(Ω) if n = 1,

{f ∈ Hµ(Ω) : (f̂ , ui)L2(Ω,µ) = 0, i = 1, . . . , n− 1} if n ≥ 2.

Then λn = min
{
R̃µ(f) : f ∈ Hn

µ(Ω)
}
. Moreover, for each n ≥ 1, if R̃µ(f) = λn

holds for some f ∈ Hn
µ(Ω), then f is a λn-eigenfunction.

Proof. For any fixed n ≥ 1, we first prove that if R̃µ(f) attains minimum at f in
Hn

µ(Ω), then f ∈ dom(E). In fact, for any f ∈ Hn
µ(Ω), let f = fE +fN be the direct

sum of f , where fE ∈ dom(E) = N⊥ and fN ∈ N . Then

⟨fE , fN ⟩H1
0 (Ω) =

∫
Ω

∇fE · ∇fN dx = 0. (4.1)

Moreover, ∥fE∥L2(Ω,µ) = ∥f̂∥L2(Ω,µ) as ∥fN ∥L2(Ω,µ) = 0. Thus,

R̃µ(f) =

∫
Ω
|∇f |2 dx∫

Ω
|f̂ |2 dµ

=

∫
Ω
|∇fE +∇fN |2 dx∫

Ω
|fE |2 dµ

=

∫
Ω
|∇fE |2 + |∇fN |2 dx∫

Ω
|fE |2 dµ

. (by (4.1))

(4.2)

Since fE ∈ Hn
µ(Ω), it follows by (4.2) that R̃µ(f) attains minimum in Hµ(Ω) if and

only if ∥fN ∥H1
0 (Ω) = 0, i.e., f = fE ∈ dom(E). Moreover, we can conclude that

min{R̃µ(f)|f ∈ Hµ} = min{Rµ(u)|u ∈ dom(E)} = λ1 (4.3)

and for n ≥ 2,

min{R̃µ(f)|f ∈ Hn
µ}

= min{Rµ(u)|u ∈ dom(E), (u, ui)L2(Ω,µ) = 0, i = 1, . . . , n− 1}
= λn.

(4.4)

Therefore, the last assertion of the lemma follows from Lemma 4.2. □

Proof of theorem 1.1. We follow [50] for the proof of (a). We use some methods
and techniques in [22] to prove (b).
(a) We divide the proof into two steps as follows: Step 1. Suppose on the contrary

that the λ1-eigenfunction u1 has a node. i.e., there exists x0 ∈ Ω such that

u1(x0) = 0. (4.5)

Let
Ω+ := {x ∈ Ω |u1(x) > 0} and Ω− := {x ∈ Ω |u1(x) < 0}.

We claim that Ω+ and Ω− are nonempty. In fact, if Ω+ = ∅, then for any x ∈ Ω,

u1(x) ≤ 0. (4.6)

Since −∆µu1 = λ1u1 and λ1 > 0, we have −∆µu1(x) = λ1u1(x) ≤ 0. Thus

∆µu1(x) ≥ 0 for all x ∈ Ω.

Hence u1 is a µ-subharmonic function. Combining this with Theorem 3.5, we see
that u1 cannot attain 0 in Ω. Thus, by (4.6),

u1(x) < 0 for all x ∈ Ω,

which contradicts (4.5). Hence Ω+ ̸= ∅. By the same argument, Ω− ̸= ∅.



EJDE-2025/12 NODAL SETS AND CONTINUITY OF EIGENFUNCTIONS 11

Step 2. Let

u+(x) =

{
u1(x), x ∈ Ω+,

0, x ∈ Ω \ Ω+,

and let u−(x) = u1(x)− u+(x). Note that |u1(x)| = u+(x)− u−(x) and

∇u+(x) =

{
∇u1(x), x ∈ Ω+,

0, x ∈ Ω\Ω+,
∇u−(x) =

{
0, x ∈ Ω\Ω−,

∇u1(x) x ∈ Ω−.

Obviously, u+ and u− belong to H1
0 (Ω). It follows by the linearity of H1

0 (Ω) that
|u1| ∈ H1

0 (Ω).

Step 3. Since |u1| ∈ H1
0 (Ω) and ∥|u1|∥L2(Ω,µ) > 0, we have |u1| ∈ Hµ and thus

R̃µ

(
|u1|

)
=

∫
Ω

∣∣∇|u1|
∣∣2 dx∫

Ω
|u1|2 dµ

=

∫
Ω+

∣∣∇u1∣∣2 dx∫
Ω
|u1|2 dµ

+

∫
Ω−

∣∣∇u1∣∣2 dx∫
Ω
|u1|2 dµ

=

∫
Ω
|∇u1|2 dx∫

Ω
|u1|2 dµ

=

∫
Ω
(−∆µu1) · u1 dµ∫

Ω
|u1|2 dµ

=
λ1

∫
Ω
u21 dµ∫

Ω
|u1|2 dµ

= λ1.

It follows from Lemma 4.2(a) that λ1 = min{Rµ(u) : u ∈ dom(E)}. Combining this
and (4.3), we have

R̃µ(|u1|) = λ1.

Hence, by Lemma 4.3, |u1| is a λ1-eigenfunction, i.e., −∆µ|u1| = λ1|u1|. Combining
this with λ1 > 0, we have

∆µ|u1(x)| ≤ 0 for all x ∈ Ω.

Hence |u1| is a µ-superharmonic function. By Remark 3.6, u1 cannot attain 0 in
Ω. Thus,

|u1(x)| > 0 for all x ∈ Ω,

i.e., |u1| does not have nodes in Ω. This contradicts Step 1 and completes the proof
of the first part of the theorem.

(b) By the proof of (a), u1(x) ̸= 0, for any x ∈ Ω. Without loss of generality, we
assume u1(x) > 0. Since un is orthogonal to u1, i.e.,∫

Ω

unu1 dµ = 0,

un must change sign in Ω. Thus un must be positive on some subdomians of Ω and
negative on some other subdomains of Ω. By the continuity of un, these subdomains
must be separated by the nodal set of un. Hence, for n ≥ 2, the nodal set of un
divides Ω into at least two subdomains.

For the second part of (b), let Zµ be defined as in (1.3). Then Ω\Zµ(un) = {x ∈
Ω |un(x) ̸= 0}. Assume Zµ(un) divides Ω intom (m ≥ 2) subdomains: Ω1, . . . ,Ωm,
where the Ωi are pairwise disjoint and separated by a subset of Zµ(un). Moreover,

Ω \ Zµ(un) = ∪m
j=1Ωj .

Let

wj(x) =

{
un(x), x ∈ Ωj ,

0, x ∈ Ω\Ωj .



12 S.-M. NGAI, M.-K. ZHANG, W.-Q. ZHAO EJDE-2025/12

Then

∇wj(x) =

{
∇un(x), x ∈ Ωj ,

0, x ∈ Ω\Ωj .

Let

w =

m∑
j=1

cjwj , (4.7)

where c1, . . . , cm are arbitrary constants. Note that w ∈ Hµ and

R̃µ(w) =

∫
Ω
|∇w|2 dx∫

Ω
|w|2 dµ

=

∑m
j=1 c

2
j

∫
Ωj

|∇un|2 dx∑m
j=1 c

2
j

∫
Ωj

|un|2 dµ

=

∑m
j=1 c

2
j

∫
Ωj
(−∆µun) · un dµ∑m

j=1 c
2
j

∫
Ωj

|un|2 dµ
=
λn

∑m
j=1 c

2
j

∫
Ωj
u2n dµ∑m

j=1 c
2
j

∫
Ωj

|un|2 dµ
= λn.

(4.8)

Since the system (4.9) below hasm−1 equations inm unknowns cj , it has a nonzero
solution {c1, . . . , cm}. Hence, we can choose the coefficients {cj}mj=1 of w in (4.7)
so that (

w, ui
)
L2(Ω,µ)

= 0, i = 1, . . . ,m− 1, (4.9)

where {ui}m−1
i=1 are the first m − 1 eigenfunctions. For this choice of {cj}mj=1,

w ∈ Hm
µ . By (4.4), we have

R̃µ(w) ≥ λm.

Combining this and (4.8), we have λm ≤ λn. Since λn < λn+rn , we have λm <
λn+rn . Thus m ≤ n+ rn − 1. Therefore, the nodal set of un divides Ω into at most
n+ rn − 1 subdomains. □

From Theorem 1.1(a), we can immediately derive the following corollary.

Corollary 4.4. The multiplicity of the first eigenvalue λ1 is 1.

Proof. Suppose, on the contrary, that the multiplicity of λ1 is not 1. Then there ex-
ists another λ1-eigenfunction v so that u1 and v are linearly independent in L2(Ω, µ).
Write v = c1u1 +w in L2(Ω, µ), where c1 is a constant and w ∈ (span(u1))

⊥. Thus∫
Ω

u1w dµ = 0. (4.10)

As in the proof of Theorem 1.1, w does not change sign in Ω. Combining this and
(4.10), we have w(x) = 0 (µ-a.e.) in Ω. Thus v = c1u1 in L2(Ω, µ), contradicting
the fact that u1 and v are linearly independent in L2(Ω, µ). This completes the
proof. □

5. Continuity of eigenfunctions

In this section, we prove Theorem 1.2. Let Ω ⊆ Rd be a bounded domain. We
recall the Green function of the classical Laplacian ∆. For u ∈ C2(Ω),

∆u =

d∑
i=1

∂2u

∂x2i
.

Let

g(x,y) =

{
− 1

2π ln |x− y| if d = 2,

−|x− y|2−d if d ≥ 3.
(5.1)



EJDE-2025/12 NODAL SETS AND CONTINUITY OF EIGENFUNCTIONS 13

The following definition of Green’s function follows from [15].

Definition 5.1. Let d ≥ 2, Ω ⊆ Rd be a bounded domain. If there exists a real
valued function hΩ(·, ·) on Ω×Ω such that for each x ∈ Ω, hΩ(x, ·) is the harmonic
minorant of g(x, ·) on Ω, i.e., hΩ(x, ·) is the greatest harmonic function satisfying

hΩ(x, ·) ≤ g(x, ·) (5.2)

on Ω, then the function

G(x,y) := g(x,y)− hΩ(x,y) (5.3)

is called the Green function of Ω.

It is known that hΩ(x,y), if it exists, is a symmetric continuous function on
Ω × Ω [25]. Hence the Green function G(x,y) is symmetric on Ω × Ω [15]. Some
basic properties of the Green function are summarized below. Fix any point x ∈ Ω.

(a) G(x, ·) is defined on Ω× Ω and G(x,x) = +∞ [15, Chapter VII.4].
(b) G(x, ·) is continuous and harmonic on Ω− {x} [15, Chapter VII.4].
(c) G(x,y) ≥ 0 for any y ∈ Ω (by (5.2)).

For a bounded domain on which the Green function exists, we provide an equivalent
definition of a regular boundary (see [15, VIII 14] or [35]).

Definition 5.2. A bounded domain Ω ⊆ Rd (d ≥ 2) on which the Green function
exists is said to have a regular boundary if for any z ∈ ∂Ω and y ∈ Ω,

lim
x→z

G(x,y) = 0.

Remark 5.3. For any open set Ω ⊆ Rd, the Green function always exists when
d ≥ 3. When d = 2, the Green function exists if R2 \ ∂Ω is not connected (see [4,
Theorm 4.1.2] and [35, 5]). Examples of domains with regular boundary include
those with smooth or Lipschitz boundaries (see [35, Section 4]).

For f ∈ C1(Ω), the unique solution of the equation in C2(Ω):

−∆u = f

u|∂Ω = 0

can be represented through the Green function G(x,y) by

u(x) =

∫
Ω

G(x,y)f(y) dy.

More details about the Green function can be found in, for example, [4, 16, 35, 10,
11, 15].

According to [25], the Green function G(x,y) for ∆, if exists, is also the Green
function for ∆µ. It means that for the equation −∆µu = f, there exists a Green
operator defined on Lp(Ω, µ) (p ≥ 1) by

(Gµf)(x) :=

∫
Ω

G(x,y)f(y) dµ(y) (5.4)

such that u = Gµf . The operator Gµ is the inverse of −∆µ [25, Theorem 1.3]. To
ensure that Gµ has good properties, we need the following assumption in [25]:

sup
x∈Ω

∫
Ω

G(x,y) dµ(y) ≤ C < +∞ for some constant C > 0. (5.5)

It is proved in [25] that the condition dim∞(µ) > d− 2 implies (5.5).



14 S.-M. NGAI, M.-K. ZHANG, W.-Q. ZHAO EJDE-2025/12

Proposition 5.4. Let Ω ⊆ Rd be a bounded domain on which the classical Green
function G(x,y) exists and let f ∈ dom(∆µ). Let µ be a finite positive Borel

measure with supp(µ) ⊆ Ω. Assume dim∞(µ) > d− 2. Then Gµf is bounded, i.e.,

there exists a constant C̃ > 0 such that |Gµf(x)| ≤ C̃ for all x ∈ Ω.

Proof. Since the case d = 1 is clear, we divide the proof into two cases: d = 2 and
d ≥ 3.

Case 1. d = 2. We claim that for each x ∈ Ω, G(x,y) ∈ L2(Ω, µ). By (5.3), it

suffices to prove that there exists some constant C̃ > 0 such that∫
Ω

(ln |x− y|)2 dµ(y) ≤ C̃ (5.6)

for all x ∈ Ω. Using the same method in the proof of [25, Proposition 4.1], one can
prove that (5.6) holds. Hence,∣∣Gµf(x)

∣∣ = ∣∣∣ ∫
Ω

G(x,y)f(y) dµ(y)
∣∣∣ ≤ ∥G(x, ·)∥L2(Ω,µ)∥f∥L2(Ω,µ). (5.7)

Case 2. d ≥ 3. Since f ∈ dom(∆µ) ⊆ H1
0 (Ω) and dim∞(µ) > d− 2, there exists a

sequence fm ∈ C∞
c (Ω) such that fm → f in L2(Ω, µ). We claim that there exists

some constant C1 > 0 such that

lim
m→∞

∣∣Gµ(f
2 − f2m)

∣∣ ≤ C1. (5.8)

To see this, by (5.3), it suffices to prove that

lim
m→∞

∫
Ω

|g(x,y)| · |f2(y)− f2m(y)| dµ(y) ≤ C1.

Note that, by Hölder’s inequality,∫
Ω

|f2 − f2m| dµ =

∫
Ω

|f − fm||f + fm| dµ

≤ ∥f − fm∥L2(Ω,µ)∥f + fm∥L2(Ω,µ) → 0

(5.9)

as m→ ∞. Let diam(Ω) = r0. We have∫
Ω

|g(x,y)||f2(y)− f2m(y)| dµ(y)

=

∫
|x−y|≤1

|x− y|−(d−2)|f2(y)− f2m(y)| dµ(y)

+

∫
1≤|x−y|≤r0

|x− y|−(d−2)|f2(y)− f2m(y)| dµ(y).

By (5.9), the second integral on the right-hand side tends to 0, since∫
1≤|x−y|≤r0

|x− y|−(d−2)|f2(y)− f2m(y)| dµ(y)

≤
∫
1≤|x−y|≤r0

|f2(y)− f2m(y)| dµ(y).
(5.10)



EJDE-2025/12 NODAL SETS AND CONTINUITY OF EIGENFUNCTIONS 15

Furthermore, we let Vk(x) := {y : 2−k ≤ |x− y| ≤ 2−(k−1)}. Then∫
|x−y|≤1

|x− y|−(d−2)|f2(y)− f2m(y)| dµ(y)

=

∞∑
k=1

∫
Vk(x)

|x− y|−(d−2)|f2(y)− f2m(y)| dµ(y)

≤ lim
N→∞

N∑
k=1

2k(d−2)

∫
Vk(x)

|f2(y)− f2m(y)| dµ(y).

(5.11)

By (5.9), for each k ∈ {1, 2, . . . , N}, there exists mN sufficiently large such that∫
Vk(x)

|f2(y)− f2mN
(y)| dµ(y) ≤ 2−2k(d−2).

Hence
N∑

k=1

2k(d−2)

∫
Vk(x)

|f2(y)− f2mN
(y)| dµ(y) ≤

N∑
k=1

2−k(d−2). (5.12)

Letting N → ∞, we have, by (5.11) and (5.12),∫
|x−y|≤1

|x− y|−(d−2)|f2(y)− f2m(y)| dµ(y) ≤
∞∑
k=1

2−k(d−2) < +∞.

Combining this and (5.10) completes the proof of the claim in (5.8). It follows from
(5.8) that there exists some sufficiently large integer N0 such that for all m > N0

and all x ∈ Ω, ∣∣Gµ(f
2(x)− f2m(x))

∣∣ ≤ C1 + 1.

It follows from Property (c) that for any x,y ∈ Ω, G(x,y) ≥ 0. Hence,∣∣Gµf
2
∣∣ ≤ ∣∣Gµf

2
N0+1

∣∣+ C2

≤
∫
Ω

∣∣G(x,y)f2N0+1

∣∣ dµ+ C2 (by (5.4))

≤ ∥f2N0+1∥L∞(Ω)

∫
Ω

G(x,y) dµ(y) + C2

≤ C∥f2N0+1∥L∞(Ω) + C2 (by (5.5))

=: C3,

where C2 = C1 + 1. This proves that∣∣Gµf
2(x)

∣∣ = ∫
Ω

G(x,y)f2(y) dµ(y) ≤ C3. (5.13)

Now, for d ≥ 3, by Hölder’s inequality, (5.5) and (5.13), we have∣∣Gµf(x)
∣∣2 =

∣∣∣ ∫
Ω

G(x,y)f(y) dµ(y)
∣∣∣2

≤
(∫

Ω

∣∣G(x,y)1/2f(y) ·G(x,y)1/2∣∣ dµ(y))2

≤
∫
Ω

G(x,y)f2(y) dµ(y) ·
∫
Ω

G(x,y) dµ(y)

≤ C · C3. (by (5.13))



16 S.-M. NGAI, M.-K. ZHANG, W.-Q. ZHAO EJDE-2025/12

Therefore
∣∣Gµf

∣∣ is bounded by
√
C · C3. Combining this and (5.7) completes the

proof. □

We give an outline of the proof of Theorem 1.2. First, we prove any λ-eigenfunction
u can be expressed as λGµu. Second, we use the properties of Green’s function,
Lebesgue’s dominated convergence theorem and ideas from [35, Proposition 1.26.7]
to study the continuity of Gµu on Ω.

Proof of Theorem 1.2. Let f ∈ dom(∆µ), we claim that f is a λ-eigenfunction of
∆µ if and only if f = λGµf . In fact, on the one hand, if f ∈ dom(∆µ) satisfies
f = λGµf , then by [25, Theorem 1.3], we have

−∆µf = −λ∆µGµf = λf,

which implies that f is a λ-eigenfunction of ∆µ. On the other hand, if f ∈ dom(∆µ)

is a λ-eigenfunction, i.e., −∆µf = λf , in view of the fact that Gµ =
(
−∆µ

)−1
(see

[25, Theorem 1.3]), we have

Gµ(λf) = Gµ(−∆µf) = f.

The linearity of Gµ implies that f = λGµf . Therefore, to prove Theorem 1.2, it
suffices to prove that for any f ∈ dom(∆µ), Gµf is continuous. We divide the proof
into three steps.

Step 1. We claim that for any ϵ > 0, there exists some f1 ∈ L2(Ω, µ) such that
|Gµf1| < ϵ. In the case d = 2, by (5.6), we see that for each y ∈ Ω, G(·,y) belongs
to L2(Ω, µ), and ∥G(·,y)∥L2(Ω,µ) has a uniform bound independent of y. Hence,

there exists a constant Ĉ1 such that for all y ∈ Ω, ∥G(·,y)∥L2(Ω,µ) ≤ Ĉ1. Since
dim∞(µ) > d−2 = 0, µ does not have point masses, i.e., µ is a continuous measure
[43, Proposition 6.5]. Therefore, for any z ∈ Ω,

lim
r→0+

∫
Br(z)

|G(x,y)|2 dµ(y) = 0 for all x ∈ Ω.

Hence, for any ϵ > 0, there exists r̃1 > 0 sufficiently small such that for all x ∈ Ω,∫
Br̃1

(z)

|G(x,y)|2 dµ(y) ≤ ϵ2. (5.14)

Now consider the case d ≥ 3. Let ϵ > 0. Then by (5.5) and the continuity of µ
again, there exists r̃2 > 0 sufficiently small such that for all x ∈ Ω,∫

Br̃2
(z)

G(x,y) dµ(y) < ϵ2. (5.15)

Let r̃ := min{r̃1, r̃2} and f1 := fχBr̃(z). For the case d = 2, by Hölder’s inequality
and (5.14), we have

|Gµf1| =
∣∣∣ ∫

Ω

G(x,y)f1(y) dµ(y)
∣∣∣

≤ ∥f1∥L2(Ω,µ)

(∫
Br̃(z)

G2(x,y) dµ(y)
)1/2

≤ ϵ∥f∥L2(Ω,µ).

(5.16)

For the case d ≥ 3, by Hölder’s inequality and (5.15), we have∣∣Gµf1
∣∣2 =

∣∣∣ ∫
Ω

G(x,y)f1(y) dµ(y)
∣∣∣2



EJDE-2025/12 NODAL SETS AND CONTINUITY OF EIGENFUNCTIONS 17

≤
∫
Br̃(z)

G(x,y)f21 (y) dµ(y) ·
∫
Br̃(z)

G(x,y) dµ(x)

≤ ϵ2
∫
Ω

G(x,y)f2(y) dµ(y).

Combining this and (5.13), we have∣∣Gµf1
∣∣ ≤ C3 · ϵ. (5.17)

Therefore, combining (5.16) and (5.17) proves the claim.

Step 2. Let f2 := f − f1. We claim that Gµf2 is continuous at z. In fact, by the
continuity of G(x,y), for any y ∈ Ω \Br̃(z), G(z,y) is continuous at z, i.e.,

lim
x→z

G(x,y) = G(z,y). (5.18)

Note that by (5.1) and (5.3) there exists δ ∈ (0, r̃/2) such that for any x ∈ Bδ(z)
and all y ∈ Ω \Br̃(z),

|G(x,y)| ≤ C4 := max
{∣∣ log r̃

2

∣∣+ C5,
( r̃
2

)2−d
+ C5

}
,

where C5 > 0 is a constant. Moreover, f2 ∈ L1(Ω, µ) as ∥f2∥L1(Ω,µ) ≤ ∥f∥L1(Ω,µ).
Therefore, by (5.18) and Lebesgue’s dominated convergence theorem, we have

lim
x→z

Gµf2(x) = lim
x→z

∫
Ω

G(x,y)f2(y) dµ(y)

= lim
x→z

∫
Ω\Br̃(z)

G(x,y)f2(y) dµ(y)

=

∫
Ω\Br̃(z)

G(z,y)f2(y) dµ(y)

= Gµf2(z).

Step 3. By Step 2, for any ϵ > 0, there exists δ > 0 such that for any z̃ ∈ Bδ(z),∣∣Gµf2(z)−Gµf2(z̃)
∣∣ < ϵ. Combining this, Step 1 and the definitions of f1 and f2,

we have

|Gµf(z)−Gµf(z̃)| = |Gµf1(z) +Gµf2(z)−Gµf1(z̃)−Gµf2(z̃)|
≤ |Gµf1(z)|+ |Gµf1(z̃)|+ |Gµf2(z)−Gµf2(z̃)|
≤ 3ϵ,

which shows that Gµf is continuous at z. Since z is arbitrary, Gµf is continuous
on Ω.

Finally, if the boundary of Ω is regular, then, applying the argument in Steps
1–3 to Ω, we can prove that the eigenfunctions of ∆µ are continuous on Ω. □

6. Examples of continuous eigenfunctions

In this section, we assume that Ω ⊆ R2. We will construct some examples of
continuous eigenfunctions of ∆µ. As mentioned in Section 1, we are interested in
the case that the measures are singular with respect to Lebesgue measure.

Let D(Ω) := C∞
c (Ω) be the vector space of test functions. Recall that a distri-

bution T : D(Ω) → C is a continuous linear map [23, 49]. For any locally integrable



18 S.-M. NGAI, M.-K. ZHANG, W.-Q. ZHAO EJDE-2025/12

function f ∈ L1
loc(Ω), define a distribution Tf as

⟨Tf , v⟩ :=
∫
Ω

fv dx, v ∈ D(Ω).

The i-th partial distributional derivative ∂T/∂xi of a distribution T is defined by〈 ∂T
∂xi

, v
〉
= −

〈
T,

∂v

∂xi

〉
, v ∈ D(Ω).

We consider the square domain Ω = (−1, 1)× (−1, 1). To describe the distribu-
tional derivative of functions we construct, we need to define the following specific
distributions in D′(Ω), where D′(Ω) is the dual space of D(Ω).

Definition 6.1. Let Ω = (−1, 1) × (−1, 1) and β ∈ (−1, 1) be a fixed constant.
Define δI,β and δβ,II as distributions in D′(Ω) so that the following equations hold
for all v(x, y) ∈ D(Ω):〈

δI,β , v(x, y)
〉
:=

∫
Ω

v(x, y) dxdδβ(y) =

∫ 1

−1

v(x, β) dx, (6.1)

〈
δβ,II, v(x, y)

〉
:=

∫
Ω

v(x, y) dδβ(x)dy =

∫ 1

−1

v(β, y) dy, (6.2)

where δβ is the Dirac measure at β defined on R.

Remark 6.2. The superscript β in δI,β and δβ,II represents the point at which
the Dirac measure takes the value 1. The position of β indicates the axis on which
the Dirac measure is defined. Hence, δβ in (6.1) is defined on the y-axis, and δβ in
(6.2) is defined on the x-axis. The Roman superscripts I and II represent dx and
dy, respectively.

It can be checked directly that δI,β and δβ,II are distributions. Moreover, the
following property holds.

Proposition 6.3. Use the above notations. For any f(x, y) ∈ C(Ω), f(x, β)δI,β

and f(β, y)δβ,II are distributions in D′(Ω).

Proof. For each v(x, y) ∈ D(Ω), by (6.2) and (6.1), we have〈
f(x, β)δI,β , v(x, y)

〉
=

∫
Ω

f(x, β)v(x, y) dδβ(y) dx =

∫ 1

−1

f(x, β)v(x, β) dx,

〈
f(β, y)δβ,II, v(x, y)

〉
=

∫
Ω

f(β, y)v(x, y) dδβ(x) dy =

∫ 1

−1

f(β, y)v(β, y) dy .

The proof can be completed by checking the linearity and continuity. □

For the above square domain Ω = (−1, 1)× (−1, 1), let µ0 be the 1-dimensional
Lebesgue measure defined on [−1, 1] × {0} and µ1 be the 1-dimensional Lebesgue
measure defined on {0} × [−1, 1], as shown in Figure 2(a). We will use µ0 and µ1

to construct a measure on Ω, which is singular respect to the Lebesgue measure on
R2.

Example 6.4. Use the above notation. Let µ := µ0 +µ1 be defined on Ω. Then µ
is singular respect to the 2-dimensional Lebesgue measure dx. Let ∆µ be defined
as in Section 2. Then

u(x, y) = 1 + |xy| − |x| − |y| (6.3)



EJDE-2025/12 NODAL SETS AND CONTINUITY OF EIGENFUNCTIONS 19

is a 2-eigenfunction of (1.2). u(x, y) (see Figure 2(b)) is continuous and has no
nodal points in Ω. Hence, it is a first eigenfunction of −∆µ in equation (1.2).

(a) (b)

Figure 2. Measures and eigenfunctions in Example 6.4.

Proof. According to [25, Proposition 2.2] and (1.2), we have, for any v(x, y) ∈ D(Ω),

−
∫
Ω

∆u(x, y)v(x, y) dx dy = λ

∫
Ω

u(x, y)v(x, y) dµ. (6.4)

We first calculate the second-order distributional derivative of u(x, y). For any
v(x, y) ∈ D(Ω), we have〈∂2T|x|

∂x2
, v(x, y)

〉
= −

〈∂T|x|
∂x

, vx(x, y)
〉
=

〈
T|x|, vxx(x, y)

〉
=

∫
Ω

|x|vxx(x, y) dx dy

=

∫ 1

−1

dy

∫ 1

0

xvxx(x, y) dx−
∫ 1

−1

dy

∫ 0

−1

xvxx(x, y) dx

= 2

∫ 1

−1

v(0, y) dy

= 2
〈
δ0,II, v(x, y)

〉
.

Hence, ∆|x| = 2δ0,II in the sense of distribution. By the same argument, we have

∆|y| = 2δI,0

and
∆|xy| = 2

(
|x|δI,0 + |y|δ0,II

)
in the sense of distribution. In summary,

∆u(x, y) = ∆(1 + |xy| − |x| − |y|) = 2(|x|δI,0 + |y|δ0,II − δI,0 − δ0,II), (6.5)

in the sense of distribution. By direct calculation, we have

−
∫
Ω

∆u(x, y)v(x, y) dx dy = −
∫
Ω

∆
(
1 + |xy| − |x| − |y|

)
v(x, y) dx dy

= 2
(∫ 1

−1

(
1− |x|

)
v(x, 0) dx+

∫ 1

−1

(
1− |y|

)
v(0, y) dy

)



20 S.-M. NGAI, M.-K. ZHANG, W.-Q. ZHAO EJDE-2025/12

= 2

∫
Ω

(
1− |x| − |y|

)
v(x, y) d(µ0 + µ1)

= 2

∫
Ω

(
1− |x| − |y|

)
v(x, y) dµ.

On the other hand,

λ

∫
Ω

u(x, y)v(x, y) dµ = λ

∫
Ω

(
1 + |xy| − |x| − |y|

)
v(x, y) d(µ0 + µ1)

= λ

∫
Ω

(
1− |x| − |y|

)
v(x, y) d(µ0 + µ1)

= λ

∫
Ω

(
1− |x| − |y|

)
v(x, y) dµ.

Combining the above two equalities and (6.4), we obtain λ = 2. Therefore, if
u(x, y) ∈ dom(∆µ), it is a 2-eigenfunction. Next, we prove that u(x, y) ∈ dom(∆µ).
We divide the proof into three steps.

Step 1. We claim that u(x, y) ∈ H1
0 (Ω). To see this, let

ξ1(x, y) =

{
sgn(x)(y − 1), y ≥ 0,

− sgn(x)(y + 1) y < 0,

ξ2(x, y) =

{
sgn(y)(x− 1), x ≥ 0,

− sgn(y)(x+ 1), x < 0.

By direct calculation, (ξ1(x, y), ξ2(x, y)) is the weak derivative of u(x, y). Moreover,
both ξ1(x, y) and ξ2(x, y) are in L2(Ω). Thus, u(x, y) ∈ H1(Ω). Note that the
square domain Ω is a Lipschitz domain. According to [34, Theorem 3.33], we have
u(x, y) ∈ H1

0 (Ω).

Step 2. We prove that u(x, y) ∈ dom(E). Let I be defined as in (2.2) and

τ(x) ∈ N :=
{
u ∈ H1

0 (Ω) : ∥I(u)∥L2(Ω,µ) = 0
}
.

Write Ω = ∪4
i=1Ωi, where Ω1 = [0, 1)× [0, 1), Ω2 = (−1, 0)× [0, 1), Ω3 = (−1, 0]×

(−1, 0], and Ω4 = (0, 1)× (−1, 0). By Step 1, we have

(u, τ)H1
0 (Ω) =

∫
Ω

∇u · ∇τ dx =

∫
Ω

(ξ1, ξ2) · ∇τ dx =

4∑
i=1

∫
Ωi

(ξ1, ξ2) · ∇τ dx.

Since τ ∈ N , we have∫
Ω1

(ξ1, ξ2) · ∇τ dx =

∫ 1

0

∫ 1

0

(y − 1, x− 1) · (τx, τy) dx dy

=

∫ 1

0

∫ 1

0

(y − 1)τx + (x− 1)τy dx dy

=

∫ 1

0

(y − 1) dy

∫ 1

0

τx dx+

∫ 1

0

(x− 1) dx

∫ 1

0

τy dy

=

∫ 1

0

(y − 1)τ(x, y)
∣∣∣1
0
dy +

∫ 1

0

(x− 1)τ(x, y)
∣∣∣1
0
dx

= −
∫ 1

0

(y − 1)τ(0, y) dy −
∫ 1

0

(x− 1)τ(x, 0) dx = 0.



EJDE-2025/12 NODAL SETS AND CONTINUITY OF EIGENFUNCTIONS 21

The last equality holds because τ = 0 dx-a.e. on {0} × (−1, 1) and (−1, 1) × {0}.
Similarly, we obtain ∫

Ωi

(ξ1, ξ2) · ∇τ dx = 0, for i = 1, 2, 3, 4.

Hence, ⟨u, τ⟩H1
0 (Ω) = 0, for τ ∈ N . Thus u(x, y) ∈ N⊥ = dom(E).

Step 3. We prove that u(x, y) ∈ dom(∆µ). Combining [25, Proposition 2.2] and
(6.5), we have∫

Ω

∇u · ∇v dx = −
∫
Ω

∆u · v dx

= 2
[ ∫ 1

−1

(
1− |x|

)
v(x, 0) dx+

∫ 1

−1

(
1− |y|

)
v(0, y) dy

]
= 2

∫
Ω

(
1− |x| − |y|

)
v(x, y) d(µ0 + µ1)

= 2

∫
Ω

(
1 + |xy| − |x| − |y|

)
v(x, y) dµ.

Moreover, f(x, y) := 2(1+ |xy|− |x|− |y|) = 2u(x, y) ∈ L2(Ω, µ). Therefore, by [25,
Proposition 2.2] again, u(x, y) ∈ dom(∆µ). □

Using the method of Example 6.4, we can construct the following two examples.

(a) (b)

Figure 3. Measures and eigenfunctions in Example 6.5.

Example 6.5. Let Ω = (−2, 2) × (−1, 1). Write Ω1 = (−2, 0] × (−1, 1) and
Ω2 = [0, 2) × (−1, 1). Let measure µ = µ0 + µ1 + µ2 be defined on Ω, µ0, µ1

and µ2 are 1-dimensional Lebesgue measures on [−2, 2] × {0}, {−1} × [−1, 1] and
{1} × [−1, 1], respectively, as shown in Figure 3(a). Then

u(x, y) =

{
1 + |(x+ 1)y| − |x+ 1| − |y|, (x, y) ∈ Ω1,

−1− |(x− 1)y|+ |x− 1|+ |y|, (x, y) ∈ Ω2

is a 2-eigenfunction that satisfies equation (1.2). u(x, y) is continuous, and the
nodal line of u divides the domain Ω into 2 subdomains (see Figure 3(b)). We omit
the proof as it can be obtained by the argument in Example 6.4.

Example 6.6. Let Ω = (0, 2n)×(−1, 1), Ωi = (2(i−1), 2i]×(−1, 1), i = 1, . . . , n−1,
Ωn = [2(n − 1), 2n) × (−1, 1), and Ω = ∪n

i=1Ωi. Let µ = µ0 + µ1 + · · · + µn be
defined on Ω, where µ0 is the 1-dimensional Lebesgue measure on [0, 2n]×{0}, and



22 S.-M. NGAI, M.-K. ZHANG, W.-Q. ZHAO EJDE-2025/12

µi is the 1-dimensional Lebesgue measure on {2i− 1}× [−1, 1], as shown in Figure
4(a). For i = 1, . . . , n, let

ψi(x, y) :=

{
(−1)i−1

(
1 +

∣∣(x− (2i− 1)
)
y
∣∣− |x− (2i− 1)| − |y|

)
, (x, y) ∈ Ωi,

0, (x, y) ∈ Ω\Ωi.

We define

u(x, y) :=

n∑
i=1

ψi(x, y).

Then u(x, y) is a 2-eigenfunction satisfying equation (1.2). u(x, y) is continuous,
and the nodal lines of the function divide Ω into n subdomains (see Figure 4(b).
The method is the same as that in the proof of Example 6.4; we omit it.

(a)

(b)

Figure 4. Measures and eigenfunctions in Example 6.6.

Acknowledgments. This research was supported by the National Natural Science
Foundation of China, grants 12271156 and 11771136, and by the Construct Program
of Key Discipline in Hunan Province. S.-M. Ngai was supported by a Faculty
Research Scholarly Pursuit Funding from Georgia Southern University.

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Sze-Man Ngai

Key Laboratory of High Performance Computing and Stochastic Information Process-
ing (HPCSIP) (Ministry of Education of China), College of Mathematics and Statistics,

Hunan Normal University, Changsha, Hunan 410081, China.
Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA
30460-8093, USA.
Beijing Institute of Mathematical Sciences and Applications, Huairou District, 101400,

Beijing, China
Email address: ngai@bimsa.cn

Meng-Ke Zhang
Key Laboratory of High Performance Computing and Stochastic Information Process-

ing (HPCSIP) (Ministry of Education of China), College of Mathematics and Statistics,

Hunan Normal University, Changsha, Hunan 410081, China
Email address: 2750600901@qq.com



EJDE-2025/12 NODAL SETS AND CONTINUITY OF EIGENFUNCTIONS 25

Wen-Quan Zhao (corresponding author)

Key Laboratory of High Performance Computing and Stochastic Information Process-

ing (HPCSIP) (Ministry of Education of China), College of Mathematics and Statistics,
Hunan Normal University, Changsha, Hunan 410081, China

Email address: zhaowq1008@hunnu.edu.cn


	1. Introduction
	2. Preliminaries
	3. Maximum principle
	4. Courant nodal domain theorem 
	5. Continuity of eigenfunctions
	6. Examples of continuous eigenfunctions 
	Acknowledgments

	References