. HEAT EQUATIONS DEFINED BY A CLASS OF FRACTAL MEASURES

(1)

HEAT EQUATIONS DEFINED BY A CLASS OF FRACTAL MEASURES

WEI TANG AND SZE-MAN NGAI

Abstract. We set up a framework to study one-dimensional heat equations defined by fractal Laplacians associated with self-similar measures with overlaps. We show that for a class of such self-similar measures, a heat equation can be discretized and the finite element method can be applied to yield a system of linear differential equations. We show that the numerical solutions converge to the actual solution and obtain the rate of convergence. We also study some properties of the solutions of the heat equation.

Contents

1. Introduction 1

2. Preliminaries 4

2.1. Dirichlet Laplacian defined by a measure 6

2.2. Gelfand triple 7

2.3. Existence and uniqueness of weak solution 8

3. The finite element method 10

4. Fractal measures defined by iterated function systems 13 4.1. Infinite Bernoulli convolution associated with the golden ratio 15 4.2. Three-fold convolution of the Cantor measure 16 4.3. A class of self-similar measures satisfying (EFT) 17 5. Neumann boundary condition and additional properties of solutions of

heat equations 18

6. Convergence of numerical approximations 23

References 26

1. Introduction

LetU ⊆R^d,d≥1, be a bounded open subset, and let µ be a positive finite Borel measure with supp(µ)⊆U and µ(U)>0. It is known (see, e.g., [11]) that µdefines a Dirichlet Laplace operator ∆_µ, if the following Poincar´e inequality for a measure

Date: September 29, 2018.

2010 Mathematics Subject Classification. Primary: 28A80, 35K05; Secondary: 74S05, 65L60, 65L20.

Key words and phrases. Fractal; Laplacian; heat equation; self-similar measure with overlaps.

The authors are supported in part by the National Natural Science Foundation of China, grant 11771136, and Construct Program of the Key Discipline in Hunan Province. The second author is also supported in part by the Hunan Province Hundred Talents Program.

1

(2)

(PI) holds: There exists some constantC > 0 such that Z

U

|u|²dµ≤C Z

U

|∇u|²dx for all u∈C_c^∞(U) (1.1) (see, e.g., [11, 15, 18]). The main purpose of this paper is to study heat equations defined by Dirichlet Laplacians ∆_µ. More precisely, we study the following non- homogeneousparabolic initial/boundary value problem (IBVP):







ut−∆µu=f on U ×[0, T], u= 0 on ∂U ×[0, T],

u=g on U × {t= 0}.

(1.2) The existence and uniqueness results (see Definition 2.6 for definition of a weak solution) follows easily from the general theory for heat equations in Hilbert spaces (see Section 2.3 and Theorem 2.4 for details). In this paper, we study the solution of equation (1.2) both theoretically and numerically.

We call a µ-measurable subsetI of U a cell (in U) if µ(I)>0. Clearly, U itself is a cell. Two cells I, J inU are measure disjoint with respect toµifµ(I∩J) = 0. Let I ⊆U be a cell. We call a finite familyP of measure disjoint cells a µ-partition of I if J ⊆ I for all J ∈ P, and µ(I) = P

J∈Pµ(J). A sequence of µ-partitions (P_k)k≥1

is refining if for any J₁ ∈P_k and anyJ₂ ∈ P_k+1, either J₂ ⊆J₁ or they are measure disjoint, i.e., each member of P_k+1 is a subset of some member of P_k. Throughout this paper,|E| denotes the diameter of a subset E ⊆Rⁿ.

In order to discretize (1.2) and obtain numerical approximations of the weak solution, we will often impose the following additional conditions on µ: there exists a sequence of refining µ-partitions (P_k)k≥1 = ({I_k,`}^N_`=0^(k))k≥1 of U such that

(P1) there exist some constantρ∈(0,1) and some integer m₀ satisfying max{|J|: J ∈P_k} ≤ρ^k−m⁰ for all k ≥1;

(P2) for any k≥1, each cell I ∈P_k is closed and connected;

(P3) for any k ≥2 and any 0≤ `≤N(k), there exist similitudes (τ_k,`,i)^N(1)_i=0 of the formτ_k,`,i(x) =r_k,`,ix+b_k,`,iand constants (c_k,`,i)^N(1)_i=0 such thatτ_k,`,i(I_1,i)⊆I_k,`, and

µ|_I_k,` =

N(1)

X

i=0

c_k,`,i·µ|_I_1,i ◦τ_k,`,i⁻¹ . (1.3) Under assumption (P3), the µ measure of each subinterval in the partition can be computed by using (1.3), making it possible to discretizing the heat equation (1.2).

Finally, we assume that µis a measure on R with supp(µ) = [a, b].

Let f(x, t) ≡ 0. Multiplying the first equation in (1.2) by v ∈ domE_D (see Sec- tion 2.1 for definition), integrating both sides with respect todµ, and then integrating

(3)

by parts, we obtain

− Z b

a

ux(x, t)v⁰(x)dx= Z b

a

ut(x, t)v(x)dµ, (1.4) where u_x(x, t) is the weak partial derivative ofu with respect tox and u_t(x, t) is the weak partial derivative with respect to t.

Theorem 1.1. Let µ be a positive finite Borel measure on R with supp(µ) = [a, b].

Assume that there exists a sequence of refining partitions (P_k)k≥1 = ({I_k,`}^N_`=0^(k))k≥1

satisfying conditions (P1)–(P3), and R

I1,`xⁱdµ, 0≤`≤N(1), i= 0,1,2, can be evaluated explicitly. Then the finite element method for equation (1.4) can be discretized into a system of first-order ordinary differential equations, which has a unique solution that can be solved numerically.

We are mainly interested in fractal measures µ. Let X be a non-empty compact subset of R^d. Throughout this paper, an iterated function system (IFS) refers to a finite family of contractive similitudes {Si}^q_i=1 defined on X, i.e.,

S_i(x) =ρ_ix+b_i, i= 1, . . . , q, (1.5) where 0 < ρ_i < 1, and b_i ∈ R^d. It is well-known that for each IFS {S_i}^q_i=1, there exists a unique non-empty compact subset F ⊆ X, called the self-similar set, such that

F =

q

[

i=1

S_i(F);

moreover, associated to each set of probability weights {w_i}^q_i=1 (i.e., w_i > 0 and Pq

i=1w_i = 1), there is a unique probability measure, called the self-similar measure, satisfying the following identity

µ=

q

X

i=1

w_iµ◦S_i⁻¹ (1.6)

(see [8, 12]). An IFS {S_i}^q_i=1 is said to satisfy the open set condition (OSC) if there exists a non-empty bounded open setOsuch thatS

iS_i(O)⊆OandS_i(O)∩S_j(O) =∅ for all i 6= j. IFSs that do not satisfy (OSC), as well as all associated self-similar measures, are said to have overlaps.

It is worth pointing out that for general self-similar measures with overlaps, it does not seem possible to discretize the heat equations (1.2) in the way described in the paper, and thus it is not clear how numerical approximations of the weak solution can be obtain. Theorem 1.1 provides a framework under which discretization can be performed.

Based on Theorem 1.1, we solve the homogeneous IBVP (1.2) numerically for three different one-dimensional self-similar measures with overlaps, namely, the inifinite

(4)

Bernoulli convolution associated with the golden ratio, the three-fold convolution of the Cantor measure, and a class of self-similar measures that we call essentially of finite type (EFT) (see [17]). These measures share the common property that the support can be partitioned into a sequence of arbitrarily small intervals whose measures can be computed explicitly.

In Section 5, we show that Theorem 1.1 also holds for heat equations defined by Neumann Laplacians (see (5.1)), and study some properties of the solutions.

The following theorem shows that the approximate solutions converge to the actual weak solution and we also obtain a rate of convergence. See Subsection 2.1 and Definition 2.1 for the definitions of k · k_dom_E_D and k · k_2,dom_E_D, respectively.

Theorem 1.2. Assume the hypotheses of Theorem 1.1, let f = 0 in equation (1.2), and fix t ∈ [0, T]. Then the approximate solutions u^m obtained by the finite element method converge in L²((a, b), µ) to the actual weak solution u. Moreover,

ku^m−uk_µ≤2 √

Tku_tk_2,domE_D +kuk_domE_D

ρ^m/2, (1.7)

where ρ is any constant in condition (P1).

We remark that Theorem 1.2 also holds for heat equations defined by Neumann Laplacians. Details are given in Section 6.

The rest of this paper is organized as follows. Section 2 summarizes some notation, definitions and results that will be needed throughout the paper, and gives the existence and unique results of the heat equation (1.2). Section 3 is devoted to the proof of Theorem 1.1. In Section 4, we apply Theorem 1.1 to three different self-similar measures with overlaps. We study some properties of the solutions of heat equations defined by Neumann Laplacians in Section 5. Finally, the proof of Theorem 1.2 is given in Section 6.

2. Preliminaries

In this section, we summarize some notation, definitions and facts that will be used throughout the rest of the paper. For a Banach space X, we denote its topological dual by X⁰. For v ∈X⁰ andu∈X, we lethv, ui=hv, ui_X⁰_,X :=v(u) denote thedual pairing of X⁰ and X.

A function s: [0, T]→X is calledsimple if it has the form s(t) =

N

X

m=1

χ_E_m(t)u_m for∈[0, T], (2.1)

(5)

where for each m = 1, . . . , N, E_m is a Lebesgue measurable subset of [0, T], u_m ∈X and χ_E_m is the characteristic function on E_m. A function u : [0, T] → X is called strongly measurable if there exist simple functions s_n : [0, T]→X such that

s_n(t)→u(t) for Lebesgue a.e. t∈[0, T] as n → ∞.

A function u : [0, T] → X is weakly measurable if for each v ∈ X⁰, the mapping t7→ hv, u(t)i is Lebesgue measurable.

A function u : [0, T] → X is almost separably valued if there exists a subset E ⊆ [0, T] with zero Lebesgue measure such that the set{u(t) :t∈[0, T]\E}is separable.

By a theorem of Petti [20], a function u : [0, T] → X is strongly measurable if and only if it is weakly measurable and almost separably valued. Since any subset of a separable Banach space X is separable, the two concepts of measurability coincide and we can use the term measurable without ambiguity.

Definition 2.1. Let X be a separable Banach space with norm k · k_X. Denote by L^p(0, T;X) the space of all measurable functions u: [0, T]→X satisfying

(1) kukL^p(0,T;X):=

RT

0 ku(t)k^p_Xdt 1/p

<∞, if 1≤p < ∞, and (2) kuk_L^∞_(0,T_;X₎:= esssup_0≤t≤Tku(t)k_X <∞, if p=∞.

If the interval [0, T] is understood, we will abbreviate these norms as kuk_p,X and kuk∞,X, respectively.

LetU ⊆R^d,d≥1, be a bounded open subset. For a functionϕ:U →R, we letϕ⁰ denote both its classical and weak derivatives. If u∈L²(0, T;X), where X is H₀¹(U) or L²(U, µ) etc., then for each fixed t, we denote by ux(x, t) (or ∇u) the classical or weak derivatives of u with respect to x.

Remark 2.1. For each 1 ≤ p ≤ ∞, L^p(0, T;X) is a Banach space; moreover, L^p²(0, T;X) ⊆L^p¹(0, T;X) if 1≤p₁ ≤ p₂ ≤ ∞. Let X be a separable Banach space with inner product (·,·)_X. If (X,(·,·)_X)is a separable Hilbert space, then L²(0, T;X) is a Hilbert space with the inner product

(u, v)_L²_(0,T;X) :=

Z T 0

(u(t), v(t))_Xdt.

Definition 2.2. Let X be a Banach space and u ∈ L¹(0, T;X). We say that v ∈ L¹(0, T;X) is the weak derivative of u with respect to t, written u_t =v, if

Z T 0

φ_t(t)u(t)dt =− Z T

0

φ(t)v(t)dt

for all scalar test functions φ∈C_c^∞(0, T).

(6)

Definition 2.3. Let X be a Banach space and X⁰ its dual. We say that a sequence {u_m}^∞_m=1 ⊆ X converges weakly to u ∈ X, written u_m * u, if hv, u_mi → hv, ui for each bounded linear functional v ∈X⁰.

For the more general definition of devatives of distributions with values in a Hilbert space, we refer the reader to [23, Section 25].

2.1. Dirichlet Laplacian defined by a measure. For convenience, we summarize the definition of the Dirichlet Laplacian on a bounded domain defined by a measure;

details can be found in [11]. Let U ⊆R^d,d ≥1, be a bounded open subset and µ be a positive finite Borel measure with supp(µ)⊆U and µ(U)>0. Let

L²(U, µ) :=

f :U →R:kfk_µ:=Z

U

|f|²dµ1/2

<∞

.

In particular, if µ is Lebesgue measure, we denote L²(U, µ) simply by L²(U). Let H¹(U) be the Sobolev space with inner product

hu, vi_H¹_(U) :=

Z

U

uv dx+ Z

U

∇u· ∇v dx,

and let H₀¹(U) denote the completion of C_c^∞(U) in the H¹(U) norm, which admits the equivalent inner product defined by

hu, vi_H¹

0(U) :=

Z

U

∇u· ∇v dx.

Note that both H¹(U) andH₀¹(U) are Hilbert spaces.

We assume thatµsatisfies (PI) (see (1.1)). Then each equivalence classu∈H₀¹(U) contains a unique (in the L²(U, µ) sense) member ˆu that belongs to L²(U, µ) and satisfies both conditions below:

(1) there exists a sequence{un}inC_c^∞(U) such thatun→uˆinH₀¹(U) andun→uˆ inL²(U, µ);

(2) ˆusatisfies inequality (1.1).

We call ˆu the L²(U, µ)-representative of u. Define a mapping ι : H₀¹(U) → L²(U, µ) by ι(u) = ˆu. ι is a bounded linear operator, but not necessarily injective. Consider the subspace N of H₀¹(U) defined as N :=

u ∈H₀¹(U) :kι(u)k_µ = 0 . Now let N^⊥ be the orthogonal complement of N in H₀¹(U). Then ι :N^⊥ →L²(U, µ) is injective.

Unless explicitly stated otherwise, we will denote theL²(U, µ)-representative ˆusimply byu.

Consider the non-negative bilinear form E(·,·) in L²(U, µ) defined by E(u, v) :=

Z

U

∇u· ∇v dx (2.2)

(7)

with domain domE_D =N^⊥, or more precisely, ι(N^⊥). (PI) implies that (E,domE_D) is a closed quadratic form inL²(U, µ). Hence there exists a non-negative self-adjoint operator A inL²(U, µ) such that

E(u, v) = A^1/2u, A^1/2v

and domE_D = dom (A^1/2)

(see, e.g., [9, Theorem 1.3.1]). We write ∆µ=−A, and call it the(Dirichlet) Laplacian with respect to µ. Let u ∈ domE_D. Then u ∈ dom ∆_µ if and only if there exists a unique f ∈ L²(U, µ) such that E(u, v) = (f, v)_µ for all v ∈ domE_D. In this case,

−∆_µu=f. Throughout this paper, we let

domE_D :=N^⊥ and k · k_domE_D :=k · k_H¹

0(U).

Some sufficient conditions for (PI) and the existence of an orthonormal basis {ϕ_n}^∞_n=1ofL²(U, µ) consisting of the eigenfunctions of−∆_µcan be found in [4,11,15].

We remark that ifn = 1, then (PI) holds for any suchµ, and thus ∆_µ is well-defined;

moreover,−∆_µ has compact resolvent.

2.2. Gelfand triple. The notion of Gelfand triple, defined below, plays an important role in our investigation of the heat equation.

Definition 2.4. Let V, H be separable Hilbert spaces with a continuous dense embedding ι : V ,→ H. By identifying H with its dual H⁰, we obtain the following continuous and dense embedding V ,→ H ∼=H⁰ ,→V⁰. Assume, in addition, that the dual pairing betweenV andV⁰ is compatible with the inner product onH, in the sense that

hv, ui_V⁰_,V = (v, u)_H for all u∈V ⊂H and v ∈H ∼=H⁰ ⊂V⁰. The triple (V, H, V⁰) is called a Gelfand triple.

We remark that sinceV is itself a Hilbert space, it is isomorphic with its dual V⁰. However, this isomorphism is in general not the same as the composition ι^∗ι : V ⊂ H =H⁰ ,→V⁰, whereι^∗ is the adjoint of ι.

LetU,µand domE_D be given as in Section 2.1. Then the spaces domE_D,L²(U, µ), (domED)⁰ form a Gelfand triple:

domE_D ,→L²(U, µ)∼= (L²(U, µ))⁰ ,→(domE_D),

where we identify L²(U, µ) with (L²(U, µ))⁰. The embeddingL²(U, µ),→(domE_D)⁰ is given by

w∈L²(U, µ)7→(w,·)_µ ∈(L²(U, µ))⁰ ⊂(domE_D)⁰.

(8)

2.3. Existence and uniqueness of weak solution. In this subsection, we consider the existence and uniqueness of weak solution of equation (1.2).

Definition 2.5. Let V be a Hilbert space, and 0< T <∞. For each integer k ≥ 0, define the Sobolev space

W₂^k(0, T;V) :=n

u: (0, T)→V measurable: dⁿu

dtⁿ ∈L²(0, T;V) for 0≤n≤ko , where the differentiation is in the distributional sense. Equip W₂^k(0, T;V) with the norm

kuk²_k,V :=

k

X

n=0

Z T 0

dⁿu dtⁿ

2 V

dt.

Let V, H be separable Hilbert spaces. Assume that the embedding V ,→ H is continuous, injective, and dense, such that V ,→ H ,→ V⁰ is a Gelfand triple. Let 0< T <∞. Fort ∈[0, T] andϕ, ψ∈V, assume that a(t;ϕ, ψ) is a sesquilinear form in ϕ, ψ. Let <(z) denote the real part of a complext number z. We also need the following conditions:

(C1) For fixed ϕ, ψ∈V, a(t;ϕ, ψ) is measurable on [0, T].

(C2) There exists some constant c >0, independent of t, such that

|a(t;ϕ, ψ)| ≤ckϕk_V · kψk_V for all t∈[0, T] and ϕ, ψ ∈V.

(C3) There exist real k₀, α≥0, independent of t and ϕ, such that

<(a(t;ϕ, ϕ)) +k₀kϕk²_H ≥αkϕk_V for all t ∈[0, T] and ϕ∈V.

It follows from condition (C2) that there exists a representation operatorL(t) :V → V⁰, such that for each t, L(t) is linear and continuous, with a(t;ϕ, ψ) = (L(t)ϕ, ψ)_H.

The proofs of Theorems 2.2 and 2.3 below can be found in [23, Sections 26–27].

Theorem 2.2. Let V and H be separable Hilbert spaces. Assume that the embedding V ,→ H is continuous, injective, and dense, such that V ,→ H ,→ V⁰ is a Gelfand triple. Assume that the sesquilinear form a(t;ϕ, ψ) satisfies conditions (C1)–(C3) above. Then for any T ∈(0,∞), f ∈L²(0, T;V⁰), and u₀ ∈H, there exists a unique function u(t)∈L²(0, T;V) with du/dt ∈L²(0, T;V⁰) such that

du

dt +L(t)u=f for t∈[0, T], and u(0) =u₀, (2.3) in the sense that

du dt, ϕ

H

+ (L(t)u, ϕ)_H = (f, ϕ)_H for all ϕ∈V.

We will also need the following additional assumption on a(t;ϕ, ψ):

(9)

(C4) for fixed ϕ, ψ ∈ V, let a(t;ϕ, ψ) be k-fold differentiable with respect to t in [0, T]. We assume that (d^j/dt^j)a(t;ϕ, ψ) is continuous in [0, T] for j = 0, . . . , k −1, (d^k/dt^k)a(t;ϕ, ψ) exists and is measurable in [0, T], and there exists a constant c0, independent of t, such that

d^j

dt^ja(t;ϕ, ψ)

≤c0kϕkV · kψkV for all j = 0, . . . , k.

The following theorem shows that the smoothness of the solution of equation (2.3) increases with that off.

Theorem 2.3. Assume the hypotheses of Theorem 2.2, and that condition(C4)above holds. Consider the parabolic equation

du

dt +L(t)u=f for t∈[0, T], and u(0) =u₀. (2.4) Assume, in addition, that f ∈ W₂^k(0, T;V⁰), u₀ ∈ H for k = 0; u₀ ∈ V, f(0) − L(0)u₀ ∈H for k= 1; u₀ ∈V, f(0)−L(0)u₀ ∈V, f⁽¹⁾(0)−L(0)f(0)∈H fork = 2;

whereas for k ≥3, we assume u0 ∈V, f(0)−L(0)u0 ∈V,

f⁽ⁱ⁾(0)−L(0)f⁽ⁱ⁻¹⁾(0)∈V, i= 1,· · · , k−2, and f^(k−1)(0)−L(0)f^(k−2)(0) ∈H.

Then the solution u of (2.4) satisfies

u∈W₂^k(0, T;V) and d^k+1u(t)

dt^k+1 ∈L²(0, T;V⁰).

We will apply Theorems 2.2 and 2.3 to the heat equation (1.2). Let U ⊆ R^d, d ≥ 1, be a bounded open subset, and let µ be a positive finite Borel measure with supp(µ)⊆ U and µ(U) > 0. Assume µ satisfies (PI). Let (E,domE_D) be defined as in (2.2), and−∆_µ be the Dirichlet Laplacian with respect to µ.

Definition 2.6. Use the notation above. Let0< T <∞. Assumef ∈L²(0, T; (domE_D)⁰) and g ∈L²(U, µ). A function u∈L²(0, T; domE_D) with u_t∈L²(0, T; (domE_D)⁰) is a weak solution of IBVP (1.2) if the following conditions are satisfied:

(1) hut, vi+E(u, v) = hf, vi for each v ∈domED and Lebesgue a.e. t∈[0, T];

(2) u(x,0) = g(x) for all x∈U.

Here h·,·i denotes the pairing between (domE_D)⁰ and domE_D.

Theorem 2.4. Use the notation in Definition 2.6. Assume f ∈ L²(0, T; (domE_D)⁰) and g ∈L²(U, µ). Then equation (1.2) has a unique weak solution.

Proof. In order to apply Theorem 2.2, we let V = domE_D, H = L²(U, µ), and set a(t;u, v) := E(u, v), which is independent oft. Conditions (C1) and (C4) clearly hold,

(10)

because the map t 7→ a(t;u, v) = E(u, v) is constant in time and real-valued. Since for all u, v ∈domE_D,

a(t;u, v) =

Z

U

∇u· ∇v dx

≤ Z

U

|∇u|²dx 1/2

· Z

U

|∇v|²dx 1/2

=kuk_domE_D · kvk_domE_D,

condition (C2) holds. Thus there exists a representation operator L : domE_D → (domE_D)⁰ such thatE(u, v) = (Lu, v)_µ, which satisfiesL=−∆_µ on dom ∆_µ.

Finally, for allt ∈[0, T] and u∈domE_D,

a(t;u, u) +kuk²_µ=E(u, u) +kuk²_µ≥ E(u, u) =kuk²_dom_E

D,

and thus condition (C3) holds with k₀ =α = 1. Consequently, the assertion follows

from Theorem 2.2.

As a consequence of Theorem 2.3, we have the following regularity result for solutions of homogeneous heat equations in our setting.

Theorem 2.5. Use the notation in Theorem 2.4 and assume the hypotheses of The- orem 2.4. Assume in addition that f = 0. Then for all k ≥ 1, the solution of the homogeneous equation (1.2) satisfies

u∈W₂^k(0, T; domED) and d^k+1u(t)

dt^k+1 ∈L² 0, T; (domED)⁰ .

3. The finite element method

In this section, we let f = 0 in equation (1.2), and use the finite element method to solve the homogeneous IBVP. Let µ be a positive finite Borel measure on R with supp(µ) = [a, b]. Assume that there exists a sequence of refining partitions (Pk)k≥1 = ({I_k,`}^N(k)_`=0 )k≥1satisfying conditions (P1)–(P3) in Section 1. Without loss of generality, we can write I_m,` = [x_m,`, x_m,`+1] form ≥1 and 0 ≤` ≤N(m)−1. It is easy to see that x_m,0 =a and x_m,N(m) =b for all m≥1.

We apply the finite element method to approximate the weak solution u(x, t) satisfying (1.4) by

u^m(x, t) =

N(m)

X

`=0

β_m,`(t)φ_m,`(x), (3.1)

where, for `= 0,1, . . . , N(m),β_m,`(t) are functions to be determined andφ_m,`(x) are the standard piecewise linearfinite element basis functions (also calledtent functions)

(11)

defined by

φ_m,`(x) =











x−xm,`−1

xm,`−xm,`−1

if x∈Im,`−1, `= 1,2, . . . , N(m), x−x_m,`+1

xm,`−xm,`+1

if x∈I_m,`, `= 0,1, . . . , N(m)−1,

0 otherwise.

(3.2)

We require u^m(x, t) to satisfy the integral form of the homogeneous heat equation Z b

a

u^m_t (x, t)φm,i(x)dµ=− Z b

a

u^m_x(x, t)·φ⁰_m,i(x)dx for i= 1, . . . , N(m)−1, (3.3) and the Dirichlet boundary conditionu^m(a, t) = u^m(b, t) = 0, where u^m_t := (u^m)_t and u^m_x := (u^m)_x. We note that φ_m,i(a) = φ_m,i(x_m,0) = 0 and φ_m,j(b) = φ_m,j(x_m,N(m)) = 0 for all i = 1, . . . , N(m), j = 0,1, . . . , N(m)− 1, and m ≥ 1. Thus β_m,0(t) = β_m,N(m)(t) = 0 for all m ≥1. Using this and substituting (3.1) into (3.3) gives

N(m)−1

X

`=1

β_m,`⁰ Z b

a

φ_m,`(x)φ_m,i(x)dµ=−

N(m)−1

X

`=1

β_m,`

Z b a

φ⁰_m,`(x)φ⁰_m,i(x)dx (3.4) for 1≤i≤N(m)−1. We define the mass matrixM =M^(m)= (M_ij^(m)) and stiffness matrix K=K^(m) = (K_ij^(m)), respectively, by

M_ij^(m) = Z b

a

φ_m,i(x)φ_m,j(x)dµ and K_ij^(m) = Z b

a

φ⁰_m,i(x)φ⁰_m,j(x)dx,

where 1≤i, j ≤N(m)−1. It follows from the definition of φ_m,j(x) that bothM and K are tridiagonal. Let

w(t) = w_m(t) :=





w_m,1(t) ... wm,N(m)−1(t)



=





β_m,1(t) ... βm,N(m)−1(t)



. (3.5)

Then (3.4) can be put into matrix form as

Mw⁰ =−Kw. (3.6)

This gives us a system of first-order linear ODEs with constant coefficients. To solve it, we need to impose initial conditions. The initial condition u(x,0) =g(x) for a ≤ x ≤b can be approximated by its linear interpolant eg(x) = PN(m)−1

i=1 g(x_m,i)φ_m,i(x).

Therefore, we set w_m,i(0) =g(x_m,i). This leads to the initial condition

w(0) =wm(0) :=wm,0 =





g(x_m,1) ... g(xm,N(m)−1)



. (3.7)

(12)

Consequently, we get the linear system





 Mdw

dt =−Kw, t >0, w(0) =w_m,0.

(3.8) Since supp(µ) = [a, b], [2, Proposition 3.1] implies that M is invertible. Thus the system in (3.8) has a unique solution. Hence, the following proposition holds.

Proposition 3.1. Assume that supp(µ) = [a, b]. Then (3.8) has a unique solution w(t). Moreover, βm,j(t)∈C(0, T) for m ≥1 and j = 1, . . . , N(m)−1.

Proof of Theorem 1.1. The assertions hold by combining the derivations above and

Proposition 3.1.

We note that the matrix K can be computed directly; we only describe how to compute M. In the following, the constants cm,i,j and similitudes τm,i,j come from condition (P3) in Section 1. From the definition of theφ_m,i’s and (1.3), we have, for 1≤i≤N(m)−1,

M_i,i^(m) = 1

(x_m,i−xm,i−1)²

N(1)

X

j=0

c_m,i−1,j Z

Im,i−1

(x−x_m,i−1)²dµ|_I_1,j ◦τ_m,i−1,j⁻¹

+ 1

(x_m,i−x_m,i+1)²

N(1)

X

j=0

cm,i,j

Z

Im,i

(x−xm,i+1)²dµ|I1,j ◦τ_m,i,j⁻¹

= (x_m,i−xm,i−1)⁻²·

N(1)

X

j=0

cm,i−1,j

Z

I1,j

τm,i−1,j(x)−xm,i−1

2

dµ

+ (x_m,i−x_m,i+1)⁻²·

N(1)

X

j=0

c_m,i,j Z

I1,j

τ_m,i,j(x)−x_m,i+12

dµ.

(3.9)

For 2≤i≤N(m)−1, we obtain M_i,i−1^(m) = −1

(x_m,i−x_m,i−1)²

N(1)

X

j=0

cm,i−1,j

Z

Im,i−1

(x−xm,i−1)(x−x_m,i)dµ|_I_1,j ◦τ_m,i−1,j⁻¹

=−(x_m,i−xm,i−1)⁻²·

N(1)

X

j=0

cm,i−1,j

Z

I1,j

τm,i−1,j(x)−xm,i−1

τm,i−1,j(x)−x_m,i dµ,

(3.10) and M_i−1,i^(m) =M_i,i−1^(m) .

Define

Jk,j :=

Z

I1,j

x^kdµ, k = 0,1,2, and j = 0, . . . , N(1). (3.11)

(13)

Since each τ_k,`,i is of the form τ_k,`,i(x) =r_k,`,ix+b_k,`,i, we can see that the matrix M is completely determined by the integralsJ_k,j, wherek = 0,1,2 and j = 0, . . . , N(1).

Hereafter, we assume that the constantJ_k,j can be evaluated explicitly for k= 0,1,2 and j = 0, . . . , N(1).

Next, we discuss the solution of the linear system (3.6). Let w_n := w(t_n), n ≥0, and use the central difference method to solve the equation (3.8). We approximate the derivative as

w⁰(t_n)≈ w_n+1−w_n

δ and w(t_n)≈ w_n+1+w_n

2 . (3.12)

Substituting (3.12) into (3.6) yields Mw_n+1−w_n

δ =−Kw_n+1+w_n

2 .

That is,w_n+1 = (2M+δK)⁻¹(2M−δK)w_n. Therefore, equation (3.6) becomes







wn+1 = (2M+δK)⁻¹(2M−δK)wn, n = 0,1,2, . . . , w₀ =w(t₀) = w(0),

t_n =nδ. .

(3.13)

To solve this system, fixδ and substitute the initial condition w₀ from (3.7) into the first equation in (3.13) to get w₁. Then w_n+1 can be computed recursively.

4. Fractal measures defined by iterated function systems

In this section, we solve the homogeneous IBVP (1.2) numerically for three different measures, namely, the infinite Bernoulli convolution associated with the golden ratio, the three-fold convolution of the Cantor measure, and a class of self-similar measures satisfying (EFT). These measures are defined by IFSs with overlaps. In the first and second cases, the measures satisfy a family of second-order self-similar identities (see definition below). These identities were first introduced by Strichartz et al. [22]

to approximate the density of the infinite Bernoulli convolution associated with the golden ratio.

Let {S_i}^q_i=1 be an IFS of contractive similitudes on R, and letµ be the associated self-similar measure. Assume that supp(µ) = [a, b]. Define an auxiliary IFS

T_j(x) =r_jx+d_j, j = 1,2, . . . , N,

where n_j ∈N, d_j ∈R. We say that µ satisfies a family ofsecond-order (self-similar) identities with respect to {T_j}^N_j=1 (see [13]) if

(1) supp(µ)⊆SN

j=1T_j(supp(µ)), and

(14)

(2) for each Borel subset A ⊆ supp(µ) and 0 ≤ i, j ≤ N, µ(T_iT_jA) can be expressed as a linear combination of {µ(T_kA) : k = 1, . . . , N}. In matrix form,





µ(T1TjA) ... µ(T_NT_jA)



=M_j





µ(T1A) ... µ(T_NA)



, j = 1, . . . , N; or, equivalently,

µ(T_iT_jA) =e_iM_j





µ(T₁A) ... µ(T_NA)



, i, j = 1, . . . , N, (4.14) wheree_i is the ith row of the N×N identity matrix and M_j is someN ×N matrix independent ofA.

Proposition 4.1. Let µ be a self-similar measure defined by an IFS of contractive similitudes on R. Assume that supp(µ) = [a, b], µ satisfies a family of second-order (self-similar) identities with respect to {T_j}^N_j=1, and {T_j}^N_j=1 satisfies (OSC). Define

P_k :=n

T_j([a, b]) :j ∈ {1, . . . , N}^ko

for k ≥1.

Then(P_k)k≥1 is a sequence of refining µ-partitions of[a, b]satisfying conditions (P1)–

(P3) in Section 1. Moreover, the matrix M is completely determined by the integrals Rb

a x^kdµ◦T_j, k = 0,1,2, j = 1, . . . , N.

Proof. It is easy to see that (P_k)k≥1 is a sequence of refining µ-partitions of [a, b]

satisfying conditions (P1) and (P2). For j = (j₁, . . . , j_m) ∈ {1, . . . , N}^m, iterating (4.14) shows that for any Borel subset A⊆supp(µ),

µ(TjA) = cj





µ(T₁A) ... µ(T_NA)



, (4.15)

where c_j := [c¹_j, . . . , c^N_j ] :=e_j₁M_j₂· · ·M_j_m, i.e., µ(T_jA) =

N

X

i=1

cⁱ_j·µ(T_iA).

It follows that

µ|_T_j_([a,b]) =

N

X

i=1

cⁱ_j ·µ|_T_i_([a,b])◦τ_j,i⁻¹,

where τ_j,i⁻¹ =T_i◦T_j⁻¹. Hence, condition (P3) holds. Since Z b

a

x^kdµ◦T_j = Z

Tj[a,.b]

(T_j⁻¹x)^kdµ and Z

Tj[a,b]

x^kdµ= Z b

a

(T_jx)^kdµ◦T_j, M is also determined by the intervals Rb

a x^kdµ◦T_j, k= 0,1,2, j = 1, . . . , N.

(15)

Let µ be a positive finite Borel measure on R with supp(µ) = [a, b]. Assume that there exists a sequence of refining partitions (P_k)k≥1 = ({I_k,`}^N(k)_`=0 )k≥1 satisfying conditions (P1)–(P3) in Section 1. In order to solve (3.6) or (3.13), we need to compute the integrals Jk,j, k = 0,1,2, j = 0, . . . , N(1), as defined in (3.11). The exact values of these integrals for the first and second measures in this section have been obtained in [2, 3]. We will find the exact values of these integrals for the third measure. The following integration formula is used repeatedly. For every continuous function f on [a, b],

Z b a

f dµ=

q

X

i=1

w_i Z b

a

f ◦S_idµ. (4.16)

Substituting the values ofJ_k,j into (3.9), we obtain the matrix M. This allows us to solve equation (3.13).

4.1. Infinite Bernoulli convolution associated with the golden ratio. We consider the infinite Bernoulli convolution associated with the golden ratio:

µ= 1

2µ◦S₁⁻¹+1

2µ◦S₂⁻¹, (4.17)

where

S₁(x) = ρx, S₂(x) =ρx+ (1−ρ), ρ= (√

5−1)/2.

We note that supp(µ) = [0,1]. Strichartz et al. [22] showed that µ satisfies a family of second-order identities with respect to the following auxiliary IFS:

T₁(x) :=ρ²x, T₂(x) :=ρ³x+ρ², T₃(x) :=ρ²x+ρ. (4.18) Moreover, µ satisfies the following second-order identities [13]: for each Borel A ⊆ [0,1],





µ(T₁T_jA) µ(T₂T_jA) µ(T₃T_jA)



=M_j





µ(T₁A) µ(T₂A) µ(T₃A)



, j = 1,2,3, (4.19) where

M₁ = 1 8





2 0 0 1 2 0 0 4 0



, M₂ = 1 4





0 1 0 0 1 0 0 1 0



, M₃ = 1 8





0 4 0 0 2 1 0 0 2



.

This can be used to compute the measure of suitable subintervals of [0,1]. In fact, if we let J = (j₁, . . . , j_m), j_i ∈ {1,2,3}, then for each BorelA ⊆[0,1],

µ(T_J(A)) =c_J







, where c_J =e_j₁M_j₂· · ·M_j_m = (c¹_J, c²_J, c³_J). (4.20) The integrals R1

0 x^kdµ◦Tj, k = 0,1,2, j = 1,2,3, have been calculated in [3, Section 5]. We can thus calculate the entries of the mass matrix M and solve the linear system (3.6). The result is shown in Figure 1.

(16)

0.0 0.2 0.4 0.6 0.8 1.0 0.2

0.4 0.6 0.8 1.0

Figure 1. Figure for numerical solutions of the heat equation defined by the infinite Bernoulli convolution associated with the golden ratio.

The initial condition is given by the function g(x) := sin((5π(x−0.4)) for x∈(0.4,0.6), and g(x) := 0 otherwise. Hereδ = 0.0001. From top to bottom, the values of t are 0.0, 0.0004, 0.002, 0.01, 0.04, 0.1, 0.2.

4.2. Three-fold convolution of the Cantor measure. We consider the following three-fold convolution of the Cantor measure studied in [13, 16, 17]. The three-fold convolution of the Cantor measure µ is the self-similar measure defined by the following IFS with overlaps (see [16]):

S_i(x) = 1 3x+ 2

3(i−1), i= 1,2,3,4, together with probability weights {1/8,3/8,3/8,1/8}. That is,

µ= 1

8µ◦S₁⁻¹ +3

8µ◦S₂⁻¹+3

8µ◦S₃⁻¹+ 1

8µ◦S₄⁻¹. (4.21) Note that supp(µ) = [0,3]. It is shown in [13] thatµsatisfies a family of second-order identities with respect to the following auxiliary IFS

T₁(x) := 1

3x, T₂(x) = 1

3x+ 1, T₃(x) = 1

3x+ 2. (4.22) In fact, for each BorelA ⊆[0,3],





µ(T₁T_jA) µ(T₂T_jA) µ(T₃T_jA)



=M_j







, j =1,2,3, where M₁, M₂,M₃ are given by

M₁ = 1 8





1 0 0 0 3 0 1 0 3



, M₂ = 1 8





0 1 0 3 0 3 0 1 0



, M₃ = 1 8





3 0 1 0 3 0 0 0 1



.

Similarly, if J = (j₁, . . . , j_m),j_i ∈ {1,2,3}, then for each BorelA⊆[0,3], µ(TJ(A)) =cJ







, where cJ =ej1Mj2· · ·Mjm = (c¹_J, c²_J, c³_J). (4.23)

(17)

The integrals R1

0 x^kdµ◦T_j, k = 0,1,2, j = 1,2,3, have been calculated in [2, Section 4.3]. We can thus calculate the entries of the mass matrix M and solve the linear system (3.6). The result is shown in Figure 2.

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.2 0.4 0.6 0.8 1.0

Figure 2. Figure for numerical solutions of the heat equation defined by the three-fold convolution of the Cantor measure. The initial condition is given by the function g(x) = sin((5π(x/3−0.4)), x ∈(1.2,1.8), and g(x) = 0 otherwise. Here δ = 0.0001. From top to bottom, the values of t are 0.0, 0.0012, 0.004, 0.01, 0.04, 0.18, 0.5.

4.3. A class of self-similar measures satisfying (EFT). In this subsection, we consider the following family of IFSs:

S₁(x) =r₁x, S₂(x) = r₂x+r₁(1−r₂), S₃(x) =r₂x+ 1−r₂, (4.24) where the contraction ratios r₁, r₂ ∈ (0,1) satisfy r₁ + 2r₂ −r₁r₂ ≤ 1, i.e., S₂(1) ≤ S₃(0). The Hausdorff dimension of the self-similar sets is computed in [14]. The mul- tifractal properties and spectral dimension of the corresponding self-similar measures are recently studied in [6, 17, 19].

Letµbe a self-similar measure defined by an IFS in (4.24) and a probability vector (pi)³_i=1. Let I1,1 :=S1(X)S

S2(X) and I1,0 :=S3(X), where X = [0,1]. In order to define a sequence of refiningµ-partitions of [0,1], we adopt the definition of an island from [17]. Let M_k :={1,2,3}^k for k ≥1 and M₀ :=∅. A closed subset I ⊆[0,1] is called alevel-k island with respect to {M_k} if the following conditions hold:

(1) there exists a finite sequence of indexesi0,i1, . . . ,ininMksuch thatSi_k(0,1)∩

S_i_k+1(0,1)6=∅ for all k= 0, . . . , n−1, andI =Sn

k=0S_i_k([0,1]);

(2) for anyj ∈ M_k\ {i₀, . . . ,i_n} and any k ∈ {0, . . . , n}, S_j(0,1)∩S_i_k(0,1) =∅.

Intuitively, for each level-k island I, I^◦ is a connected component of SM_k(0,1) :=

S

i∈M_kSi(0,1) (see Figure 3). For k ≥1, define P_k:=

I :I is a level-k island with respect to {M_k} . (4.25) We note that P₁ = {I_1,1, I_1,0} (see Figure 3). It is easy to see that (P_k)k≥1 is a sequence of refining µ-partitions of [0,1] satisfying condition (P2). By [17, Lemma

(18)

3.5] and the proof of [17, Example 3.3], (P_k)_k≥1 satisfies conditions (P1) and (P3) with ρ=|I_1,1| and m₀ = 0.

0 r X 1

k = 1

I_1,1

I_1,0

r r r

r r

rr r r

r r r

k = 2

r r

rr r r r rr r

rrr r r r r r

r r rr r r r r r

k = 3

Figure 3. µ-partitions Pk for k = 1,2,3, where Pk is defined as in (4.25). Cells that are labeled consist of line segments enclosed by a box. The figure is drawn with r₁ = 1/2 and r₂ = 1/3.

If S₂(1) = S₃(0), then supp(µ) = [0,1]. In particular, we let r₁ = 1/2, r₂ = 1/3.

We can calculate the integrals R

I1,ix^kdµ, where i= 0,1 and k= 0,1,2. Ifp1 =p2 = p₃ = 1/3, then

Z

I1,0

dµ= 1/3,

Z

I1,0

x dµ= 28 99,

Z

I1,0

x²dµ= 8 33, Z

I1,1

dµ= 2/3,

Z

I1,1

x dµ= 26 99,

Z

I1,1

x²dµ= 4 33. Similarly, if p₁ = 2/3, p₂ =p₃ = 1/6, then

Z

I1,0

dµ= 1/6,

Z

I1,0

x dµ= 23 180,

Z

I1,0

x²dµ= 64 645, Z

I1,1

dµ= 5/6,

Z

I1,1

x dµ= 31 180,

Z

I1,1

x²dµ= 38 645.

We can thus calculate the entries of the mass matrixMand solve the linear system (3.6). The result is shown in Figure 4.

5. Neumann boundary condition and additional properties of solutions of heat equations

Letµbe a positive finite Borel measure onRwith supp(µ) = [a, b]. It is well known (see, e.g., [1]) that µ defines a Neumann Laplacian ∆^N_µ such that ∆^N_µu = f if and

(19)

0.0 0.2 0.4 0.6 0.8 1.0 0.2

0.4 0.6 0.8 1.0

(a) p₁=p₂=p₃= 1/3

0.0 0.2 0.4 0.6 0.8 1.0

0.2 0.4 0.6 0.8 1.0

(b) p₁= 2/3, p₂=p₃= 1/6

Figure 4. Numerical solutions of the heat equation corresponding to the self-similar measure defined by the IFS in (4.24) with probability vector (p_i)³_i=1. The initial condition is given byg(x) := sin((5π(x−0.4)) for x ∈ (0.4,0.6), and g(x) := 0 otherwise. Again δ = 0.0001. From top to bottom, the values oftare 0.0, 0.0012, 0.004, 0.01, 0.04, 0.1, 0.2.

only if

E(u, v) = − Z b

a

f v dµ for all v ∈H¹(a, b),

where E(·,·) is defined as in (2.2). In this section, we consider the following heat equation defined by Neumann Laplacians ∆^N_µ:







u_t−∆^N_µu=f on (a, b)×[0, T], u⁰_x = 0 on{a, b} ×[0, T], u=g on (a, b)× {t= 0}.

(5.1) We first note that the existence and uniqueness results of the heat equation 5.1 can be proved by using a method similar to that for equation (1.2).

Letf(x, t)≡0 in (5.1). Similar to (1.4), we obtain

− Z b

a

u_x(x, t)v⁰(x)dx= Z b

a

u_t(x, t)v(x)dµ for all v ∈domE_N. (5.2)

In the rest of this section, we assume the hypotheses of Section 3 and use the notation in Section 3. As in Section 3, we use the finite element method to approximate the weak solution u(x, t) satisfying (5.2) by

u^m(x, t) =

N(m)

X

`=0

β_m,`(t)φ_m,`(x). (5.3)

Thus we require u^m(x, t) to satisfy equation (3.3) and the Neumann boundary condition u^m_x(a, t) = u^m_x(b, t) = 0. The Neumann boundary condition implies that β_m,0 = β_m,1 and β_m,N(m) = β_m,N(m)−1, which is different from the Dirichlet case.

(20)

Consequently, we can obtain an analogue of (3.4):

β_m,1⁰ Z b

a

φm,0(x)φm,i(x)dµ+ Z b

a

φm,1(x)φm,i(x)

+

N(m)−2

X

`=2

β_m,`⁰ Z b

a

φm,`(x)φm,i(x)dµ +β_m,N(m)−1⁰ Z b

a

φ_m,N(m)−1(x)φ_m,i(x)dµ+ Z b

a

φ_m,N(m)(x)φ_m,i(x)

=−β_m,1Z b a

φ⁰_m,0(x)φ⁰_m,i(x)dµ+ Z b

a

φ⁰_m,1(x)φ⁰_m,i(x)

−

N(m)−2

X

`=2

βm,`

Z b a

φ⁰_m,`(x)φ⁰_m,i(x)dx

−β_m,N(m)−1Z b a

φ⁰_m,N(m)−1(x)φ⁰_m,i(x)dµ+ Z b

a

φ⁰_m,N(m)(x)φ⁰_m,i(x)

for 1≤i≤N(m)−1. This system can also be put into a matrix form as (3.6). Using the same derivations as in Scetion 3, we can conclude that (5.2) can be discretized into a system of first-order ordinary differential equations, which has a unique solution that can be solved numerically. That is, Theorem 1.1 holds for the heat equation in (5.1). The numerical solutions of the heat equation (5.1) corresponding to the three different self-similar measures in Section 4 are shown in Figures 5–7, respectively.

0.0 0.2 0.4 0.6 0.8 1.0

0.2 0.4 0.6 0.8 1.0

Figure 5. Numerical solutions of the heat equation (5.1) defined by the infinite Bernoulli convolution associated with the golden ratio. The initial condition is given by the function g(x) := sin((5π(x−0.4)) for x∈ (0.4,0.6), and g(x) := 0 otherwise. Here δ = 0.0001. From top to bottom, the values oft are 0.0, 0.0004, 0.002, 0.01, 0.02 0.04, 0.1.

The following proposition shows that for the system (5.1), energy is conserved.

Proposition 5.1. Letµbe a positive finite Borel measure onRwithsupp(µ) = [a, b], and let u be the weak solution of the heat equation (5.1). Then

h(t) :=

Z b a

u(x, t)dµ(x)

is a constant function of t.

(21)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.2

0.4 0.6 0.8 1.0

Figure 6. Numerical solutions of the heat equation defined by the three-fold convolution of the Cantor measure in (4.21). The initial condition is given by the functiong(x) = sin((5π(x/3−0.4)), x∈(1.2,1.8), and g(x) = 0 otherwise. Here δ = 0.0001. From top to bottom, the values of t are 0.0, 0.0012, 0.004, 0.01, 0.04, 0.1, 0.2.

0.0 0.2 0.4 0.6 0.8 1.0

0.2 0.4 0.6 0.8 1.0

(a) p1=p2=p3= 1/3

0.0 0.2 0.4 0.6 0.8 1.0

0.2 0.4 0.6 0.8 1.0

(b) p1= 2/3, p2=p3= 1/6

Figure 7. Numerical solutions of the heat equation corresponding to the self-similar measure defined by the IFS in (4.24) with probability vector (p_i)³_i=1. The initial condition is given byg(x) := sin((5π(x−0.4)) for x ∈ (0.4,0.6), and g(x) := 0 otherwise. Again δ = 0.0001. From top to bottom, the values of t are 0.0, 0.0012, 0.004, 0.01, 0.02, 0.05, 0.2.

Proof. Fix anyt ∈[0, T]. Then for anyδ >0, there exists ξ ∈(0, δ) such that h(t+δ)−h(t)

δ =

Z b a

u(x, t+δ)−u(x, t)

δ dµ(x)

= Z b

a

u_t(x, t+ξ)dµ(x).

(5.4)

Since u is the weak solution of the heat equation (5.1), (5.4) implies that h(t+δ)−h(t)

δ =

Z b a

∆^N_µu(x, t+ξ)dµ(x)

=− Z b

a

∂u

∂x(x, t+ξ)· ∂

∂x(1)dx= 0,

where in the second equality, we have used the fact that 1 belongs to the domain of the form corresponding to −∆^N_µ. Hence, the assertion follows.