# Semilinear Evolution Problems with Ventcel-Type Conditions on Fractal Boundaries

Authors: Maria Rosaria Lancia; Paola Vernole
Journal: International Journal of Partial Differential Equations
 (2014)

---

## Abstract

A semilinear parabolic transmission problem with Ventcel's boundary conditions on a fractal interfaceS or the corresponding prefractal interface S
h is studied. Regularity results for the solution in both cases are proved. The asymptotic behaviour of the solutions of the approximating problems to the solution of limit fractal problem is analyzed.


---

## Body

## 1. Introduction

In this paper we study the parabolic semilinear second-order transmission problem which we formally state as(1)
(
P
¯
)
{
u
t
(
t
,
P
)
-
Δ
u
(
t
,
P
)
=
J
(
u
(
t
,
P
)
)
in
[
0
,
T
]
×
Q
i
,
-
c
0
Δ
L
u
(
t
,
P
)
=
[
∂
u
(
t
,
P
)
∂
n
]
on
⁡
[
0
,
T
]
×
L
,
u
(
t
,
P
)
=
0
on
⁡
[
0
,
T
]
×
∂
Q
,
u
1
(
t
,
P
)
=
u
2
(
t
,
P
)
on
⁡
[
0
,
T
]
×
L
,
u
(
t
,
P
)
=
0
on
⁡
[
0
,
T
]
×
∂
L
u
(
0
,
P
)
=
ϕ
on
⁡
Q
,
where Q is the bounded open set (
-
1,1
)
2
×
(
0,1
), and L is a “cylindrical” layer dividing the set Q into two subsets Q
1 and Q
2 (see Figure 2). When L is the Koch-type surface S
=
K
×
I, where K is the snowflake and I
=
[
0,1
] (see Section 2), E
L is the energy functional E
S introduced in (12); when L is the prefractal surface S
h, E
L is the energy functional E
S
h introduced in (24). J is a nonlinear function from a subset of L
2
(
Q
) into L
2
(
Q
). u
i denotes the restriction of u to Q
i, [
u
]
=
u
1
-
u
2 denotes the jump of u across L, Δ
L denotes the Laplace operator defined on the layer L (see (12) in Section 3), and [
∂
u
/
∂
n
]
=
∂
u
1
/
∂
n
1
+
∂
u
2
/
∂
n
2 denotes the jump of the normal derivatives across L, to be intended in a suitable sense.More precisely, we assume thatJ
(
u
) is a nonlinear mapping from L
2
p
(
Q
) to L
2
(
Q
) for any fixed p
>
1, locally Lipschitz; that is, Lipschitz on bounded sets in L
2
p
(
Q
) with Lipschitz constant l
(
r
) when restricted to B
(
0
,
r
)
⊂
L
2
p
(
Q
), satisfying suitable growth conditions (see conditions (i) and (ii) in Section 4). Examples of this type of nonlinearity include, for example, J
(
u
)
=
u
|
u
|
p
-
1
,
p
>
1 which occur in combustion theory (see [1]) and in the Navier-Stokes system (see [2]).In the recent years there has been an increasing interest in the study of linear transmission problems across irregular layers of fractal type and the corresponding prefractal layers [3–7]. Problems of this type are also known in the literature as problems with Ventcel’s boundary conditions [8] or second-order transmission conditions. Fractal layers can provide new interesting settings in those model problems, in which the surface absorption of tension, electric conduction, or flow is the relevant effect. The literature on semilinear equations on smooth domains is extensive (see e.g., [9–13] and the recent review in [14]); the fractal case is more awkward (see e.g., [15–19]).In our case one has to take into account that the diffusion phenomenon takes place both across the smooth domainQ and the cylindrical layer L; this fact has a counterpart in the structure of the energy functional E
[
u
] and hence on problem (
P
¯
). In [18] the authors proved local existence and uniqueness results of the “mild” solution of an abstract evolution transmission problem across a prefractal or fractal interface (see (36) and (37)).In this paper we give a strong interpretation of the abstract problem studied in [18],;namely, we prove that the solution of the abstract problem solves problem (
P
¯
) in a suitable sense (see Theorems 22 and 20).The results on the strong interpretation in the prefractal case are deduced by proving regularity results for the solutions of elliptic problems in polyhedral domains. It turns out that the restrictionu
h
i of the solution u
h to Q
h
i belongs to suitable weighted Sobolev spaces (see the proof of Theorem 22). This regularity result is important not only in itself but also in the numerical approximation procedure; to this regard, see [20]. Following this point of view, it is also important to study the asymptotic behaviour of the solutions of the prefractal problems.The proof of the convergence of the solution of the prefractal problems to the one of the (limit) fractal problem relies on the convergence, in the Mosco’s sense, of the energy forms which, in turn, implies the convergence of semigroups in the strong operator topology ofL
2
(
Q
) (see Theorem 16). The plan of the paper is as follows. In Section 2 we describe the geometry of the problem; in Section 3 we introduce the Dirichlet energy forms and the associated semigroups and we recall the results on the convergence of the approximating energy forms (see [21] for details). In Section 4 we recall existence and uniqueness results for the local mild solution as well as global existence and regularity results. In Section 5 we prove that the solution of the abstract Cauchy problems (
P
) and (
P
h
) solves problem (
P
¯
) in the fractal and prefractal cases, respectively, (see Theorems 22 and 20). In Section 6 we prove the convergence of the solutions of the approximating problems to the solution of the limit fractal problem in a suitable functional space. In Appendices A and B, for the reader convenience, we introduce the functional spaces and traces involved.


## 2. Geometry of the Fractal LayersS and S
h

In the paper by|
P
-
P
0
| we denote the Euclidean distance in ℝ
D and the Euclidean balls by B
(
P
0
,
r
)
=
{
P
∈
ℝ
D
:
|
P
-
P
0
|
<
r
},  P
0
∈
ℝ
D,  r
>
0. By the Koch snowflake F, we will denote the union of three coplanar Koch curves (see [22]) K
1, K
2, and K
3 as shown in Figure 1. We assume that the junction points A
1, A
3, and A
5 are the vertices of a regular triangle with unit side length; that is, |
A
1
-
A
3
|
=
|
A
1
-
A
5
|
=
|
A
3
-
A
5
|
=
1. In this section we briefly recall the essential notions on the geometry; for details see [18].Figure 1
Decomposition of the snowflake.Two different viewpoints of the domainQ and the layer S.
(a)
(b)The Hausdorff dimension of the Koch snowflake is given byd
f
=
log
⁡
⁡
4
/
log
⁡
⁡
3. This fractal is no longer self-similar (and hence not nested).One can define, in a natural way, a finite Borel measureμ
F supported on F by
(2)
μ
F
:
=
μ
1
+
μ
2
+
μ
3
,
where μ
i denotes the normalized d
f-dimensional Hausdorff measure, restricted to K
i, i
=
1,2
,
3.The measureμ
F has the property that there exist two positive constants c
1 and c
2:
(3)
c
1
r
d
≤
μ
F
(
B
(
P
,
r
)
∩
F
)
≤
c
2
r
d
,
∀
P
∈
F
,
where d
=
d
f
=
log
⁡
⁡
4
/
log
⁡
⁡
3 and B
(
P
,
r
) denotes the Euclidean ball in ℝ
2. As μ
F is supported on F, it is not ambiguous to write in (3) μ
F
(
B
(
P
,
r
)
) in place of μ
F
(
B
(
P
,
r
)
∩
F
). In the terminology of Appendices A and B, we say that F is a d-set with d
=
d
f.Remark 1.
The Koch snowflake can be also regarded as a fractal manifold (see [23] Section  2.2).LetQ denote a bounded open set in ℝ
3; in our basic model, Q denotes the parallelepiped Q
=
(
-
1,1
)
2
×
(
0,1
) and S denotes a “cylindrical” layer in Q of the type S
=
F
×
I, where I
=
[
0,1
] and F is the Koch snowflake. We assume that S is located in a median position inside Q and divides Q in two subsets Q
1 and Q
2 (see Figure 2).We give a pointP
∈
S the Cartesian coordinates P
=
(
x
,
y
), where x
=
(
x
1
,
x
2
) are the coordinates of the orthogonal projection of P on the plane containing F and y is the coordinate of the orthogonal projection of P on the y-line containing the interval I: P
=
(
x
,
y
)
∈
S, x
=
(
x
1
,
x
2
)
∈
F, y
∈
I.One can define, in a natural way, a finite Borel measurem supported on S as the product measure
(4)
d
m
=
d
μ
F
d
y
,
where d
y denotes the one-dimensional Lebesgue measure on I. The measure m has the property that there exist two positive constants c
1 and c
2:
(5)
c
1
r
d
≤
m
(
B
(
P
,
r
)
∩
S
)
≤
c
2
r
d
,
∀
P
∈
S
,
where d
=
d
f
+
1
=
log
⁡
⁡
12
/
log
⁡
⁡
3 and B
(
P
,
r
) denotes the Euclidean ball in ℝ
3. As m is supported on S, it is not ambiguous to write in (5) m
(
B
(
P
,
r
)
) in place of m
(
B
(
P
,
r
)
∩
S
). Thus S turns out to be a d-set with d
=
d
f
+
1 (see Appendices A and B).ByS
h, we denote the prefractal layer of the type S
h
=
F
h
×
I, h
=
1,2
,
…
,
F
h is the piecewise linear prefractal approximation of F at the step h. S
h is a surface of polyhedral type. S
h divides Q in two subsets Q
h
i,  i
=
1,2.We give a pointP
∈
S
h the Cartesian coordinates P
=
(
x
,
y
), where x
=
(
x
1
,
x
2
) are the coordinates of the orthogonal projection of P on the plane containing F
h and y is the coordinate of the orthogonal projection P on the y-line containing the interval I.


## 3. Energy Forms and Semigroups Associated

### 3.1. The Energy FormE

In this section we introduce the energy functional onS. We first define the energy functional on the cross section F by integrating its Lagrangian on F. For the concept of Lagrangian on fractals, that is, the notion of a measure-valued local energy, we refer to [24, 25]. Here for the sake of simplicity we only mention that the Lagrangian on K, ℒ
K, is a measure-valued map on 𝒟
(
F
)
×
𝒟
(
F
) which is bilinear symmetric and positive (ℒ
K
[
u
] is a positive measure.) The measure-valued Lagrangian takes on the fractal K the role of the Euclidean Lagrangian d
ℒ
(
u
,
v
)
=
∇
u
·
∇
v
d
x. Note that in the case of the Koch curve, the Lagrangian is absolutely continuous with respect to the measure μ; on the contrary, this is not true on most fractals (see [24]). In [23] the Lagrangian ℒ
F on the snowflake F has been defined by using its representation as a fractal manifold. Here we do not give details on the construction and definition of ℒ
F and we refer to Section  4 in [23] for details; in particular in Definition 4.5 a Lagrangian measure ℒ
F on F and the corresponding energy form ℰ
F as
(6)
ℰ
F
(
u
,
v
)
=
∫
F
d
ℒ
F
(
u
,
v
)
with domain 𝒟
(
F
) have been introduced. The domain 𝒟
(
F
), which is a Hilbert space with norm
(7)
(
∥
u
∥
L
2
(
F
,
μ
F
)
2
+
ℰ
F
(
u
,
u
)
)
1
/
2
,
has been characterized in terms of the domains of the energy forms on K
i (see [23] Theorem 4.6).In the following, we will omit the subscriptF, the Lagrangian measure will be simply denoted by ℒ
(
u
,
v
), and we will set ℒ
[
u
]
=
ℒ
(
u
,
u
); an analogous notation will be adopted for the energies.We define the energy formsE
S on the fractal layer S
=
F
×
I by setting
(8)
E
S
[
u
]
=
σ
1
∫
I
∫
F
ℒ
x
[
u
]
(
d
x
)
d
y
+
σ
2
∫
F
∫
I
|
D
y
u
|
2
d
y
μ
F
(
d
x
)
,
where σ
1 and σ
2 are positive constants. Here ℒ
x
(
·
,
·
)
(
d
x
) denotes the measure-valued Lagrangian (of the energy form ℰ
F of F with domain 𝒟
(
F
)) now acting on u
(
x
,
y
) and v
(
x
,
y
) as function of x
∈
F for a.e. y
∈
I; μ
F
(
d
x
) is the d
f-Hausdorff measure acting on each section F of S for a.e. y
∈
I with d
f
=
log
⁡
⁡
4
/
log
⁡
⁡
3; D
y
(
·
) denotes the derivative in the y direction.The formE
S is defined for u
∈
𝒟
(
S
), where 𝒟
(
S
) is the closure in the intrinsic norm
(9)
∥
u
∥
𝒟
(
S
)
=
(
E
S
[
u
]
+
∥
u
∥
L
2
(
S
,
m
)
2
)
1
/
2
of the set
(10)
C
0
(
S
)
∩
L
2
(
(
0,1
)
;
𝒟
(
F
)
)
∩
H
0
1
(
(
0,1
)
;
L
2
(
F
)
)
,
where L
2
(
F
)
=
L
2
(
F
,
μ
F
(
d
x
)
).In the following, we will also use the formE
S
(
u
,
v
) which is obtained from E
S
[
u
] by the polarization identity:
(11)
E
S
(
u
,
v
)
=
1
2
{
E
S
[
u
+
v
]
-
E
S
[
u
]
-
E
S
[
v
]
}
,
u
,
v
∈
𝒟
(
S
)
.Proposition 2.
In the previous notations and assumptions, the formE
S with domain 𝒟
(
S
) is a regular Dirichlet form in L
2
(
S
,
m
) and the space 𝒟
(
S
) is a Hilbert space under the intrinsic norm (9).The proof can be carried on as in Proposition 3.1 of [26]. For the definition and properties of regular Dirichlet forms, we refer to [25]. We now define the Laplace operator on S. As (
E
S
,
𝒟
(
S
)
) is a closed, bilinear form on L
2
(
S
,
m
), there exists (see Chapter 6, Theorem  2.1 in [27]) a unique self-adjoint, nonpositive operator Δ
S on L
2
(
S
,
m
)—with domain 𝒟
(
Δ
S
)
⊆
𝒟
(
S
) dense in L
2
(
S
,
m
)—such that
(12)
E
S
(
u
,
v
)
=
-
∫
S
‍
(
Δ
S
u
)
v
d
m
,
E
S
u
∈
𝒟
(
Δ
S
)
,
v
∈
𝒟
(
S
)
.
Let (
𝒟
(
S
)
)
′ denote the dual of the space 𝒟
(
S
). We now introduce the Laplace operator on the fractal S as a variational operator from 𝒟
(
S
)
→
(
𝒟
(
S
)
)
′ by
(13)
E
S
(
z
,
w
)
=
-
〈
Δ
S
z
,
w
〉
(
𝒟
(
S
)
)
′
,
𝒟
(
S
)
for z
∈
𝒟
(
S
) and for all w
∈
𝒟
(
S
), where 〈
·
,
·
〉
(
𝒟
(
S
)
)
′
,
𝒟
(
S
) is the duality pairing between (
𝒟
(
S
)
)
′ and 𝒟
(
S
). We use the same symbol Δ
S to define the Laplace operator both as a self-adjoint operator in (12) and as a variational operator in (13). It will be clear from the context to which case we refer.In the next, we will also use the spectral dimensionν of S. We find that if r
(
λ
) is the number of eigenvalues associated with E
S smaller than λ, then r
(
λ
)
~
λ
ν
/
2. It can be shown that in our case ν
=
2 (see [28, 29]). We stress the fact that in the fractal case ν
<
d
<
D, while in the Euclidean setting ν
=
d.Consider now the space of functionsu
:
Q
→
ℝ as
(14)
V
(
Q
,
S
)
=
{
u
∈
H
0
1
(
Q
)
:
u
|
S
∈
𝒟
(
S
)
}
.
Here we denote by the symbol f
|
S the trace γ
0
f of f to S (see Appendices A and B).The spaceV
(
Q
,
S
) is nontrivial; see Proposition 3.3 of [4]. We now introduce the energy form
(15)
E
[
u
]
=
∫
Q
|
D
u
|
2
d
Q
+
c
0
E
S
[
u
|
S
]
defined on the domain V
(
Q
,
S
). Here and in the following, d
Q denotes the 3-dimensional Lesbesgue measure and E
(
u
,
v
) denotes the corresponding bilinear form
(16)
E
(
u
,
v
)
=
∫
Q
D
u
D
v
d
Q
+
c
o
E
S
(
u
|
S
,
v
|
S
)
defined on V
(
Q
,
S
)
×
V
(
Q
,
S
).As in Theorem 3.2 of [26], the following result can be proved.Proposition 3.
The formE defined in (15) is a regular Dirichlet form in L
2
(
Q
) and the space V
(
Q
,
S
) is a Hilbert space equipped with the scalar product
(17)
(
u
,
v
)
V
(
Q
,
S
)
=
E
(
u
,
v
)
.We denote by∥
u
∥
V
(
Q
,
S
) the norm in V
(
Q
,
S
), associated with (17), that is
(18)
∥
u
∥
V
(
Q
,
S
)
=
(
c
0
E
S
[
u
|
S
]
+
∫
Q
‍
|
D
u
|
2
d
Q
)
1
/
2
.
As in Propositions (
3.6
) and (
3.1
) in [4], the following result can be proved.Proposition 4.
The space𝒟
(
S
) is embedded in B
β
,
0
2,2
,
β
=
d
f
/
2.Proposition 5.
The space𝒟
(
S
) is embedded in B
α
2,2, α
<
1.As(
E
,
V
(
Q
,
S
)
) is a closed bilinear form on L
2
(
Q
) with domain V
(
Q
,
S
) dense in L
2
(
Q
), there exists (see Chapter 6 Theorem  2.1 in [27]) a unique self-adjoint nonpositive operator A on L
2
(
Q
) with domain 𝒟
(
A
)
⊆
V
(
Q
,
S
) dense in L
2
(
Q
) such that
(19)
E
(
u
,
v
)
=
-
∫
Q
A
u
v
d
Q
,
u
∈
𝒟
(
A
)
,
v
∈
V
(
Q
,
S
)
.
Moreover in Theorem 13.1 of [25] it is proved that to each closed symmetric form E a family of linear operators {
G
α
,
α
>
0
} can be associated with the property
(20)
E
(
G
α
u
,
v
)
+
α
(
G
α
u
,
v
)
=
(
u
,
v
)
,
u
∈
L
2
(
Q
)
,
v
∈
V
(
Q
,
S
)
,
and this family is a strongly continuous resolvent with generator A, which also generates a strongly continuous semigroup {
T
(
t
)
}
t
≥
0.For the reader’s convenience, we recall here the main properties of the semigroup{
T
(
t
)
}
t
≥
0; the reader is referred to Proposition  3.5 in [21] for the proof.Proposition 6.
Let{
T
(
t
)
}
t
≥
0 be the semigroup generated by the operator A associated with the energy form in (19). Then {
T
(
t
)
}
t
≥
0 is an analytic contraction positive preserving semigroup in L
2
(
Q
).Remark 7.
It is well known that the symmetric and contraction analytic semigroupT
(
t
) uniquely determines analytic semigroups on the space L
p
,
1
≤
p
<
∞ (see Theorem 1.4
.
1 [30]) which we still denote by T
(
t
) and by A
p its infinitesimal generator.From Theorem  2.11 in [31], the following estimate on the decay of the heat semigroup holds.Proposition 8.
There exists a positive constantM such that
(21)
∥
T
(
t
)
∥
ℒ
(
L
1
→
L
∞
)
≤
{
M
t
-
n
/
2
,
f
o
r
e
v
e
r
y
t
∈
(
0,1
]
M
t
-
ν
/
2
,
f
o
r
e
v
e
r
y
t
∈
[
1
,
∞
)
.
One will consider the case n
=
3 and ν
=
2; here ν is the spectral dimension of S.From interpolation theory results, it can be proved (see Section  3.1 in [18]) that
(22)
∥
T
(
t
)
∥
L
2
→
L
2
p
≤
{
M
(
1
/
2
)
-
(
1
/
2
p
)
t
-
(
n
/
4
)
(
1
-
1
/
p
)
,
for  every
t
∈
(
0,1
]
M
(
1
/
2
)
-
(
1
/
2
p
)
t
-
(
ν
/
4
)
(
1
-
1
/
p
)
,
for  every
t
∈
[
1
,
∞
)
.


### 3.2. The Energy FormsE
S
h

ByQ we denote the parallelepiped as defined in Section 3 and by S
h we denote the prefractal layer of the type S
h
=
F
h
×
I, h
=
1,2
,
…, F
h is the prefractal approximation of F at the step h (see Section 2). S
h divides Q in two subsets Q
h
i, i
=
1,2.We first construct the energy formsE
S
h on the prefractal layers S
h
=
F
h
×
I, h
∈
ℕ. By ℓ we denote the natural arc-length coordinate on each edge of F
h and we introduce the coordinates x
1
=
x
1
(
ℓ
), x
2
=
x
2
(
ℓ
), and y
=
y on every affine “face” S
h
(
j
) of S
h. By d
ℓ we denote the one-dimensional measure given by the arc-length ℓ and by d
σ are denote the surface measure on each face S
h
(
j
) of S
h; that is, d
σ
=
d
ℓ
d
y. We define E
S
h
[
u
] by setting
(23)
E
S
h
[
u
]
=
∑
j
‍
(
∫
S
h
(
j
)
(
σ
h
1
|
D
ℓ
u
|
2
+
σ
h
2
|
D
y
u
|
2
)
d
σ
)
,
where σ
h
1 and σ
h
2 are positive constants and u
∈
H
1
(
S
h
), the Sobolev space of functions on the piecewise affine set S
h (see Appendices A and B). By Fubini theorem, we can write this functional in the form
(24)
E
S
h
[
u
]
=
σ
h
1
∫
I
(
∫
F
h
|
D
ℓ
u
|
2
d
ℓ
)
d
y
+
σ
h
2
∫
F
h
(
∫
I
|
D
y
u
|
2
d
y
)
d
ℓ
.
We denote the corresponding bilinear form by E
S
h
(
u
,
v
). In the sequel we denote by the symbol f
|
S
h the trace γ
0
f to S
h.Consider now the space of functionsu
:
Q
→
ℝ as
(25)
V
(
Q
,
S
h
)
=
{
u
∈
H
0
1
(
Q
)
:
u
|
S
h
∈
H
0
1
(
S
h
)
}
;
it is not trivial as it contains 𝒟
(
Q
).Consider now the energy form(26)
E
(
h
)
[
u
]
=
∫
Q
‍
|
D
u
|
2
d
Q
+
E
S
h
[
u
|
S
h
]
,
defined on the domain V
(
Q
,
S
h
).ByE
(
h
)
(
u
,
v
) we will denote the corresponding bilinear form
(27)
E
(
h
)
(
u
,
v
)
=
∫
‍
D
u
D
v
d
Q
+
E
S
h
(
u
|
S
h
,
v
|
S
h
)
defined on V
(
Q
,
S
h
)
×
V
(
Q
,
S
h
).Theorem 9.
The formE
(
h
), defined in (26), with domain V
(
Q
,
S
h
) is a regular Dirichlet form in L
2
(
Q
) and the space V
(
Q
,
S
h
) is a Hilbert space equipped with the scalar product
(28)
(
u
,
v
)
V
(
Q
,
S
h
)
=
E
(
h
)
(
u
,
v
)
.For the proof, see Theorem  4.1 in [4].We denote by∥
u
∥
V
(
Q
,
S
h
) the corresponding energy norm in V
(
Q
,
S
h
); that is,
(29)
∥
u
∥
V
(
Q
,
S
h
)
=
(
∫
Q
‍
|
D
u
|
2
d
Q
+
E
S
h
[
u
|
S
h
]
)
1
/
2
.
Proceeding as in Section 3.1 we denote by {
G
α
h
,
α
>
0
}, A
h, and {
T
h
(
t
)
}
t
≥
0 the resolvents, the generators, and the semigroups associated to E
(
h
), for every h
∈
ℕ, respectively.As in Proposition6, the following result can be proved.Proposition 10.
Let{
T
h
(
t
)
}
t
≥
0 be the semigroup generated by the operator A
h associated with the energy form in (27). Then {
T
h
(
t
)
}
t
≥
0 is an analytic contraction positive preserving semigroup in L
2
(
Q
).By proceeding as in Remark7, one can show that for every h
∈
ℕ the symmetric and contraction analytic semigroup T
h
(
t
) uniquely determines analytic semigroups on the space L
p
,
1
<
p
<
∞ (see Theorem 1.4
.
1 [30]) which we still denote by T
h
(
t
) and by A
p
h its infinitesimal generator.The following estimate on the decay of the heat semigroup holds (see e.g., [32]).Proposition 11.
There exists a positive constantM
¯ such that
(30)
∥
T
h
(
t
)
∥
L
1
→
L
∞
≤
{
M
-
t
-
n
/
2
,
f
o
r
e
v
e
r
y
t
∈
(
0,1
]
M
-
t
-
ν
/
2
,
f
o
r
e
v
e
r
y
t
∈
[
1
,
∞
)
,
where M
¯ does not depend on h. One considers the cases n
=
3 and ν
=
2; here ν is the Euclidean dimension of S.As before by interpolation results it can be proved that(31)
∥
T
h
(
t
)
∥
L
2
→
L
2
p
≤
{
M
-
(
1
/
2
)
-
(
1
/
2
p
)
t
-
(
n
/
4
)
(
1
-
1
/
p
)
,
for  every
t
∈
(
0,1
]
M
-
(
1
/
2
)
-
(
1
/
2
p
)
t
-
(
ν
/
4
)
(
1
-
1
/
p
)
,
for  every
t
∈
[
1
,
∞
)
.


### 3.3. The Convergence of Forms and Semigroups

We now recall the results proved in [21] on the convergence of the approximating energy forms E
(
h
) to the fractal energy E. In this asymptotic behaviour, the factors σ
h
1 and σ
h
2 have a key role and can be regarded as a sort of renormalization factors of the approximating energies. These factors take into account the nonrectifiability of the curve F and hence the irregularity of the surface S and in particular the effect of the D-dimensional length intrinsic to the curve; for details, see [6]. The convergence of functional is here intended in the sense of the M-convergence which we define below.


#### 3.3.1. TheM-Convergence of Forms

We recall, for the sake of completeness, the definition ofM-convergence of forms introduced by Mosco in [33].We extend the formE defined in (15) and E
(
h
) defined in (26) on the whole space L
2
(
Q
) by defining
(32)
E
[
u
]
=
+
∞
for  every
u
∈
L
2
(
Q
)
V
(
Q
,
S
)
,
E
(
h
)
[
u
]
=
+
∞
for  every
u
∈
L
2
(
Q
)
V
(
Q
,
S
h
)
.Definition 12.
A sequence of form{
E
(
h
)
} M-converges to a form E in L
2
(
Q
) if(a)for  every
{
v
h
} converging weakly to u in L
2
(
Q
)
(33)
lim
⁡
inf
⁡
h
→
∞
⁡
E
(
h
)
[
v
h
]
≥
E
[
u
]
;
(b)for  every
u
∈
L
2
(
Q
) there exists {
w
h
} converging strongly to u in L
2
(
Q
) such that
(34)
lim
⁡
sup
⁡
h
→
∞
⁡
E
(
h
)
[
w
h
]
≤
E
[
u
]
.Definition 13.
The sequence of forms{
E
(
h
)
} is asymptotically compact in L
2
(
Q
) if every sequence {
u
h
} with
(35)
lim
⁡
inf
⁡
h
→
∞
⁡
E
(
h
)
[
u
h
]
+
∫
Q
|
u
h
|
2
d
Q
<
∞
has a subsequence strongly convergent in L
2
(
Q
).Proposition 14.
The sequence of forms (26) is asymptotically compact in L
2
(
Q
).Remark 15.
We point out that, as the sequence of forms (26) is asymptotically compact in L
2
(
Q
), M-convergence is equivalent to the Γ-convergence (see Lemma  2.3.2 in [34]) and thus we can take in (a) v
h strongly converging to u in L
2
(
Q
).Theorem 16.
Letσ
h
1
=
σ
1
c
0
(
3
d
f
-
1
)
h and σ
h
2
=
σ
2
c
0
(
3
1
-
d
f
)
h; then the sequence of forms {
E
(
h
)
} defined in (26) M-converges in the space L
2
(
Q
) to the form E defined in (15). The sequence of semigroups {
T
h
(
t
)
} associated with the form E
(
h
) converges to the semigroup T
(
t
) associated with the form E in the strong operator topology of L
2
(
Q
) uniformly on every interval [
0
,
t
1
].


## 3.1. The Energy FormE

In this section we introduce the energy functional onS. We first define the energy functional on the cross section F by integrating its Lagrangian on F. For the concept of Lagrangian on fractals, that is, the notion of a measure-valued local energy, we refer to [24, 25]. Here for the sake of simplicity we only mention that the Lagrangian on K, ℒ
K, is a measure-valued map on 𝒟
(
F
)
×
𝒟
(
F
) which is bilinear symmetric and positive (ℒ
K
[
u
] is a positive measure.) The measure-valued Lagrangian takes on the fractal K the role of the Euclidean Lagrangian d
ℒ
(
u
,
v
)
=
∇
u
·
∇
v
d
x. Note that in the case of the Koch curve, the Lagrangian is absolutely continuous with respect to the measure μ; on the contrary, this is not true on most fractals (see [24]). In [23] the Lagrangian ℒ
F on the snowflake F has been defined by using its representation as a fractal manifold. Here we do not give details on the construction and definition of ℒ
F and we refer to Section  4 in [23] for details; in particular in Definition 4.5 a Lagrangian measure ℒ
F on F and the corresponding energy form ℰ
F as
(6)
ℰ
F
(
u
,
v
)
=
∫
F
d
ℒ
F
(
u
,
v
)
with domain 𝒟
(
F
) have been introduced. The domain 𝒟
(
F
), which is a Hilbert space with norm
(7)
(
∥
u
∥
L
2
(
F
,
μ
F
)
2
+
ℰ
F
(
u
,
u
)
)
1
/
2
,
has been characterized in terms of the domains of the energy forms on K
i (see [23] Theorem 4.6).In the following, we will omit the subscriptF, the Lagrangian measure will be simply denoted by ℒ
(
u
,
v
), and we will set ℒ
[
u
]
=
ℒ
(
u
,
u
); an analogous notation will be adopted for the energies.We define the energy formsE
S on the fractal layer S
=
F
×
I by setting
(8)
E
S
[
u
]
=
σ
1
∫
I
∫
F
ℒ
x
[
u
]
(
d
x
)
d
y
+
σ
2
∫
F
∫
I
|
D
y
u
|
2
d
y
μ
F
(
d
x
)
,
where σ
1 and σ
2 are positive constants. Here ℒ
x
(
·
,
·
)
(
d
x
) denotes the measure-valued Lagrangian (of the energy form ℰ
F of F with domain 𝒟
(
F
)) now acting on u
(
x
,
y
) and v
(
x
,
y
) as function of x
∈
F for a.e. y
∈
I; μ
F
(
d
x
) is the d
f-Hausdorff measure acting on each section F of S for a.e. y
∈
I with d
f
=
log
⁡
⁡
4
/
log
⁡
⁡
3; D
y
(
·
) denotes the derivative in the y direction.The formE
S is defined for u
∈
𝒟
(
S
), where 𝒟
(
S
) is the closure in the intrinsic norm
(9)
∥
u
∥
𝒟
(
S
)
=
(
E
S
[
u
]
+
∥
u
∥
L
2
(
S
,
m
)
2
)
1
/
2
of the set
(10)
C
0
(
S
)
∩
L
2
(
(
0,1
)
;
𝒟
(
F
)
)
∩
H
0
1
(
(
0,1
)
;
L
2
(
F
)
)
,
where L
2
(
F
)
=
L
2
(
F
,
μ
F
(
d
x
)
).In the following, we will also use the formE
S
(
u
,
v
) which is obtained from E
S
[
u
] by the polarization identity:
(11)
E
S
(
u
,
v
)
=
1
2
{
E
S
[
u
+
v
]
-
E
S
[
u
]
-
E
S
[
v
]
}
,
u
,
v
∈
𝒟
(
S
)
.Proposition 2.
In the previous notations and assumptions, the formE
S with domain 𝒟
(
S
) is a regular Dirichlet form in L
2
(
S
,
m
) and the space 𝒟
(
S
) is a Hilbert space under the intrinsic norm (9).The proof can be carried on as in Proposition 3.1 of [26]. For the definition and properties of regular Dirichlet forms, we refer to [25]. We now define the Laplace operator on S. As (
E
S
,
𝒟
(
S
)
) is a closed, bilinear form on L
2
(
S
,
m
), there exists (see Chapter 6, Theorem  2.1 in [27]) a unique self-adjoint, nonpositive operator Δ
S on L
2
(
S
,
m
)—with domain 𝒟
(
Δ
S
)
⊆
𝒟
(
S
) dense in L
2
(
S
,
m
)—such that
(12)
E
S
(
u
,
v
)
=
-
∫
S
‍
(
Δ
S
u
)
v
d
m
,
E
S
u
∈
𝒟
(
Δ
S
)
,
v
∈
𝒟
(
S
)
.
Let (
𝒟
(
S
)
)
′ denote the dual of the space 𝒟
(
S
). We now introduce the Laplace operator on the fractal S as a variational operator from 𝒟
(
S
)
→
(
𝒟
(
S
)
)
′ by
(13)
E
S
(
z
,
w
)
=
-
〈
Δ
S
z
,
w
〉
(
𝒟
(
S
)
)
′
,
𝒟
(
S
)
for z
∈
𝒟
(
S
) and for all w
∈
𝒟
(
S
), where 〈
·
,
·
〉
(
𝒟
(
S
)
)
′
,
𝒟
(
S
) is the duality pairing between (
𝒟
(
S
)
)
′ and 𝒟
(
S
). We use the same symbol Δ
S to define the Laplace operator both as a self-adjoint operator in (12) and as a variational operator in (13). It will be clear from the context to which case we refer.In the next, we will also use the spectral dimensionν of S. We find that if r
(
λ
) is the number of eigenvalues associated with E
S smaller than λ, then r
(
λ
)
~
λ
ν
/
2. It can be shown that in our case ν
=
2 (see [28, 29]). We stress the fact that in the fractal case ν
<
d
<
D, while in the Euclidean setting ν
=
d.Consider now the space of functionsu
:
Q
→
ℝ as
(14)
V
(
Q
,
S
)
=
{
u
∈
H
0
1
(
Q
)
:
u
|
S
∈
𝒟
(
S
)
}
.
Here we denote by the symbol f
|
S the trace γ
0
f of f to S (see Appendices A and B).The spaceV
(
Q
,
S
) is nontrivial; see Proposition 3.3 of [4]. We now introduce the energy form
(15)
E
[
u
]
=
∫
Q
|
D
u
|
2
d
Q
+
c
0
E
S
[
u
|
S
]
defined on the domain V
(
Q
,
S
). Here and in the following, d
Q denotes the 3-dimensional Lesbesgue measure and E
(
u
,
v
) denotes the corresponding bilinear form
(16)
E
(
u
,
v
)
=
∫
Q
D
u
D
v
d
Q
+
c
o
E
S
(
u
|
S
,
v
|
S
)
defined on V
(
Q
,
S
)
×
V
(
Q
,
S
).As in Theorem 3.2 of [26], the following result can be proved.Proposition 3.
The formE defined in (15) is a regular Dirichlet form in L
2
(
Q
) and the space V
(
Q
,
S
) is a Hilbert space equipped with the scalar product
(17)
(
u
,
v
)
V
(
Q
,
S
)
=
E
(
u
,
v
)
.We denote by∥
u
∥
V
(
Q
,
S
) the norm in V
(
Q
,
S
), associated with (17), that is
(18)
∥
u
∥
V
(
Q
,
S
)
=
(
c
0
E
S
[
u
|
S
]
+
∫
Q
‍
|
D
u
|
2
d
Q
)
1
/
2
.
As in Propositions (
3.6
) and (
3.1
) in [4], the following result can be proved.Proposition 4.
The space𝒟
(
S
) is embedded in B
β
,
0
2,2
,
β
=
d
f
/
2.Proposition 5.
The space𝒟
(
S
) is embedded in B
α
2,2, α
<
1.As(
E
,
V
(
Q
,
S
)
) is a closed bilinear form on L
2
(
Q
) with domain V
(
Q
,
S
) dense in L
2
(
Q
), there exists (see Chapter 6 Theorem  2.1 in [27]) a unique self-adjoint nonpositive operator A on L
2
(
Q
) with domain 𝒟
(
A
)
⊆
V
(
Q
,
S
) dense in L
2
(
Q
) such that
(19)
E
(
u
,
v
)
=
-
∫
Q
A
u
v
d
Q
,
u
∈
𝒟
(
A
)
,
v
∈
V
(
Q
,
S
)
.
Moreover in Theorem 13.1 of [25] it is proved that to each closed symmetric form E a family of linear operators {
G
α
,
α
>
0
} can be associated with the property
(20)
E
(
G
α
u
,
v
)
+
α
(
G
α
u
,
v
)
=
(
u
,
v
)
,
u
∈
L
2
(
Q
)
,
v
∈
V
(
Q
,
S
)
,
and this family is a strongly continuous resolvent with generator A, which also generates a strongly continuous semigroup {
T
(
t
)
}
t
≥
0.For the reader’s convenience, we recall here the main properties of the semigroup{
T
(
t
)
}
t
≥
0; the reader is referred to Proposition  3.5 in [21] for the proof.Proposition 6.
Let{
T
(
t
)
}
t
≥
0 be the semigroup generated by the operator A associated with the energy form in (19). Then {
T
(
t
)
}
t
≥
0 is an analytic contraction positive preserving semigroup in L
2
(
Q
).Remark 7.
It is well known that the symmetric and contraction analytic semigroupT
(
t
) uniquely determines analytic semigroups on the space L
p
,
1
≤
p
<
∞ (see Theorem 1.4
.
1 [30]) which we still denote by T
(
t
) and by A
p its infinitesimal generator.From Theorem  2.11 in [31], the following estimate on the decay of the heat semigroup holds.Proposition 8.
There exists a positive constantM such that
(21)
∥
T
(
t
)
∥
ℒ
(
L
1
→
L
∞
)
≤
{
M
t
-
n
/
2
,
f
o
r
e
v
e
r
y
t
∈
(
0,1
]
M
t
-
ν
/
2
,
f
o
r
e
v
e
r
y
t
∈
[
1
,
∞
)
.
One will consider the case n
=
3 and ν
=
2; here ν is the spectral dimension of S.From interpolation theory results, it can be proved (see Section  3.1 in [18]) that
(22)
∥
T
(
t
)
∥
L
2
→
L
2
p
≤
{
M
(
1
/
2
)
-
(
1
/
2
p
)
t
-
(
n
/
4
)
(
1
-
1
/
p
)
,
for  every
t
∈
(
0,1
]
M
(
1
/
2
)
-
(
1
/
2
p
)
t
-
(
ν
/
4
)
(
1
-
1
/
p
)
,
for  every
t
∈
[
1
,
∞
)
.


## 3.2. The Energy FormsE
S
h

ByQ we denote the parallelepiped as defined in Section 3 and by S
h we denote the prefractal layer of the type S
h
=
F
h
×
I, h
=
1,2
,
…, F
h is the prefractal approximation of F at the step h (see Section 2). S
h divides Q in two subsets Q
h
i, i
=
1,2.We first construct the energy formsE
S
h on the prefractal layers S
h
=
F
h
×
I, h
∈
ℕ. By ℓ we denote the natural arc-length coordinate on each edge of F
h and we introduce the coordinates x
1
=
x
1
(
ℓ
), x
2
=
x
2
(
ℓ
), and y
=
y on every affine “face” S
h
(
j
) of S
h. By d
ℓ we denote the one-dimensional measure given by the arc-length ℓ and by d
σ are denote the surface measure on each face S
h
(
j
) of S
h; that is, d
σ
=
d
ℓ
d
y. We define E
S
h
[
u
] by setting
(23)
E
S
h
[
u
]
=
∑
j
‍
(
∫
S
h
(
j
)
(
σ
h
1
|
D
ℓ
u
|
2
+
σ
h
2
|
D
y
u
|
2
)
d
σ
)
,
where σ
h
1 and σ
h
2 are positive constants and u
∈
H
1
(
S
h
), the Sobolev space of functions on the piecewise affine set S
h (see Appendices A and B). By Fubini theorem, we can write this functional in the form
(24)
E
S
h
[
u
]
=
σ
h
1
∫
I
(
∫
F
h
|
D
ℓ
u
|
2
d
ℓ
)
d
y
+
σ
h
2
∫
F
h
(
∫
I
|
D
y
u
|
2
d
y
)
d
ℓ
.
We denote the corresponding bilinear form by E
S
h
(
u
,
v
). In the sequel we denote by the symbol f
|
S
h the trace γ
0
f to S
h.Consider now the space of functionsu
:
Q
→
ℝ as
(25)
V
(
Q
,
S
h
)
=
{
u
∈
H
0
1
(
Q
)
:
u
|
S
h
∈
H
0
1
(
S
h
)
}
;
it is not trivial as it contains 𝒟
(
Q
).Consider now the energy form(26)
E
(
h
)
[
u
]
=
∫
Q
‍
|
D
u
|
2
d
Q
+
E
S
h
[
u
|
S
h
]
,
defined on the domain V
(
Q
,
S
h
).ByE
(
h
)
(
u
,
v
) we will denote the corresponding bilinear form
(27)
E
(
h
)
(
u
,
v
)
=
∫
‍
D
u
D
v
d
Q
+
E
S
h
(
u
|
S
h
,
v
|
S
h
)
defined on V
(
Q
,
S
h
)
×
V
(
Q
,
S
h
).Theorem 9.
The formE
(
h
), defined in (26), with domain V
(
Q
,
S
h
) is a regular Dirichlet form in L
2
(
Q
) and the space V
(
Q
,
S
h
) is a Hilbert space equipped with the scalar product
(28)
(
u
,
v
)
V
(
Q
,
S
h
)
=
E
(
h
)
(
u
,
v
)
.For the proof, see Theorem  4.1 in [4].We denote by∥
u
∥
V
(
Q
,
S
h
) the corresponding energy norm in V
(
Q
,
S
h
); that is,
(29)
∥
u
∥
V
(
Q
,
S
h
)
=
(
∫
Q
‍
|
D
u
|
2
d
Q
+
E
S
h
[
u
|
S
h
]
)
1
/
2
.
Proceeding as in Section 3.1 we denote by {
G
α
h
,
α
>
0
}, A
h, and {
T
h
(
t
)
}
t
≥
0 the resolvents, the generators, and the semigroups associated to E
(
h
), for every h
∈
ℕ, respectively.As in Proposition6, the following result can be proved.Proposition 10.
Let{
T
h
(
t
)
}
t
≥
0 be the semigroup generated by the operator A
h associated with the energy form in (27). Then {
T
h
(
t
)
}
t
≥
0 is an analytic contraction positive preserving semigroup in L
2
(
Q
).By proceeding as in Remark7, one can show that for every h
∈
ℕ the symmetric and contraction analytic semigroup T
h
(
t
) uniquely determines analytic semigroups on the space L
p
,
1
<
p
<
∞ (see Theorem 1.4
.
1 [30]) which we still denote by T
h
(
t
) and by A
p
h its infinitesimal generator.The following estimate on the decay of the heat semigroup holds (see e.g., [32]).Proposition 11.
There exists a positive constantM
¯ such that
(30)
∥
T
h
(
t
)
∥
L
1
→
L
∞
≤
{
M
-
t
-
n
/
2
,
f
o
r
e
v
e
r
y
t
∈
(
0,1
]
M
-
t
-
ν
/
2
,
f
o
r
e
v
e
r
y
t
∈
[
1
,
∞
)
,
where M
¯ does not depend on h. One considers the cases n
=
3 and ν
=
2; here ν is the Euclidean dimension of S.As before by interpolation results it can be proved that(31)
∥
T
h
(
t
)
∥
L
2
→
L
2
p
≤
{
M
-
(
1
/
2
)
-
(
1
/
2
p
)
t
-
(
n
/
4
)
(
1
-
1
/
p
)
,
for  every
t
∈
(
0,1
]
M
-
(
1
/
2
)
-
(
1
/
2
p
)
t
-
(
ν
/
4
)
(
1
-
1
/
p
)
,
for  every
t
∈
[
1
,
∞
)
.


## 3.3. The Convergence of Forms and Semigroups

We now recall the results proved in [21] on the convergence of the approximating energy forms E
(
h
) to the fractal energy E. In this asymptotic behaviour, the factors σ
h
1 and σ
h
2 have a key role and can be regarded as a sort of renormalization factors of the approximating energies. These factors take into account the nonrectifiability of the curve F and hence the irregularity of the surface S and in particular the effect of the D-dimensional length intrinsic to the curve; for details, see [6]. The convergence of functional is here intended in the sense of the M-convergence which we define below.


### 3.3.1. TheM-Convergence of Forms

We recall, for the sake of completeness, the definition ofM-convergence of forms introduced by Mosco in [33].We extend the formE defined in (15) and E
(
h
) defined in (26) on the whole space L
2
(
Q
) by defining
(32)
E
[
u
]
=
+
∞
for  every
u
∈
L
2
(
Q
)
V
(
Q
,
S
)
,
E
(
h
)
[
u
]
=
+
∞
for  every
u
∈
L
2
(
Q
)
V
(
Q
,
S
h
)
.Definition 12.
A sequence of form{
E
(
h
)
} M-converges to a form E in L
2
(
Q
) if(a)for  every
{
v
h
} converging weakly to u in L
2
(
Q
)
(33)
lim
⁡
inf
⁡
h
→
∞
⁡
E
(
h
)
[
v
h
]
≥
E
[
u
]
;
(b)for  every
u
∈
L
2
(
Q
) there exists {
w
h
} converging strongly to u in L
2
(
Q
) such that
(34)
lim
⁡
sup
⁡
h
→
∞
⁡
E
(
h
)
[
w
h
]
≤
E
[
u
]
.Definition 13.
The sequence of forms{
E
(
h
)
} is asymptotically compact in L
2
(
Q
) if every sequence {
u
h
} with
(35)
lim
⁡
inf
⁡
h
→
∞
⁡
E
(
h
)
[
u
h
]
+
∫
Q
|
u
h
|
2
d
Q
<
∞
has a subsequence strongly convergent in L
2
(
Q
).Proposition 14.
The sequence of forms (26) is asymptotically compact in L
2
(
Q
).Remark 15.
We point out that, as the sequence of forms (26) is asymptotically compact in L
2
(
Q
), M-convergence is equivalent to the Γ-convergence (see Lemma  2.3.2 in [34]) and thus we can take in (a) v
h strongly converging to u in L
2
(
Q
).Theorem 16.
Letσ
h
1
=
σ
1
c
0
(
3
d
f
-
1
)
h and σ
h
2
=
σ
2
c
0
(
3
1
-
d
f
)
h; then the sequence of forms {
E
(
h
)
} defined in (26) M-converges in the space L
2
(
Q
) to the form E defined in (15). The sequence of semigroups {
T
h
(
t
)
} associated with the form E
(
h
) converges to the semigroup T
(
t
) associated with the form E in the strong operator topology of L
2
(
Q
) uniformly on every interval [
0
,
t
1
].


## 3.3.1. TheM-Convergence of Forms

We recall, for the sake of completeness, the definition ofM-convergence of forms introduced by Mosco in [33].We extend the formE defined in (15) and E
(
h
) defined in (26) on the whole space L
2
(
Q
) by defining
(32)
E
[
u
]
=
+
∞
for  every
u
∈
L
2
(
Q
)
V
(
Q
,
S
)
,
E
(
h
)
[
u
]
=
+
∞
for  every
u
∈
L
2
(
Q
)
V
(
Q
,
S
h
)
.Definition 12.
A sequence of form{
E
(
h
)
} M-converges to a form E in L
2
(
Q
) if(a)for  every
{
v
h
} converging weakly to u in L
2
(
Q
)
(33)
lim
⁡
inf
⁡
h
→
∞
⁡
E
(
h
)
[
v
h
]
≥
E
[
u
]
;
(b)for  every
u
∈
L
2
(
Q
) there exists {
w
h
} converging strongly to u in L
2
(
Q
) such that
(34)
lim
⁡
sup
⁡
h
→
∞
⁡
E
(
h
)
[
w
h
]
≤
E
[
u
]
.Definition 13.
The sequence of forms{
E
(
h
)
} is asymptotically compact in L
2
(
Q
) if every sequence {
u
h
} with
(35)
lim
⁡
inf
⁡
h
→
∞
⁡
E
(
h
)
[
u
h
]
+
∫
Q
|
u
h
|
2
d
Q
<
∞
has a subsequence strongly convergent in L
2
(
Q
).Proposition 14.
The sequence of forms (26) is asymptotically compact in L
2
(
Q
).Remark 15.
We point out that, as the sequence of forms (26) is asymptotically compact in L
2
(
Q
), M-convergence is equivalent to the Γ-convergence (see Lemma  2.3.2 in [34]) and thus we can take in (a) v
h strongly converging to u in L
2
(
Q
).Theorem 16.
Letσ
h
1
=
σ
1
c
0
(
3
d
f
-
1
)
h and σ
h
2
=
σ
2
c
0
(
3
1
-
d
f
)
h; then the sequence of forms {
E
(
h
)
} defined in (26) M-converges in the space L
2
(
Q
) to the form E defined in (15). The sequence of semigroups {
T
h
(
t
)
} associated with the form E
(
h
) converges to the semigroup T
(
t
) associated with the form E in the strong operator topology of L
2
(
Q
) uniformly on every interval [
0
,
t
1
].


## 4. Evolution Problems: Existence and Convergence of the Solutions

In this Section we recall the results on existence and uniqueness of the solution of the abstract problems(
P
) and (
P
h
) (see below) and the asymptotic behaviour of the solutions of the abstract problems. In Section 5 we will show that the solutions of the abstract problems solve (
P
¯
) in both cases. We refer the reader to [18].We consider the abstract Cauchy problems as(36)
(
P
)
{
d
u
(
t
)
d
t
=
A
u
(
t
)
+
J
(
u
(
t
)
)
,
0
≤
t
≤
T
u
(
0
)
=
ϕ
and for every h
∈
ℕ
(37)
(
P
h
)
{
d
u
h
(
t
)
d
t
=
A
h
u
h
(
t
)
+
J
(
u
h
(
t
)
)
,
0
≤
t
≤
T
u
h
(
0
)
=
ϕ
h
,
where A
:
𝒟
(
A
)
⊂
L
2
(
Q
)
→
L
2
(
Q
) and A
h
:
𝒟
(
A
h
)
⊂
L
2
(
Q
)
→
L
2
(
Q
) are the generators associated, respectively, to the energy form E and the energy forms E
(
h
) introduced in (15) and (26), T is a fixed positive real number, and ϕ and ϕ
h are given functions in L
2
(
Q
). We assume that J is a mapping from L
2
p
(
Q
)
→
L
2
(
Q
), p
>
1 locally Lipschitz, that is, Lipschitz on bounded sets in L
2
p
(
Q
); we let l
(
r
) denote the Lipschitz constant of J:
(38)
∥
J
(
u
)
-
J
(
v
)
∥
L
2
(
Q
)
≤
l
(
r
)
∥
u
-
v
∥
L
2
p
(
Q
)
,
where ∥
u
∥
L
2
p
(
Q
)
≤
r
,
∥
v
∥
L
2
p
(
Q
)
≤
r. We also assume that J
(
0
)
=
0. This assumption is not necessary in all that follows, but it simplifies the calculations (see [11]). In order to prove the local existence theorem, we make the following assumptions on the growth of l
(
r
) when r
→
∞.We set for brevitya
:
=
(
n
/
4
)
(
1
-
(
1
/
p
)
); we note that 0
<
a
<
1, for n
≤
4, and p
>
1.(i)
There exists0
<
b
<
a such that l
(
r
)
=
𝒪
(
r
(
1
-
a
)
/
b
)
,
r
→
∞.
(ii)
Consider(39)
∫
τ
+
∞
l
(
r
)
r
-
1
/
a
d
r
<
∞
for everyτ
>
0.We note that (ii) implies (i) for all0
<
b
<
a since l
(
r
) is nondecreasing and
(40)
∫
r
2
r
‍
l
(
s
)
s
-
1
/
a
d
s
≥
r
l
(
r
)
(
2
r
)
-
1
/
a
.
Thus l
(
r
)
r
1
-
(
1
/
a
) is bounded as r
→
∞ which implies (i) for 0
<
b
<
a.In Theorem 5.1 of [18], the following local existence theorem has been proved.Theorem 17.
Let condition(i) hold. Let K
>
0 be sufficiently small if ϕ
∈
L
2
(
Q
) and
(41)
lim
⁡
sup
⁡
t
→
0
⁡
∥
t
b
T
(
t
)
ϕ
∥
L
2
p
<
K
.
There is a T
>
0 and a
(42)
u
∈
C
(
[
0
,
T
]
;
L
2
(
Q
)
)
with u
(
0
)
=
ϕ satisfying(1)
u
∈
C
(
(
0
,
T
]
;
L
2
p
(
Q
)
), and ∥
t
b
u
(
t
)
∥
L
2
p
<
2
K;
(2)
for everyt
∈
[
0
,
T
],
(43)
u
(
t
)
=
T
(
t
)
ϕ
+
∫
0
t
T
(
t
-
s
)
J
(
u
(
s
)
)
d
s
with the integral being both anL
2-valued and L
2
p-valued Bochner integral;
(3)
ifv
:
(
0
,
T
1
]
→
L
2
p is strongly measurable with T
1
≤
T
,
∥
t
b
v
(
t
)
∥
L
2
p
≤
2
K and also satisfies (43), then u
(
t
)
=
v
(
t
), for every t
∈
(
0
,
T
1
].
Let condition(ii) hold; there exist a T
>
0 and a unique u
(
t
)
∈
C
(
[
0
,
T
]
;
L
2
(
Q
)
) with u
(
0
)
=
ϕ satisfying(1)
u
∈
C
(
(
0
,
T
]
;
L
2
p
(
Q
)
)
(44)
lim
⁡
sup
⁡
t
→
0
⁡
∥
t
a
u
(
t
)
∥
L
2
p
<
∞
;
(2)
for everyt
∈
[
0
,
T
], u
(
t
) satisfies (43) with the integral being both an L
2-valued and L
2
p-valued Bochner integral;
(3)
ifv
:
(
0
,
T
1
]
→
L
2
p is strongly measurable with T
1
≤
T
,
∥
t
a
v
(
t
)
∥
L
2
p bounded and also satisfies (43), then u
(
t
)
=
v
(
t
), for every t
∈
(
0
,
T
1
].The claim of the Theorem is proved by a contraction mapping argument on suitable spaces of continuous functions with values in Banach space.By exploiting the analyticity of the semigroupT
(
t
) both on L
2
(
Q
) and L
2
p
(
Q
), the following regularity result for the maximal solution holds (see Theorem 5.3 [18]).Theorem 18.
Under the assumptions of Theorem17, one has that the solution u
(
t
) can be continuously extended to a maximal interval (
0
,
T
ϕ
) as a solution of (43), until ∥
u
(
t
)
∥
L
2
p
(
Q
)
→
∞ as t
→
T
ϕ, and it is a classical solution; that is,
(45)
u
∈
C
1
(
(
0
,
T
ϕ
)
;
L
2
(
Q
)
)
∩
C
(
(
0
,
T
ϕ
)
;
𝒟
(
A
)
)
and satisfies
(46)
d
u
(
t
)
d
t
=
A
u
(
t
)
+
J
(
u
)
,
f
o
r
e
v
e
r
y
t
∈
(
0
,
T
ϕ
)
.For every fixedh
∈
ℕ, the claims of Theorems 17 and 18 hold for problem (
P
h
) with the obvious changes.We now recall the convergence results of the sequence of the approximating solutions{
u
h
} when h goes to infinity (see Theorem  6.2 in [18]).Theorem 19.
Letu and u
h be the mild solutions of problems (
P
) and (
P
h
); let σ
h
1 and σ
h
2 be as in Theorem 16. In the notations and assumptions of Theorem 17, one has the following;(a)
let assumption (i) hold; letϕ
h and ϕ belong to L
q
(
Q
) with q
=
2
p
n
/
(
n
+
4
p
b
) and ϕ
h
→
ϕ in Lq
(
Q
); then
(47)
u
h
→
u
i
n
C
(
[
0
,
T
]
;
L
2
(
Q
)
)
,
sup
⁡
t
∈
[
0
,
T
]
⁡
∥
t
b
[
u
(
t
)
-
u
h
(
t
)
]
∥
L
2
p
(
Q
)
→
0
;
(b)
if assumption (ii) holds andϕ
h
→
ϕ in L
2
(
Q
), then
(48)
u
h
→
u
i
n
C
(
[
0
,
T
]
;
L
2
(
Q
)
)
sup
⁡
t
∈
[
0
,
T
]
⁡
∥
t
a
[
u
(
t
)
-
u
h
(
t
)
]
∥
L
2
p
(
Q
)
→
0
witha
=
n
/
4
(
1
-
1
/
p
).


## 5. Strong Formulation of the Transmission Problems

### 5.1. The Fractal Layer

Theorem 20.
Letu be the solution of problem (
P
). Then one has, for every fixed t
∈
(
0
,
T
],
(49)
u
t
(
t
,
P
)
-
Δ
u
(
t
,
P
)
=
J
(
u
(
t
,
P
)
)
u
t
(
t
P
D
)
f
o
r
a
.
e
.
P
∈
Q
i
i
=
1,2
,
∂
u
i
∂
n
i
∈
(
B
β
,
0
2,2
(
S
)
)
′
,
β
=
d
f
2
,
i
=
1,2
,
-
c
0
〈
Δ
S
u
|
S
,
z
〉
(
𝒟
(
S
)
)
′
𝒟
(
S
)
=
〈
[
∂
u
∂
n
]
,
z
〉
(
𝒟
(
S
)
)
′
𝒟
(
S
)
,
f
o
r
e
v
e
r
y
z
∈
𝒟
(
S
)
,
u
(
t
,
P
)
=
0
f
o
r
P
∈
∂
Q
,
u
1
=
u
2
i
n
B
d
f
/
2
2,2
(
S
)
,
u
(
0
,
P
)
=
ϕ
(
P
)
i
n
L
2
(
Q
)
,
where u
i is the restriction of u to Q
i, ∂
u
i
/
∂
n
i, i
=
1,2 is the inward “normal derivative,” to be defined in a suitable sense, [
∂
u
/
∂
n
] = (
∂
u
1
/
∂
n
1
)
+
(
∂
u
2
/
∂
n
2
) is the jump of the normal derivative, and Δ
S is the fractal Laplacian. Moreover ∂
u
i
/
∂
n
i
∈
C
(
(
0
,
T
]
;
(
B
β
,
0
2,2
(
S
)
)
′
).Proof.
Letφ
(
P
) be an arbitrary function in V
(
Q
,
S
) such that φ
(
P
)
|
Q
i
∈
𝒟
(
Q
i
); by multiplying for φ (36) in (
P
) and integrating over Q we have
(50)
∫
Q
u
t
(
t
,
P
)
φ
(
P
)
d
Q
=
∫
Q
A
u
(
t
,
P
)
φ
(
P
)
d
Q
+
∫
Q
J
(
u
(
t
,
P
)
)
φ
(
P
)
d
Q
.
From (19) and taking into account that φ
∈
𝒟
(
Q
i
), we have
(51)
∫
Q
i
u
t
(
t
,
P
)
φ
(
P
)
d
Q
=
-
∫
Q
i
D
u
(
t
,
P
)
D
φ
(
P
)
d
Q
+
∫
Q
i
J
(
u
(
t
,
P
)
)
φ
(
P
)
d
Q
=
∫
Q
i
Δ
u
(
t
,
P
)
φ
(
P
)
d
Q
+
∫
Q
i
J
(
u
(
t
,
P
)
)
φ
(
P
)
d
Q
.
From the arbitrariness of φ, we have that, for fixed t
∈
(
0
,
T
],
(52)
u
t
(
t
,
P
)
=
Δ
u
(
t
,
P
)
+
J
(
u
(
t
,
P
)
)
holds  in
𝒟
′
(
Q
i
)
.
From the density of 𝒟
(
Q
i
) in L
2
(
Q
i
) and since J
(
u
(
t
,
·
)
)
∈
L
2
(
Q
), we obtain the first assertion in (49). From this equality, we obtain Δ
u
(
t
,
P
)
=
u
t
(
t
,
P
)
-
J
(
u
(
t
,
P
)
) and since the right-hand side belongs to C
(
(
0
,
T
]
;
L
2
(
Q
i
)
) we deduce that Δ
u
(
t
,
P
)
∈
C
(
(
0
,
T
]
;
L
2
(
Q
i
)
); hence u
(
t
,
·
)
∈
C
(
(
0
,
T
]
;
V
(
Q
i
)
), where
(53)
V
(
Q
i
)
=
{
u
∈
H
0
1
(
Q
)
;
Δ
u
i
∈
L
2
(
Q
i
)
}
;
here the Laplacian is intended in the distributional sense. By proceeding as in (3.26) of [4], we prove that, for every fixed t, the normal derivative ∂
u
i
/
∂
n
i is in the dual (
B
β
,
0
2,2
(
S
)
)
′ of the space (
B
β
,
0
2,2
(
S
)
), where β
=
d
f
/
2 and
(54)
〈
∂
u
i
∂
n
i
,
v
|
S
〉
(
(
B
β
,
0
2,2
(
S
)
)
′
)
,
(
(
B
β
,
0
2,2
(
S
)
)
)
=
∫
Q
i
D
u
(
t
,
P
)
D
v
(
P
)
d
Q
+
∫
Q
i
v
(
P
)
Δ
u
i
(
t
,
P
)
d
Q
for every t
∈
(
0
,
T
] and every v
∈
H
0
1
(
Q
). By proceeding as in Section 6.1 of [21], we can prove that ∂
u
i
/
∂
n
i
∈
C
(
(
0
,
T
]
;
(
B
β
,
0
2,2
(
S
)
)
′
).
From Proposition4 and proceeding as in Section 6 of [3], it can be proved that the transmission condition
(55)
-
c
0
Δ
S
u
|
S
=
[
∂
u
∂
n
]
holds  in
(
𝒟
(
S
)
)
′
.
That is, for every t
∈
(
0
,
T
],
(56)
-
c
0
〈
Δ
S
u
|
S
,
z
〉
(
𝒟
(
S
)
)
′
𝒟
(
S
)
=
〈
[
∂
u
∂
n
]
,
z
〉
(
𝒟
(
S
)
)
′
𝒟
(
S
)
.
As a consequence of Theorem 20, the solution of problem (
P
) is the solution of the following transmission problem. For every t
∈
(
0
,
T
],(j)(57)
u
t
i
-
Δ
u
i
=
J
(
u
i
)
in
L
2
(
Q
i
)
,
i
=
1,2
,
(jj)(58)
-
c
0
Δ
S
u
=
[
∂
u
∂
n
]
in
(
𝒟
(
S
)
)
′
,
(jjj)(59)
u
=
0
in
H
1
/
2
(
∂
Q
)
,
(jv)(60)
u
1
=
u
2
in
B
d
f
/
2
2,2
(
S
)
,
(v)(61)
u
=
0
in
B
(
d
f
-
1
)
/
2
2,2
(
∂
S
)
.Remark 21.
Actually from Proposition6, one deduces that equalities (jv) and (v), respectively, hold in B
α
2,2
(
S
) and in B
α
-
(
1
/
2
)
2,2
(
∂
S
) with α
<
1.


### 5.2. The Prefractal Layer

Theorem 22.
Letu
h be the solution of problem (
P
h
). Then one has, for every fixed t
∈
(
0
,
T
],
(62)
(
u
h
)
t
(
t
,
P
)
-
Δ
u
h
(
t
,
P
)
=
J
(
u
h
(
t
,
P
)
)
f
o
r
P
∈
Q
h
i
,
a
.
e
.
i
=
1,2
,
∂
u
h
i
∂
n
i
∈
L
2
(
S
h
)
,
i
=
1,2
,
-
Δ
S
h
u
|
S
h
=
[
∂
u
h
∂
n
]
,
i
n
L
2
(
S
h
)
,
u
(
t
,
P
)
=
0
f
o
r
P
∈
∂
Q
,
u
h
1
=
u
h
2
i
n
H
0
1
(
S
h
)
,
u
h
(
0
,
P
)
=
ϕ
(
P
)
i
n
L
2
(
Q
)
,
where u
h
i is the restriction of u
h to Q
h
i, [
∂
u
h
/
∂
n
]
=
(
∂
u
h
1
/
∂
n
1
)
+
(
∂
u
h
2
/
∂
n
2
) is the jump of the normal derivatives across S
h, n
i, i
=
1,2, is the inward normal vector, and Δ
S
h
=
σ
h
1
D
ℓ
2
+
σ
h
2
D
y
2 is the piecewise tangential Laplacian associated to the Dirichlet form E
S
h. Moreover ∂
u
h
i
/
∂
n
i
∈
C
(
(
0
,
T
]
;
L
2
(
S
h
)
).Proof.
The first equality in (62) easily follows by proceeding as in Theorem 20. From this, it follows that, for every t
∈
[
0
,
T
],
(63)
u
h
(
t
,
·
)
∈
V
(
Q
h
i
)
=
{
u
∈
H
0
1
(
Q
)
;
Δ
u
h
i
∈
L
2
(
Q
h
i
)
}
.
For every fixed t
∈
(
0
,
T
], let u
h
i denote the restriction of the solution u
h to Q
h
i. By usual duality arguments (see Appendix  4 in [35]), the normal derivatives ∂
u
h
i
/
∂
n
i, i
=
1,2 belong to the dual space of H
0,0
1
/
2
(
S
h
). By proceeding as in Section 6.2 of [21], it is possible to prove that ∂
u
h
i
/
∂
n
i
∈
C
(
(
0
,
T
]
;
(
H
0,0
1
/
2
(
S
h
)
)
′
).
Then, by the Green formula for Lipschitz domains, one can prove that(64)
〈
-
Δ
S
h
u
|
S
h
,
z
〉
(
H
0,0
1
/
2
(
S
h
)
)
′
H
0,0
1
/
2
(
S
h
)
=
〈
[
∂
u
h
∂
n
]
,
z
〉
(
H
0,0
1
/
2
(
S
h
)
)
′
H
0,0
1
/
2
(
S
h
)
.
That is, the transmission condition
(65)
-
Δ
S
h
u
|
S
h
=
[
∂
u
h
∂
n
]
holds in the dual of H
0,0
1
/
2
(
S
h
) (see Proposition  2.2 in [5] for details). In order to prove that ∂
u
h
i
/
∂
n
i
∈
L
2
(
S
h
), we proceed as in Section 4.2 of [4]. Let us consider, for each fixed t
∈
(
0
,
T
], the weak solutions w
h
i and w
^
h
i in H
1
(
Q
h
i
) of the following auxiliary problems:
(66)
Δ
w
^
h
i
=
0
in
Q
h
i
,
w
^
h
i
=
u
h
on
⁡
∂
Q
h
i
,
(67)
-
Δ
w
h
i
=
-
(
u
h
i
)
t
+
J
(
u
h
i
)
in
Q
h
i
,
w
h
i
=
0
on
⁡
∂
Q
h
i
,
The regularity of u
h
i follows from the regularity of w
h
i and w
^
h
i since
(68)
u
h
i
=
w
h
i
+
w
^
h
i
.
From a regularity result of Jerison and Kenig (see Theorems 2 and 3 of [36]), we deduce that
(69)
∂
w
^
h
i
∂
n
i
∈
L
2
(
S
h
)
and ∥
∂
w
^
h
i
/
∂
n
i
∥
≤
C
(
h
)
∥
u
h
∥
V
(
Q
;
S
h
)
≤
C
(
h
)
∥
u
h
∥
D
(
A
h
).
As to the solutionw
h
i of (67), we preliminary observe that the right-hand side in the first equation of (67) belongs to L
2
(
Q
h
i
). From Proposition  4.5 in [4], it follows that
(70)
w
h
i
∈
H
s
i
(
Q
h
i
)
,
i
=
1,2
,
where 1
<
s
1
<
8
/
5 and 1
<
s
2
<
7
/
4; hence
(71)
D
α
w
h
1
∈
H
(
3
/
5
)
-
ϵ
(
Q
h
1
)
,
D
α
w
h
2
∈
H
(
3
/
4
)
-
ϵ
(
Q
h
2
)
,
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
.
|
α
|
=
1
for every ϵ
>
0; then by trace results (see Proposition A.1), we obtain, for i
=
1,2,
(72)
∂
w
h
i
∂
n
i
∈
L
2
(
S
h
)
and ∥
∂
w
h
i
/
∂
n
i
∥
L
2
(
S
h
)
≤
c
(
h
)
∥
-
(
u
h
i
)
t
+
J
(
u
h
i
)
∥
L
2
(
Q
). It follows from (67), (68), and (69) that ∂
u
h
i
/
∂
n
i
∈
L
2
(
S
h
), i
=
1,2; hence the jump belongs to L
2
(
S
h
). As H
0,0
1
/
2
(
S
h
) is dense in L
2
(
S
h
) (see e.g., [37]), we deduce that the transmission condition (64) actually holds in the L
2-sense and in particular Δ
S
h
u
h
∈
L
2
(
S
h
). The proof that ∂
u
h
i
/
∂
n
i
∈
C
(
(
0
,
T
]
;
L
2
(
S
h
)
) easily follows from (69), (72), and the fact that u
h, A
h
u
h, J
(
u
h
), and (
u
h
i
)
t belong to C
(
(
0
,
T
]
;
L
2
(
Q
)
).
From Theorem22, it follows that the solution of problem (
P
h
) is the solution of the following transmission problem. For every t
∈
(
0
,
T
],(j)(73)
u
t
i
=
Δ
u
i
+
J
(
u
i
)
in
L
2
(
Q
h
i
)
,
i
=
1,2
,
(jj)(74)
-
Δ
S
h
u
=
[
∂
u
∂
n
]
in
L
2
(
S
h
)
,
(jjj)(75)
u
=
0
in
H
1
/
2
(
∂
Q
)
,
(jv)(76)
u
1
=
u
2
in
H
1
(
S
h
)
,
(v)(77)
u
=
0
in
H
1
/
2
(
∂
S
h
)
.


## 5.1. The Fractal Layer

Theorem 20.
Letu be the solution of problem (
P
). Then one has, for every fixed t
∈
(
0
,
T
],
(49)
u
t
(
t
,
P
)
-
Δ
u
(
t
,
P
)
=
J
(
u
(
t
,
P
)
)
u
t
(
t
P
D
)
f
o
r
a
.
e
.
P
∈
Q
i
i
=
1,2
,
∂
u
i
∂
n
i
∈
(
B
β
,
0
2,2
(
S
)
)
′
,
β
=
d
f
2
,
i
=
1,2
,
-
c
0
〈
Δ
S
u
|
S
,
z
〉
(
𝒟
(
S
)
)
′
𝒟
(
S
)
=
〈
[
∂
u
∂
n
]
,
z
〉
(
𝒟
(
S
)
)
′
𝒟
(
S
)
,
f
o
r
e
v
e
r
y
z
∈
𝒟
(
S
)
,
u
(
t
,
P
)
=
0
f
o
r
P
∈
∂
Q
,
u
1
=
u
2
i
n
B
d
f
/
2
2,2
(
S
)
,
u
(
0
,
P
)
=
ϕ
(
P
)
i
n
L
2
(
Q
)
,
where u
i is the restriction of u to Q
i, ∂
u
i
/
∂
n
i, i
=
1,2 is the inward “normal derivative,” to be defined in a suitable sense, [
∂
u
/
∂
n
] = (
∂
u
1
/
∂
n
1
)
+
(
∂
u
2
/
∂
n
2
) is the jump of the normal derivative, and Δ
S is the fractal Laplacian. Moreover ∂
u
i
/
∂
n
i
∈
C
(
(
0
,
T
]
;
(
B
β
,
0
2,2
(
S
)
)
′
).Proof.
Letφ
(
P
) be an arbitrary function in V
(
Q
,
S
) such that φ
(
P
)
|
Q
i
∈
𝒟
(
Q
i
); by multiplying for φ (36) in (
P
) and integrating over Q we have
(50)
∫
Q
u
t
(
t
,
P
)
φ
(
P
)
d
Q
=
∫
Q
A
u
(
t
,
P
)
φ
(
P
)
d
Q
+
∫
Q
J
(
u
(
t
,
P
)
)
φ
(
P
)
d
Q
.
From (19) and taking into account that φ
∈
𝒟
(
Q
i
), we have
(51)
∫
Q
i
u
t
(
t
,
P
)
φ
(
P
)
d
Q
=
-
∫
Q
i
D
u
(
t
,
P
)
D
φ
(
P
)
d
Q
+
∫
Q
i
J
(
u
(
t
,
P
)
)
φ
(
P
)
d
Q
=
∫
Q
i
Δ
u
(
t
,
P
)
φ
(
P
)
d
Q
+
∫
Q
i
J
(
u
(
t
,
P
)
)
φ
(
P
)
d
Q
.
From the arbitrariness of φ, we have that, for fixed t
∈
(
0
,
T
],
(52)
u
t
(
t
,
P
)
=
Δ
u
(
t
,
P
)
+
J
(
u
(
t
,
P
)
)
holds  in
𝒟
′
(
Q
i
)
.
From the density of 𝒟
(
Q
i
) in L
2
(
Q
i
) and since J
(
u
(
t
,
·
)
)
∈
L
2
(
Q
), we obtain the first assertion in (49). From this equality, we obtain Δ
u
(
t
,
P
)
=
u
t
(
t
,
P
)
-
J
(
u
(
t
,
P
)
) and since the right-hand side belongs to C
(
(
0
,
T
]
;
L
2
(
Q
i
)
) we deduce that Δ
u
(
t
,
P
)
∈
C
(
(
0
,
T
]
;
L
2
(
Q
i
)
); hence u
(
t
,
·
)
∈
C
(
(
0
,
T
]
;
V
(
Q
i
)
), where
(53)
V
(
Q
i
)
=
{
u
∈
H
0
1
(
Q
)
;
Δ
u
i
∈
L
2
(
Q
i
)
}
;
here the Laplacian is intended in the distributional sense. By proceeding as in (3.26) of [4], we prove that, for every fixed t, the normal derivative ∂
u
i
/
∂
n
i is in the dual (
B
β
,
0
2,2
(
S
)
)
′ of the space (
B
β
,
0
2,2
(
S
)
), where β
=
d
f
/
2 and
(54)
〈
∂
u
i
∂
n
i
,
v
|
S
〉
(
(
B
β
,
0
2,2
(
S
)
)
′
)
,
(
(
B
β
,
0
2,2
(
S
)
)
)
=
∫
Q
i
D
u
(
t
,
P
)
D
v
(
P
)
d
Q
+
∫
Q
i
v
(
P
)
Δ
u
i
(
t
,
P
)
d
Q
for every t
∈
(
0
,
T
] and every v
∈
H
0
1
(
Q
). By proceeding as in Section 6.1 of [21], we can prove that ∂
u
i
/
∂
n
i
∈
C
(
(
0
,
T
]
;
(
B
β
,
0
2,2
(
S
)
)
′
).
From Proposition4 and proceeding as in Section 6 of [3], it can be proved that the transmission condition
(55)
-
c
0
Δ
S
u
|
S
=
[
∂
u
∂
n
]
holds  in
(
𝒟
(
S
)
)
′
.
That is, for every t
∈
(
0
,
T
],
(56)
-
c
0
〈
Δ
S
u
|
S
,
z
〉
(
𝒟
(
S
)
)
′
𝒟
(
S
)
=
〈
[
∂
u
∂
n
]
,
z
〉
(
𝒟
(
S
)
)
′
𝒟
(
S
)
.
As a consequence of Theorem 20, the solution of problem (
P
) is the solution of the following transmission problem. For every t
∈
(
0
,
T
],(j)(57)
u
t
i
-
Δ
u
i
=
J
(
u
i
)
in
L
2
(
Q
i
)
,
i
=
1,2
,
(jj)(58)
-
c
0
Δ
S
u
=
[
∂
u
∂
n
]
in
(
𝒟
(
S
)
)
′
,
(jjj)(59)
u
=
0
in
H
1
/
2
(
∂
Q
)
,
(jv)(60)
u
1
=
u
2
in
B
d
f
/
2
2,2
(
S
)
,
(v)(61)
u
=
0
in
B
(
d
f
-
1
)
/
2
2,2
(
∂
S
)
.Remark 21.
Actually from Proposition6, one deduces that equalities (jv) and (v), respectively, hold in B
α
2,2
(
S
) and in B
α
-
(
1
/
2
)
2,2
(
∂
S
) with α
<
1.


## 5.2. The Prefractal Layer

Theorem 22.
Letu
h be the solution of problem (
P
h
). Then one has, for every fixed t
∈
(
0
,
T
],
(62)
(
u
h
)
t
(
t
,
P
)
-
Δ
u
h
(
t
,
P
)
=
J
(
u
h
(
t
,
P
)
)
f
o
r
P
∈
Q
h
i
,
a
.
e
.
i
=
1,2
,
∂
u
h
i
∂
n
i
∈
L
2
(
S
h
)
,
i
=
1,2
,
-
Δ
S
h
u
|
S
h
=
[
∂
u
h
∂
n
]
,
i
n
L
2
(
S
h
)
,
u
(
t
,
P
)
=
0
f
o
r
P
∈
∂
Q
,
u
h
1
=
u
h
2
i
n
H
0
1
(
S
h
)
,
u
h
(
0
,
P
)
=
ϕ
(
P
)
i
n
L
2
(
Q
)
,
where u
h
i is the restriction of u
h to Q
h
i, [
∂
u
h
/
∂
n
]
=
(
∂
u
h
1
/
∂
n
1
)
+
(
∂
u
h
2
/
∂
n
2
) is the jump of the normal derivatives across S
h, n
i, i
=
1,2, is the inward normal vector, and Δ
S
h
=
σ
h
1
D
ℓ
2
+
σ
h
2
D
y
2 is the piecewise tangential Laplacian associated to the Dirichlet form E
S
h. Moreover ∂
u
h
i
/
∂
n
i
∈
C
(
(
0
,
T
]
;
L
2
(
S
h
)
).Proof.
The first equality in (62) easily follows by proceeding as in Theorem 20. From this, it follows that, for every t
∈
[
0
,
T
],
(63)
u
h
(
t
,
·
)
∈
V
(
Q
h
i
)
=
{
u
∈
H
0
1
(
Q
)
;
Δ
u
h
i
∈
L
2
(
Q
h
i
)
}
.
For every fixed t
∈
(
0
,
T
], let u
h
i denote the restriction of the solution u
h to Q
h
i. By usual duality arguments (see Appendix  4 in [35]), the normal derivatives ∂
u
h
i
/
∂
n
i, i
=
1,2 belong to the dual space of H
0,0
1
/
2
(
S
h
). By proceeding as in Section 6.2 of [21], it is possible to prove that ∂
u
h
i
/
∂
n
i
∈
C
(
(
0
,
T
]
;
(
H
0,0
1
/
2
(
S
h
)
)
′
).
Then, by the Green formula for Lipschitz domains, one can prove that(64)
〈
-
Δ
S
h
u
|
S
h
,
z
〉
(
H
0,0
1
/
2
(
S
h
)
)
′
H
0,0
1
/
2
(
S
h
)
=
〈
[
∂
u
h
∂
n
]
,
z
〉
(
H
0,0
1
/
2
(
S
h
)
)
′
H
0,0
1
/
2
(
S
h
)
.
That is, the transmission condition
(65)
-
Δ
S
h
u
|
S
h
=
[
∂
u
h
∂
n
]
holds in the dual of H
0,0
1
/
2
(
S
h
) (see Proposition  2.2 in [5] for details). In order to prove that ∂
u
h
i
/
∂
n
i
∈
L
2
(
S
h
), we proceed as in Section 4.2 of [4]. Let us consider, for each fixed t
∈
(
0
,
T
], the weak solutions w
h
i and w
^
h
i in H
1
(
Q
h
i
) of the following auxiliary problems:
(66)
Δ
w
^
h
i
=
0
in
Q
h
i
,
w
^
h
i
=
u
h
on
⁡
∂
Q
h
i
,
(67)
-
Δ
w
h
i
=
-
(
u
h
i
)
t
+
J
(
u
h
i
)
in
Q
h
i
,
w
h
i
=
0
on
⁡
∂
Q
h
i
,
The regularity of u
h
i follows from the regularity of w
h
i and w
^
h
i since
(68)
u
h
i
=
w
h
i
+
w
^
h
i
.
From a regularity result of Jerison and Kenig (see Theorems 2 and 3 of [36]), we deduce that
(69)
∂
w
^
h
i
∂
n
i
∈
L
2
(
S
h
)
and ∥
∂
w
^
h
i
/
∂
n
i
∥
≤
C
(
h
)
∥
u
h
∥
V
(
Q
;
S
h
)
≤
C
(
h
)
∥
u
h
∥
D
(
A
h
).
As to the solutionw
h
i of (67), we preliminary observe that the right-hand side in the first equation of (67) belongs to L
2
(
Q
h
i
). From Proposition  4.5 in [4], it follows that
(70)
w
h
i
∈
H
s
i
(
Q
h
i
)
,
i
=
1,2
,
where 1
<
s
1
<
8
/
5 and 1
<
s
2
<
7
/
4; hence
(71)
D
α
w
h
1
∈
H
(
3
/
5
)
-
ϵ
(
Q
h
1
)
,
D
α
w
h
2
∈
H
(
3
/
4
)
-
ϵ
(
Q
h
2
)
,
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
.
|
α
|
=
1
for every ϵ
>
0; then by trace results (see Proposition A.1), we obtain, for i
=
1,2,
(72)
∂
w
h
i
∂
n
i
∈
L
2
(
S
h
)
and ∥
∂
w
h
i
/
∂
n
i
∥
L
2
(
S
h
)
≤
c
(
h
)
∥
-
(
u
h
i
)
t
+
J
(
u
h
i
)
∥
L
2
(
Q
). It follows from (67), (68), and (69) that ∂
u
h
i
/
∂
n
i
∈
L
2
(
S
h
), i
=
1,2; hence the jump belongs to L
2
(
S
h
). As H
0,0
1
/
2
(
S
h
) is dense in L
2
(
S
h
) (see e.g., [37]), we deduce that the transmission condition (64) actually holds in the L
2-sense and in particular Δ
S
h
u
h
∈
L
2
(
S
h
). The proof that ∂
u
h
i
/
∂
n
i
∈
C
(
(
0
,
T
]
;
L
2
(
S
h
)
) easily follows from (69), (72), and the fact that u
h, A
h
u
h, J
(
u
h
), and (
u
h
i
)
t belong to C
(
(
0
,
T
]
;
L
2
(
Q
)
).
From Theorem22, it follows that the solution of problem (
P
h
) is the solution of the following transmission problem. For every t
∈
(
0
,
T
],(j)(73)
u
t
i
=
Δ
u
i
+
J
(
u
i
)
in
L
2
(
Q
h
i
)
,
i
=
1,2
,
(jj)(74)
-
Δ
S
h
u
=
[
∂
u
∂
n
]
in
L
2
(
S
h
)
,
(jjj)(75)
u
=
0
in
H
1
/
2
(
∂
Q
)
,
(jv)(76)
u
1
=
u
2
in
H
1
(
S
h
)
,
(v)(77)
u
=
0
in
H
1
/
2
(
∂
S
h
)
.


## 6. Convergence Results

Now we are interested in the behavior of the sequence{
u
h
} when h goes to ∞.Theorem 23.
Letu and u
h be the solutions of problems (
P
) and (
P
h
) according to Theorem 19. Let σ
h
1 and σ
h
2 be as in Theorem 16. For every fixed positive ϵ, one has(i)
J
(
u
h
) converges to J
(
u
) in L
2
(
[
ϵ
,
T
]
×
Q
);
(ii)
{
d
u
h
/
d
t
} weakly converges to d
u
/
d
t in L
2
(
[
ϵ
,
T
]
×
Q
);
(iii)
{
A
h
u
h
} weakly converges to A
u in L
2
(
[
ϵ
,
T
]
×
Q
);
(iv)
{
u
h
} converges to u in L
2
(
[
ϵ
,
T
]
;
H
0
1
(
Q
)
).Proof.
We prove condition (i), that is,(78)
∥
j
(
u
(
t
,
P
)
)
-
J
(
u
h
(
t
,
P
)
)
∥
L
2
(
(
ϵ
,
T
]
×
Q
)
→
0
.
From (38), we have
(79)
∫
ϵ
T
∥
J
(
u
(
t
,
P
)
)
-
J
(
u
h
(
t
,
P
)
)
∥
L
2
(
Q
)
2
d
t
≤
∫
ϵ
T
l
2
(
R
)
∥
u
(
t
)
-
u
h
(
t
)
∥
L
2
p
(
Q
)
2
d
t
≤
(
T
-
ϵ
)
sup
⁡
t
∈
[
ϵ
,
T
]
l
2
(
R
)
∥
u
(
t
)
-
u
h
(
t
)
∥
L
2
p
(
Q
)
2
.
From Theorem 19 (a), we have
(80)
sup
⁡
t
∈
[
0
,
T
]
∥
t
b
[
u
(
t
)
-
u
h
(
t
)
]
∥
L
2
p
(
Q
)
→
0
.
And hence, for every fixed ϵ
>
0,
(81)
sup
⁡
t
∈
[
ϵ
,
T
]
∥
[
u
(
t
)
-
u
h
(
t
)
]
∥
L
2
p
(
Q
)
→
0
.
This concludes the proof of condition (i).
We now prove condition (ii). From the local Lipschitz continuity ofJ
(
u
) and the Hölder continuity of u
h
(
t
) in (
ϵ
,
T
) into L
2
p, one can prove that ∥
J
(
u
h
)
∥
C
θ
(
[
ϵ
,
T
]
;
L
2
(
Q
)
) is bounded by a constant which does not depend on h; actually the constants depend only on the constants of the semigroups which in turn do not depend on h. From this, together with Theorem 18, we have that there exists a constant c independent of h such that
(82)
∥
u
h
∥
C
1
(
[
ϵ
,
T
)
;
L
2
(
Q
)
)
+
∥
u
h
∥
C
0
(
[
ϵ
,
T
]
;
D
(
A
)
)
≤
c
∥
J
(
u
h
)
∥
C
θ
(
[
ϵ
,
T
]
;
L
2
(
Q
)
)
≤
c
l
(
r
)
∥
u
h
∥
C
θ
(
[
ϵ
,
T
]
;
L
2
p
(
Q
)
)
.
Thus in particular it holdssup
⁡
t
∈
[
ϵ
,
T
]
⁡
∥
(
u
h
)
t
∥
L
2
(
Q
)
≤
c; thus, for every fixed t
∈
[
ϵ
,
T
], ∥
(
u
h
)
t
∥
L
2
(
Q
)
≤
c.
From (82), it follows that for each h, d
u
h
/
d
t belongs to L
2
(
[
ϵ
,
T
]
×
Q
) and ∥
d
u
h
/
d
t
∥
L
2
(
[
ϵ
,
T
]
×
Q
)
2
≤
c.
From the boundedness of the sequence{
d
u
h
/
d
t
} in L
2
(
[
ϵ
,
T
]
×
Q
), it follows that there exists a subsequence, which we denote with {
d
u
h
/
d
t
} and a function v
∈
L
2
(
[
ϵ
,
T
]
×
Q
) such that {
d
u
h
/
d
t
} weakly converges to v in L
2
(
[
ϵ
,
T
]
×
Q
) as h goes to ∞.
In order to prove (ii), it is enough to prove thatv
=
d
u
/
d
t.
SinceC
1
(
[
ϵ
,
T
]
×
Q
) is dense in L
2
(
[
ϵ
,
T
]
×
Q
), for every φ
∈
C
1
(
[
ϵ
,
T
]
×
Q
), we have
(83)
lim
⁡
n
→
∞
⁡
∫
Q
∫
ϵ
T
d
u
n
d
t
(
t
,
P
)
φ
(
t
,
P
)
d
t
d
Q
=
∫
Q
∫
ϵ
T
v
(
t
,
P
)
φ
(
t
,
P
)
d
t
d
Q
.
Integrating by parts the left-hand side, we get(84)
∫
Q
∫
ϵ
T
d
u
n
d
t
(
t
,
P
)
φ
(
t
,
P
)
d
t
d
Q
=
∫
Q
[
u
n
(
T
,
P
)
φ
(
T
,
P
)
-
u
n
(
0
,
P
)
φ
(
0
,
P
)
]
d
Q
-
∫
Q
∫
ϵ
T
u
n
(
t
,
P
)
d
φ
d
t
(
t
,
P
)
d
t
d
Q
.
From (47) or (48), we have
(85)
lim
⁡
n
→
∞
⁡
∫
Q
∫
ϵ
T
d
u
n
d
t
(
t
,
P
)
φ
(
t
,
P
)
d
t
d
Q
=
∫
Q
[
u
(
T
,
P
)
φ
(
T
,
P
)
-
u
(
0
,
P
)
φ
(
0
,
P
)
]
d
Q
-
∫
Q
∫
ϵ
T
u
(
t
,
P
)
d
φ
d
t
(
t
,
P
)
d
t
d
Q
=
∫
Q
∫
ϵ
T
d
u
d
t
(
t
,
P
)
φ
(
t
,
P
)
d
t
d
Q
.
From the uniqueness of weak limit, we get v
=
d
u
/
d
t a.e.. From the convergence of the sequence {
u
h
} to u in L
2
(
[
ϵ
,
T
]
×
Q
) and the weak convergence of the subsequence {
d
u
n
/
d
t
} to d
u
/
d
t in L
2
(
[
ϵ
,
T
]
×
Q
), we deduce that the whole sequence {
d
u
h
/
d
t
} weakly converges to d
u
/
d
t in L
2
(
[
ϵ
,
T
]
×
Q
).
We now prove condition (iii). It is an easy consequence of (i) and (ii). In factA
h
u
h
=
(
d
u
h
/
d
t
)
-
J
(
u
h
); taking the weak limit in L
2
(
[
ϵ
,
T
]
×
Q
), we get the thesis.
We now prove condition (iv). From (i), (iii), and the property of the scalar product inL
2
(
[
ϵ
,
T
]
×
Q
), we get that
(86)
lim
⁡
h
→
∞
⁡
(
A
h
u
h
,
u
h
)
L
2
(
[
ϵ
,
T
]
×
Q
)
=
(
A
u
,
u
)
L
2
(
[
ϵ
,
T
]
×
Q
)
.
That is,
(87)
lim
⁡
h
→
∞
⁡
∫
ϵ
T
d
t
∫
Q
A
h
u
h
(
t
,
P
)
u
h
(
t
,
P
)
d
Q
=
∫
ϵ
T
d
t
∫
Q
A
u
(
t
,
P
)
u
(
t
,
P
)
d
Q
.
From the relation between a Dirichlet form and the associated generator, it follows that
(88)
lim
⁡
h
→
∞
⁡
∫
ϵ
T
E
(
h
)
[
u
h
]
d
t
=
∫
ϵ
T
E
[
u
]
d
t
.
There exists a constant c such that
(89)
∫
ϵ
T
∫
Q
|
D
u
h
(
t
,
P
)
|
2
d
Q
d
t
+
∫
ϵ
T
E
S
h
[
u
h
]
d
t
≤
c
,
00000000000000000000000000
for
every
h
∈
ℕ
.
Hence
(90)
∥
D
u
h
∥
L
2
(
[
ϵ
,
T
]
×
Q
)
≤
c
.
There exists a subsequence D
u
n weakly converging to w in L
2
(
[
ϵ
,
T
]
×
Q
)
3. We now prove that
(91)
∫
ϵ
T
∫
Q
D
u
h
(
t
,
P
)
D
u
d
Q
d
t
→
∫
ϵ
T
∫
Q
|
D
u
(
t
,
P
)
|
2
d
Q
d
t
.
From Theorem 19, it follows in particular that u
n converges to u in L
2
(
[
ϵ
,
T
]
×
Q
); hence w
=
D
u and u
h
⇀
u in L
2
(
[
ϵ
,
T
]
;
H
0
1
(
Q
); in particular (91) holds. We now prove assertion (iv) as
(92)
0
≤
∫
ϵ
T
∥
u
h
(
t
)
-
u
(
t
)
∥
H
0
1
(
Q
)
2
d
t
=
∫
ϵ
T
d
t
∫
Q
|
D
u
h
(
t
,
P
)
-
D
u
(
t
,
P
)
|
2
d
Q
=
∫
ϵ
T
d
t
∫
Q
(
|
D
u
h
(
t
,
P
)
|
2
+
|
D
u
(
t
,
P
)
|
2
-
2
D
u
h
(
t
,
P
)
D
u
(
t
,
P
)
)
d
Q
=
∫
ϵ
T
(
E
(
h
)
[
u
h
]
-
E
S
h
[
u
h
]
)
d
t
+
∫
ϵ
T
d
t
∫
Q
(
|
D
u
(
t
,
P
)
|
2
-
2
D
u
h
(
t
,
P
)
D
u
(
t
,
P
)
)
d
Q
.
Taking the upper limit as h
→
∞, we have
(93)
0
≤
lim
⁡
sup
⁡
⁡
∥
u
h
(
t
)
-
u
(
t
)
∥
L
2
(
[
ϵ
,
T
]
;
H
0
1
(
Q
)
)
2
≤
∫
ϵ
T
E
[
u
]
d
t
(94)
-
lim
⁡
inf
⁡
h
→
∞
⁡
∫
ϵ
T
E
S
h
[
u
h
]
d
t
+
∫
ϵ
T
d
t
∫
Q
|
D
u
(
t
,
P
)
|
2
d
Q
-
2
∫
ϵ
T
d
t
∫
Q
|
D
u
(
t
,
P
)
|
2
d
Q
(95)
≤
c
0
∫
ϵ
T
E
S
[
u
]
d
t
-
lim
⁡
inf
⁡
h
→
∞
⁡
∫
ϵ
T
E
S
h
[
u
h
]
d
t
≤
0
,
where the last inequality follows from (
4.9
) in [21]. Hence the sequence {
u
h
} converges to u in L
2
(
[
ϵ
,
T
]
;
H
0
1
(
Q
)
) and therefore {
D
u
h
} converges to D
u in L
2
(
[
ϵ
,
T
]
;
(
L
2
(
Q
)
)
3
).Proposition 24.
Letu and u
h be the solutions of problems (
P
) and (
P
h
), respectively. Then u and u
h
∈
H
1
(
[
ϵ
,
T
]
×
Q
).Proof.
We prove the thesis foru. From Theorem 18, it follows that u
∈
C
(
[
ϵ
,
T
]
;
𝒟
(
A
)
) and (
d
u
/
d
t
)
∈
C
(
[
ϵ
,
T
]
;
L
2
(
Q
)
). Since 𝒟
(
A
)
⊂
V
(
Q
,
S
)
⊂
H
0
1
(
Q
), we obtain u
∈
C
(
[
ϵ
,
T
]
;
H
0
1
(
Q
)
); hence D
u
∈
C
(
[
ϵ
,
T
]
;
(
L
2
(
Q
)
)
3
). The thesis follows as C
(
[
ϵ
,
T
]
;
L
2
(
Q
)
)
⊂
L
2
(
[
ϵ
,
T
]
×
Q
). The result for u
h can be proved analogously.



---

*Source: 461046-2014-01-22.xml*