Semilinear Evolution Problems with Ventcel-Type Conditions on Fractal Boundaries

A semilinear parabolic transmission problem with Ventcel's boundary conditions on a fractal interface or the corresponding prefractal interface 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.


Introduction
In this paper we study the parabolic semilinear second-order transmission problem which we formally state as where is the bounded open set (−1, 1) 2 × (0, 1), and is a "cylindrical" layer dividing the set into two subsets 1 and 2 (see Figure 2). When is the Koch-type surface = × , where is the snowflake and = [0,1] (see Section 2), is the energy functional introduced in (12); when is the prefractal surface ℎ , is the energy functional ℎ introduced in (24). is a nonlinear function from a subset of 2 ( ) into 2 ( ). denotes the restriction of to , [ ] = 1 − 2 denotes the jump of across , Δ denotes the Laplace operator defined on the layer (see (12) in Section 3), and [ / ] = 1 / 1 + 2 / 2 denotes the jump of the normal derivatives across , to be intended in a suitable sense.
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][4][5][6][7]. Problems of this type are also known in the literature as problems with Ventcel's boundary conditions [8] or secondorder 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][10][11][12][13] and the recent review in [14]); the fractal case is more awkward (see e.g., [15][16][17][18][19]).
In our case one has to take into account that the diffusion phenomenon takes place both across the smooth domain 2 International Journal of Partial Differential Equations and the cylindrical layer ; this fact has a counterpart in the structure of the energy functional [ ] and hence on problem ( ). 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 ( ) 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 restriction ℎ of the solution ℎ to ℎ 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 of 2 ( ) (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 ( ) and ( ℎ ) solves problem ( ) 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.

Geometry of the Fractal Layers and ℎ
In the paper by | − 0 | we denote the Euclidean distance in R and the Euclidean balls by ( 0 , ) = { ∈ R : | − 0 | < }, 0 ∈ R , > 0. By the Koch snowflake , we will denote the union of three coplanar Koch curves (see [22]) 1 , 2 , and 1 , 3 , and 5 are the vertices of a regular triangle with unit side length; that is, In this section we briefly recall the essential notions on the geometry; for details see [18]. The Hausdorff dimension of the Koch snowflake is given by = 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 measure supported on by where denotes the normalized -dimensional Hausdorff measure, restricted to , = 1, 2, 3.
The measure has the property that there exist two positive constants 1 and 2 : where = = log 4/ log 3 and ( , ) denotes the Euclidean ball in R 2 . As is supported on , it is not ambiguous to write in (3) ( ( , )) in place of ( ( , ) ∩ ). In the terminology of Appendices A and B, we say that is a -set with = .
Remark 1. The Koch snowflake can be also regarded as a fractal manifold (see [23] Section 2.2).

Let
denote a bounded open set in R 3 ; in our basic model, denotes the parallelepiped = (−1, 1) 2 × (0, 1) and denotes a "cylindrical" layer in of the type = × , where = [0, 1] and is the Koch snowflake. We assume that is located in a median position inside and divides in two subsets 1 and 2 (see Figure 2).
We give a point ∈ the Cartesian coordinates = ( , ), where = ( 1 , 2 ) are the coordinates of the orthogonal projection of on the plane containing and is the coordinate of the orthogonal projection of on the -line containing the interval : = ( , ) ∈ , = ( 1 , 2 ) ∈ , ∈ . One can define, in a natural way, a finite Borel measure supported on as the product measure where denotes the one-dimensional Lebesgue measure on . The measure has the property that there exist two positive constants 1 and 2 : where = + 1 = log 12/ log 3 and ( , ) denotes the Euclidean ball in R 3 . As is supported on , it is not ambiguous to write in (5) ( ( , )) in place of ( ( , )∩ ).
International Journal of Partial Differential Equations Thus turns out to be a -set with = +1 (see Appendices A and B).
We give a point ∈ ℎ the Cartesian coordinates = ( , ), where = ( 1 , 2 ) are the coordinates of the orthogonal projection of on the plane containing ℎ and is the coordinate of the orthogonal projection on the -line containing the interval .

Energy Forms and Semigroups Associated
3.1. The Energy Form . In this section we introduce the energy functional on . We first define the energy functional on the cross section by integrating its Lagrangian on . 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 , L , is a measure-valued map on D( ) × D( ) which is bilinear symmetric and positive (L [ ] is a positive measure.) The measure-valued Lagrangian takes on the fractal the role of the Euclidean Lagrangian L( , V) = ∇ ⋅ ∇V . Note that in the case of the Koch curve, the Lagrangian is absolutely continuous with respect to the measure ; on the contrary, this is not true on most fractals (see [24]). In [23] the Lagrangian L on the snowflake has been defined by using its representation as a fractal manifold. Here we do not give details on the construction and definition of L and we refer to Section 4 in [23] for details; in particular in Definition 4.5 a Lagrangian measure L on and the corresponding energy form E as with domain D( ) have been introduced. The domain D( ), which is a Hilbert space with norm has been characterized in terms of the domains of the energy forms on (see [23] Theorem 4.6).
In the following, we will omit the subscript , the Lagrangian measure will be simply denoted by L( , V), and we will set L[ ] = L( , ); an analogous notation will be adopted for the energies.
We define the energy forms on the fractal layer = × by setting where 1 and 2 are positive constants. Here L (⋅, ⋅)( ) denotes the measure-valued Lagrangian (of the energy form E of with domain D( )) now acting on ( , ) and V( , ) as function of ∈ for a.e. ∈ ; ( ) is the -Hausdorff measure acting on each section of for a.e. ∈ with = log 4/ log 3; (⋅) denotes the derivative in the direction.
The form is defined for ∈ D( ), where D( ) is the closure in the intrinsic norm of the set where 2 ( ) = 2 ( , ( )).
In the following, we will also use the form ( , V) which is obtained from [ ] by the polarization identity: , V ∈ D ( ) . 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 . As ( , D( )) is a closed, bilinear form on 2 ( , ), there exists (see Chapter 6, Theorem 2.1 in [27]) a unique selfadjoint, nonpositive operator Δ on 2 ( , )-with domain Let (D( )) denote the dual of the space D( ). We now introduce the Laplace operator on the fractal as a variational operator from D( ) → (D( )) by for ∈ D( ) and for all ∈ D( ), where ⟨⋅, ⋅⟩ (D( )) ,D( ) is the duality pairing between (D( )) and D( ). We use the same symbol Δ to define the Laplace operator both as a selfadjoint 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 dimension ] of . We find that if ( ) is the number of eigenvalues associated with smaller than , then ( ) ∼ ]/2 . It can be shown that in our case ] = 2 (see [28,29]). We stress the fact that in the fractal case ] < < , while in the Euclidean setting ] = .
Consider now the space of functions : → R as Here we denote by the symbol | the trace 0 of to (see Appendices A and B). The space ( , ) is nontrivial; see Proposition 3.3 of [4]. We now introduce the energy form defined on the domain ( , ). Here and in the following, denotes the 3-dimensional Lesbesgue measure and ( , V) denotes the corresponding bilinear form defined on ( , ) × ( , ). As in Theorem 3.2 of [26], the following result can be proved. (15) is a regular Dirichlet form in 2 ( ) and the space ( , ) is a Hilbert space equipped with the scalar product

Proposition 3. The form defined in
We denote by ‖ ‖ ( , ) the norm in ( , ), associated with (17), that is As in Propositions (3.6) and (3.1) in [4], the following result can be proved.
As ( , ( , )) is a closed bilinear form on 2 ( ) with domain ( , ) dense in 2 ( ), there exists (see Chapter 6 Theorem 2.1 in [27]) a unique self-adjoint nonpositive operator Moreover in Theorem 13.1 of [25] it is proved that to each closed symmetric form a family of linear operators { , > 0} can be associated with the property and this family is a strongly continuous resolvent with generator , which also generates a strongly continuous For the reader's convenience, we recall here the main properties of the semigroup { ( )} ≥0 ; the reader is referred to Proposition 3.5 in [21] for the proof. Proposition 6. Let { ( )} ≥0 be the semigroup generated by the operator A associated with the energy form in (19). Then { ( )} ≥0 is an analytic contraction positive preserving semigroup in 2 ( ).

Remark 7.
It is well known that the symmetric and contraction analytic semigroup ( ) uniquely determines analytic semigroups on the space , 1 ≤ < ∞ (see Theorem 1.4.1 [30]) which we still denote by ( ) and by 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 constant such that
One will consider the case = 3 and ] = 2; here ] is the spectral dimension of .
From interpolation theory results, it can be proved (see Section 3.1 in [18]) that International Journal of Partial Differential Equations 5 3.2. The Energy Forms ℎ . By we denote the parallelepiped as defined in Section 3 and by ℎ we denote the prefractal layer of the type ℎ = ℎ × , ℎ = 1, 2, . . ., ℎ is the prefractal approximation of at the step ℎ (see Section 2). ℎ divides in two subsets ℎ , = 1, 2.
We first construct the energy forms ℎ on the prefractal layers ℎ = ℎ × , ℎ ∈ N. By ℓ we denote the natural arclength coordinate on each edge of ℎ and we introduce the coordinates 1 = 1 (ℓ), 2 = 2 (ℓ), and = on every affine "face" ( ) ℎ of ℎ . By ℓ we denote the one-dimensional measure given by the arc-length ℓ and by are denote the surface measure on each face where 1 ℎ and 2 ℎ are positive constants and ∈ 1 ( ℎ ), the Sobolev space of functions on the piecewise affine set ℎ (see Appendices A and B). By Fubini theorem, we can write this functional in the form We denote the corresponding bilinear form by ℎ ( , V). In the sequel we denote by the symbol | ℎ the trace 0 to ℎ . Consider now the space of functions : → R as it is not trivial as it contains D( ). Consider now the energy form defined on the domain ( , ℎ ). By (ℎ) ( , V) we will denote the corresponding bilinear form defined on ( , ℎ ) × ( , ℎ ).
By proceeding as in Remark 7, one can show that for every ℎ ∈ N the symmetric and contraction analytic semigroup ℎ ( ) uniquely determines analytic semigroups on the space , 1 < < ∞ (see Theorem 1.4.1 [30]) which we still denote by ℎ ( ) and by 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 constant such that
where does not depend on h. One considers the cases = 3 and ] = 2; here ] is the Euclidean dimension of .
As before by interpolation results it can be proved that 3.3. The Convergence of Forms and Semigroups. We now recall the results proved in [21] on the convergence of the approximating energy forms (ℎ) to the fractal energy . In this asymptotic behaviour, the factors 1 ℎ and 2 ℎ 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 and hence the irregularity of the surface and in particular the effect of the -dimensional length intrinsic to the curve; for details, see [6]. The convergence of functional is here intended in the sense of the -convergence which we define below.

The -Convergence of Forms.
We recall, for the sake of completeness, the definition of -convergence of forms introduced by Mosco in [33].

Remark 15.
We point out that, as the sequence of forms (26) is asymptotically compact in 2 ( ), -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 in 2 ( ).

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 ( ) and ( 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 ( ) in both cases. We refer the reader to [18]. We consider the abstract Cauchy problems as and for every ℎ ∈ N where : D( ) ⊂ 2 ( ) → 2 ( ) and ℎ : D( ℎ ) ⊂ 2 ( ) → 2 ( ) are the generators associated, respectively, to the energy form and the energy forms (ℎ) introduced in (15) and (26), is a fixed positive real number, and and ℎ are given functions in 2 ( ). We assume that is a mapping from 2 ( ) → 2 ( ), > 1 locally Lipschitz, that is, Lipschitz on bounded sets in 2 ( ); we let ( ) denote the Lipschitz constant of : where ‖ ‖ 2 ( ) ≤ , ‖V‖ 2 ( ) ≤ . We also assume that (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 ( ) when → ∞. We set for brevity := ( /4)(1 − (1/ )); we note that 0 < < 1, for ≤ 4, and > 1.

Theorem 18. Under the assumptions of Theorem 17, one has that the solution ( ) can be continuously extended to a maximal interval (0, ) as a solution of
For every fixed ℎ ∈ N, the claims of Theorems 17 and 18 hold for problem ( ℎ ) with the obvious changes.
We now recall the convergence results of the sequence of the approximating solutions { ℎ } when h goes to infinity (see Theorem 6.2 in [18]).
From Theorem 22, it follows that the solution of problem ( ℎ ) is the solution of the following transmission problem. For every ∈ (0, ], (jv) (v)
Proof. We prove condition (i), that is, From (38), we have ∫ ( ( , )) − ( ℎ ( , )) And hence, for every fixed > 0, This concludes the proof of condition (i). We now prove condition (ii). From the local Lipschitz continuity of ( ) and the Hölder continuity of ℎ ( ) in ( , ) into 2 , one can prove that ‖ ( ℎ )‖ ([ , ]; 2 ( )) 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 International Journal of Partial Differential Equations do not depend on ℎ. From this, together with Theorem 18, we have that there exists a constant independent of ℎ such that   We now prove condition (iii). It is an easy consequence of (i) and (ii). In fact ℎ ℎ = ( ℎ / ) − ( ℎ ); taking the weak limit in 2 ([ , ] × ), we get the thesis.
We now prove condition (iv). From (i), (iii), and the property of the scalar product in 2 ([ , ] × ), we get that That is, From the relation between a Dirichlet form and the associated generator, it follows that There exists a constant such that Hence There exists a subsequence weakly converging to in 2 ([ , ] × ) 3 . We now prove that From Theorem 19, it follows in particular that converges to in 2 ([ , ] × ); hence = and ℎ ⇀ in 2 ([ , ]; 1 0 ( ); in particular (91) holds. We now prove assertion (iv) as International Journal of Partial Differential Equations

Appendices
Here we recall some definitions of functional spaces and trace results.

A. Sobolev Spaces
Let be a polyhedral domain; just to fix the ideas, the parallelepiped is as in Section 2. For every integer ℎ ≥ 1, let ℎ be the prefractal surface approximating the Koch-type surface and let us denote every affine "face" of ℎ by ( ) ℎ ; ℎ divides into two subsets 1 ℎ and 2 ℎ . By (⋅), > 1 we denote the Lebesgue space with respect to the Lebesgue measure on subsets of R 3 , which will be left to the context whenever that does not create ambiguity. Let T be a closed set of R 3 ; by (T) we denote the space of continuous functions on T; by 0 (T) we denote the space of continuous functions vanishing on T. Let G be an open set of R 3 ; by 1 (G) we denote the usual Sobolev spaces (see Necas [38]); 1 0 (G) is the closure of D(G) (the smooth functions with compact support on G), with respect to the ‖ ⋅ ‖ 1 -norm. In the following, we will make use of trace spaces on boundaries of polyhedral domains of R 3 . By 1 0 ( ℎ ) we denote the closure in 1 ( ℎ ) of the set V vanishes in a neighborhood of ℎ } .
It is to be pointed out that the Sobolev space ( ℎ ) (defined in [38]) coincides, with equivalent norms, with the trace space defined in Buffa and Ciarlet in [37] (see also [39] for the case of polygonal boundaries).
When > 1, the trace spaces on nonsmooth boundaries can be defined in different ways; we now recall two trace theorems, specialized to our case, referring to [40] and [41] for a more general discussion.
For in 1 (G), we put at every point ∈ G, where the limit exists. It is known that the limit (A.2) exists at quasi every ∈ G with respect to the (1, 2)-capacity [42]. We now recall the results of Theorem 3.1 in [36] specialized to our case, referring to [41] for a more general discussion. (ii) there is a continuous linear operator Ext from 1/2 (Γ) to 1 (G), such that 0 ∘ Ext is the identity operator in 1/2 (Γ).
Such a is called a -measure on T.
Proposition B.2. The set is a -set with = . The measure is a -measure. The layer is a -set with = + 1. The measure is a -measure.
See [23,26]. We now come to the definition of the Besov spaces. Actually there are many equivalent definitions of these spaces; see, for instance, [43,44]. We recall here the one which best fits our aims and we will restrict ourselves to the case positive and noninteger, = = 2; the general setting is being much more involved; see [44].
Let T be a -set in R . Let > 0 be noninteger, = [ ] the integer part of , and a -dimensional multi-index of length | | ≤ .
Let us note that for 0 < < 1 the norm ‖ ‖ 2,2 (T) can be written as For the proof, we refer to Theorem 1 of Chapter VII in [44]; see also [43].

Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.