CONVERGENCE AND DENSITY RESULTS FOR PARABOLIC QUASI-LINEAR VENTTSEL’ PROBLEMS IN FRACTAL DOMAINS

. In this paper we study a quasi-linear evolution equation with nonlinear dynamical boundary conditions in a three dimensional fractal cylindrical domain Q , whose lateral boundary is a fractal surface S . We consider suitable approximating pre-fractal problems in the corresponding pre-fractal varying domains. After proving existence and uniqueness results via standard semigroup approach, we prove density results for the domains of energy functionals deﬁned on Q and S . Then we prove that the pre-fractal solutions converge in a suitable sense to the limit fractal one via the Mosco convergence of the energy functionals.


Introduction.
Recently there has been a growing interest in the study of particular boundary value problems, taking place in irregular (e.g. fractal) domains. This is due to the fact that many industrial processes and natural phenomena occur across irregular media, and fractal geometries are a useful tool in order to model these geometries (see [43], [44]).
Evolution problems with dynamical boundary conditions on domains with fractal boundaries are known in literature as Venttsel' problems (see [47] and [2]). This kind of boundary conditions is of great interest in applications, since they arise in problems such phase-transition phenomena, fluid diffusion, climatology and nonlinear cooling effects on the boundary (see for example [15,16,19,42] and the references listed in). There is a huge literature on linear and nonlinear Venttsel' problems, see [14], [28], [27], [35,30,31,34,32,33] (see also [10], [11] and [12] for the numerical approximation). The goal of this paper is to adapt the framework of [26] in order to extend the convergence results in [14] for a quasilinear Venttsel' problem to the three dimensional case. In the two dimensional case, one considers a fractal nonlinear energy functional and its natural approximating pre-fractal energy functionals. By using the notion of Mosco convergence (see [39,40]) of energy functionals adapted by Tölle to the nonlinear framework in varying Hilbert spaces (see [45]), the authors are able to prove the convergence of the pre-fractal solutions to the limit fractal one. The problem when passing to the three dimensional case 66 SIMONE CREO AND VALERIO REGIS DURANTE is twofold. First, since we consider the case of the p-Laplace operator for p ≥ 2, in two dimensions from Sobolev embedding theorem we have the immersion of W 1,p in the space of continuous functions; in dimension three, this does not hold anymore. Secondly, in two dimensions a complete characterization of the energy space on the fractal curve in terms of Lipschitz spaces holds; in particular, these spaces are subsets of the set of Hölder continuous functions on the fractal (see [18], [36] and [9]). In the three dimensional case, to our knowledge, this characterization does not hold anymore. Therefore functions in the domain of the energy functional have to be approximated in an appropriate way by smoother functions. We then prove density results which will turn crucial in order to prove the M-convergence of the energy functionals.
More precisely, we consider a cylindrical fractal surface S = F ×I, where F is the Koch snowflake and I = [0, 1], and for every h ∈ N its natural pre-fractal approximation S h = F h × I. We denote by Q the three-dimensional open bounded cylinder having as lateral boundary S and, for every h ∈ N, by Q h the approximating prefractal domains which are an increasing sequence exhausting Q. We introduce the energy functionals Φ p and Φ (h) p on the fractal and pre-fractal sets respectively, and we denote by V (Q, S) the domain of the fractal energy form. These functionals are proper, convex and weakly lower semicontinuous. We preliminary prove that we can approximate functions in V (Q, S) with functions in V (Q, S) ∩ C(Q) (see Theorem 6.4). The key result is the M-convergence of the pre-fractal energy functionals Φ (h) p to the fractal energy functional Φ p . This is equivalent to the G-convergence of the subdifferrentials of pre-fractal functionals (which we denote by A h ) to the subdifferential of the fractal functional (denoted by A); moreover, also the nonlinear semigroups generated by −A h converge to the nonlinear semigroup associated to −A.
We consider then the following two abstract Cauchy problems, for T > 0 fixed: and we give existence and uniqueness results for such problems. We give a characterization of A and A h in order to prove that the solutions of problems (P h ) and (P ) solve in a suitable sense a homogeneous parabolic equation for the p-Laplace operator with nonlinear Venttsel' boundary conditions (see problems (P h ) and (P ) below). We point out that the existence and uniqueness of strong solutions for problems (P h ) and (P ) can be proved also for the nonhomogeneous problems (see Theorem 2.7 in [28] for the fractal case in two dimensions), but in this case the asymptotic behavior of the solutions is still an open problem. In the homogeneous case, we are able to prove that the solutions of the pre-fractal problems converge to the limit fractal one. The plan of the paper is the following. In Section 2 we introduce some notions on fractal sets. In Section 3 we present some properties of Sobolev spaces and Besov spaces. In Section 4 we give the definition of varying Hilbert spaces. In Section 5 we introduce the energy functionals in both the pre-fractal and the fractal case. In Section 6 we prove some density results. In Section 7 we prove the M-convergence of the functionals. In Section 8 we introduce the nonlinear Venttsel' boundary value 4 , with respect to the same ratio 1 3 (see [18]). Let V (1) 0 h . It holds that K 1 = V (1) . Now let K 0 denote the unit segment whose endpoints are A 1 and A 3 . We set K i1.
In a similar way, it is possible to approximate K 2 , K 3 by the sequences (V h ) h≥0 , and denote their limits by V (2) , V (3) .
In order to approximate F , we define the increasing sequence of finite sets of The Hausdorff dimension of the Koch snowflake is given by D f = ln 4 ln 3 . One can define, in a natural way, a finite Borel measure µ supported on F by where µ i denotes the normalized D f -dimensional Hausdorff measure, restricted to K i , i = 1, 2, 3. In the following we denote by the closed polygonal curve approximating F at the (h + 1)-th step.
we denote the open bounded set having as boundary F h . We denote by Q h the three-dimensional cylindrical domain having S h as "lateral surface" and the sets Ω h × {0} and Ω h × {1} as bases.
In an analogous way, we define the cylindrical-type surface S = F × I and we denote by Ω the open bounded two-dimensional domain with boundary F . As above, by Q we denote the open cylindrical domain having S as lateral surface and the sets Ω × {0} and Ω × {1} as bases (see Figure 2).
We denote the points of S and S h by the couple P = (x, y), where x = (x 1 , x 2 ) are the coordinates of the orthogonal projection of P on the plain containing F and F h respectively (for S and S h ) and y is the coordinate of the orthogonal projection of P on the interval [0, 1], that is (x 1 , x 2 ) ∈ F (or (x 1 , x 2 ) ∈ F h for the pre-fractal case) and y ∈ I.
We introduce on S the measure where dL 1 is the one-dimensional Lebesgue measure on I. By R we denote the open equilateral triangle whose midpoints are the vertices 3. Functional spaces. By L p (·) we denote the Lebesgue space with respect to the Lebesgue measure dL 3 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 C(T ) we denote the space of continuous functions on T and C 0,β (T ) is the space of Hölder continuous functions on T , 0 < β < 1. Let G be an open set of R 3 , by W s,p (G), where s ∈ R + , we denote the (possibly fractional) Sobolev spaces (see [41]). D(G) is the space of infinitely differentiable functions with compact support on G.
By we denote the arc-length coordinate on each edge F h and we introduce the coordinates x 1 = x 1 ( ), x 2 = x 2 ( ), y = y on every affine face S h . In the following, we will make use of trace spaces on polygonal and polyhedral boundaries. By W 1,p (F h ) we denote (see [7]) the set In the sequel, we consider W 1,p (F h ) with the norm By W r,p (F h ), 0 < r ≤ 1 we denote the Sobolev space on F h , defined by local Lipschitz charts as in [41]. We denote by W 1,p (S h ) the Sobolev space (on the polyhedral domain S h ) of functions for which the norm is finite [41]. We now introduce the notions of d-set and trace.
if there exist a Borel measure µ with supp µ = M and two positive constants c 1 and c 2 such that We point out that, from Definition 3.1, it follows that F is a D f -set, the measure µ F is a D f -measure, S is a (D f + 1)-set and the measure g defined in (2.3) is a (D f + 1)-measure.
at every point P ∈ G where the limit exists.
It is known that the limit exists at quasi every P ∈ G with respect to the (s, p)capacity (see [1] From now on, we set β = 1 − 2−D f p . We now define the Besov space on S only for this particular β, which is the case of our interest. For a general treatment see [22].

Definition 3.4.
We say that f ∈ B p,p β (S) if f ∈ L p (S, g) and it holds We now recall a trace theorem.
1. There exists a linear and continuous operator γ 0 : W 1,p (Q) → B p,p α (Γ). 2. There exists a linear and continuous operator Ext : For the proof we refer to Theorem 1 of Chapter VII in [22], see also [46]. In the case Γ = S, then the smoothness index α is equal to 1 − . In the following we denote by u| S and u| S h the trace of u on S and S h respectively. Sometimes we will omit the trace subscript and the interpretation will be left to the context.
The following theorem characterizes the trace on S h of a function in W β,p (R 3 ) (see [1] for a general treatment of Sobolev spaces).
where Cβ is independent of h.
Proof. We point out that every u ∈ Wβ ,p (R 3 ) can be expressed in the following way: where Gβ is the Bessel kernel of orderβ (see [22]). Then by Hölder inequality we have where 0 < a < 1 will be chosen later. Now, by using Lemma 1 on page 104 in [22], we get Moreover, since S h is a 2-set with constant c 2 = C 3 δ −1 h (see Definition 3.1), again from Lemma 1 on page 104 in [22] we get Hence, by choosing a in order to satisfy (3.3) and (3.4), by using Fubini's Theorem we get The following theorem is a consequence of Theorem 1 in Chapter V of [22].
It is possible to prove that the domains Q h are (ε, δ) domains with parameters ε and δ independent of the (increasing) number of sides of S h . Thus by the extension theorem for (ε, δ) domains due to Jones (Theorem 1 in [20]) we obtain the following Theorem 3.8, which provides an extension operator from W 1,p (Q h ) to the space W 1,p (R 3 ) whose norm is independent of h.
Theorem 3.8. There exists a bounded linear extension operator Ext J : with C J independent of h.
Theorem 3.9. There exists a linear extension operator Ext such that, for anỹ withCβ depending onβ.

4.
Convergence of Hilbert spaces. We introduce the notion of convergent Hilbert spaces that we will use in the next sections. For further details and proofs of the theorems see [24] and [23]. The Hilbert spaces we consider are real and separable.

Definition 4.2 (Strong convergence in H). A sequence of vectors {u
Remark 1. We note that the strong convergence implies the weak convergence (see [24]).
Let us recall some characterizations of the strong convergence of a sequence of vectors {u h } h∈N in H.
x ∈ D(A) and y ∈ A(x)) there exists [x n , y n ] ∈ A n , n ∈ N such that x n → x and y n → y strongly in H.
In the following we denote by L 2 (Q, m) the Lesbegue space with respect to the measure m with dm = dL 3 + dg, (4.1) where g is the measure defined in (2.3), and by the space L 2 (Q, m h ) the Lebesgue space with respect to the measure m h with where χ Q h and χ S h denote the characteristic function of Q h and S h respectively.
Throughout the paper we consider H = L 2 (Q, m) where m is the measure in (4.1), and the sequence For the proof, see Proposition 4.1 in [35], where C and Z h in Definition 4.1 are respectively C(Q) and the identity operator on C(Q).

5.
Energy functionals. From now on, let p > 2 (for the case p = 2, we refer to [26] and [27]). By proceeding as in [8], we construct a p-energy form on F (which has the role of Euclidean p-Lagrangian dL(u, v) = |∇u| p−2 ∇u∇v dL 3 ) by defining a p-Lagrangian measure L p F on F . The corresponding p-energy form on F is given by As in [9] the following result can be proved.
Remark 2. We point out that, for p > ln 4 ln 4−ln 3 , the Hölder exponent η in Proposition 5.2 is greater than one. In this case, for the Koch snowflake F , from Corollary 4.2 in [9], the space C 0,η (F ) does not degenerate to the space of constant functions.
We now define the energy form on S: We now give an embedding result for the domain D(S). Unlike the two dimensional case where there is a characterization of the functions in D(F ) in terms of the so-called Lipschitz spaces (see Theorem 4.1 in [9]), for D(S) we do not have such characterization, but the following result holds.
Proposition 5.3. D(S) is continuously embedded in B p,p β (S), for any 0 <β < 1. Proof. We follow the proof in [25], adapted to our case.
We recall that Following [37], we define From Theorem 4.1 in [9] and Proposition 3, Chapter V in [22], it holds that D(F ) = B p,∞ D f (F ). Moreover, this last space is continuously embedded in B p,p D f −ε (F ) for ε > 0 (see Proposition 5, Chapter VIII in [22]). Hence, from the definition of D(S), we deduce that D(S) ⊂ B p,p D f −ε,1 (S). Moreover, the embedding is continuous, i.e. there exists a positive constant C such that From the definition of E S and of the norm in D(F ), we get i.e. the thesis. For any Banach space X and for any 0 <β < 1 Hence if 0 <β < 1 where the last equivalence can be proved following [37]. We now prove that there exists a positive constant C such for every 0 <β < 1 Indeed, from the above remarks, we get Let b be a continuous and strictly positive function on Q. We consider the energy functional Φ p defined as follows: For the proof see Proposition 2.3 in [28]. We now set . We introduce the energy functional on the pre-fractal domain: By proceeding as in Proposition 2.3 in [28], we can prove the following result. The following results hold. Proof. One can adapt the proof of Theorem 5.3 in [26] with small suitable changes.
We now state the main Theorem of the section.
Theorem 6.4. Let Q, S and V (Q, S) be defined as in Section 2 and Section 5 respectively. For every u ∈ V (Q, S), there exists ψ n ∈ V (Q, S) C(Q) such that: In order to prove this Theorem, we need a preliminary proposition on trace and extension operators. Proposition 6.5. Let β be as in Section 3. Let γ 0 and Ext be the trace and the extension operator defined in Theorem 3.5 respectively. Then . Proof. One can adapt the proof of Proposition 5.5 in [26] with the obvious changes when considering the case p ≥ 2 instead of p = 2.
We are now ready to prove Theorem 6.4.
We now prove that ϕ n − u W 1,p (Q) → 0. Indeed, from Theorem 3.5 and the inclusion of D(S) in B p,p β (S) (see Proposition 5.3), ϕ n − u W 1,p (Q) ≤ C 1 ϕ n − u| S B p,p β (S) ≤ ϕ n − u| S D(S) → 0 from the density Theorem 6.3. Now let us consider the function u − u. This function belongs to W 1,p (Q) and it is such that (u − u)| ∂Q = 0, then u − u ∈ W 1,p 0 (Q) (see Theorem 3 in [48]). There Let {ψ n,m } denote the doubly indexed sequence of function { ϕ n −η m }. The sequence {ψ n,m } belongs to W 1,p (Q) C(Q). From Corollary 1.16 in [3] we deduce that {ψ m,n } converges to u in W 1,p (Q) as n → ∞. In fact there exists an increasing mapping n → m(n), tending to ∞ as n → ∞, such that Hence by applying Corollary 1.16 in [3] to the right hand side of the above inequality it follows that The two terms in the sum tend to zero when m, n → ∞, then lim n→∞ ψ n,m(n) − u W 1,p (Q) = 0, (6.5) and also lim n→∞ ψ n,m(n) − u W 1,p (Q) = 0. Hence we conclude that ψ n,m(n) − u W 1,p (Q) → 0 when n → ∞.
From now on we denote by ψ n = ψ n,m(n) . We now prove (2), that is The first term in (6.6) tends to zero when n → ∞ since We now prove that also the second term in (6.6) tends to zero: and the last quantity tends to zero from the density of D(0, 1; D(F )) in D(S). This proves that ψ n → u in L p (Q, m). We now prove (3): Hence the theorem is proved.
We remark that we can prove a result similar to Theorem 6.4 also for the prefractal case. We define the space Similarly to Proposition 6.1, we can prove that D(0, 1; W 1,p (F h )) is dense in W (h) (0, 1). But it turns out that We also point out that we can prove as in Theorem 6.2 that D(0, 1; W 1,p (F h )) ⊂ C(S h ). Hence the following result holds.
Theorem 6.6. For every u ∈ V (Q, S h ) there exists ψ n ∈ V (Q, S h ) ∩ C(Q) such that: . From the trace Theorem 3.3 there exists an extensionφ n belonging to W 1,p (Q h ); then, from Theorem 3.8, there exists an extensionφ n ∈ W 1,p (R 3 ). We point out that, since ϕ n ∈ C(S h ), as in Proposition 6.5 we can prove that the extension of ϕ n is continuous on Q. We set ψ n :=φ n | Q , hence ψ n ∈ W 1,p (Q). From Theorem 3.8 and Theorem 3.3 we get and the last quantity tends to 0 for n → ∞ from the density of D(0, 1; W 1,p (F h )) in W 1,p (S h ). As to (2), the following holds from (1) and the density of D(0, 1; W 1,p (F h )) in W 1,p (S h ): We now come to (3): Hence the thesis follows.
Remark 3. The results obtained so far in this paper still hold if we consider the more general case of fractal mixtures. Since our aim is to prove convergence results (see Sections 7 and 8), we have to consider the equilateral case instead of the mixture, since for the mixture case we are not able to make an appropriate triangulation of the domain and this tool is crucial to prove the M-convergence.
The main theorem of this section is the following.
be defined as in (5.8) and (5.10) respectively. Then Φ (h) p M-converges to the functional Φ p . We preliminary state the following propositions. For the proof see Proposition 6.6 in [27].
Proof. The proof follows from Proposition 3.7 in [14].
We are now ready to prove Theorem 7.2.
Proof. We prove conditions a) and b) in Definition 7.1.

Proof of condition a).
Let v h ∈ H h be a weakly converging sequence in H to u ∈ H. We can suppose that v h ∈ V (Q, S h ) and (otherwise the thesis follows trivially). Then there exists a c independent of h such that (7.1) Let us suppose that v h is continuous on Q. From (7.1), in particular we have that v h W 1,p (Q h ) < c. For every h ∈ N from Theorem 3.8 there exists a bounded linear operator Ext : We now setv h = Ext v h | Q . Thenv h ∈ W 1,p (Q) and v h W 1,p (Q) ≤ c C, hence there exists a subsequence, still denoted byv h , weakly converging tov in W 1,p (Q). We point out thatv h strongly converges tov in L p (Q) and also in L 2 (Q) since p ≥ 2. From Proposition 7.3, v h weakly converges to u in L 2 (Q). We prove that v = u L 3 -a.e., that is For every > 0 there exists h ∈ N such that each term in the sum of the right-hand side of (7.2) is less than /3. Sincev h →v in L 2 (Q) and v h u in L 2 (Q) we deduce our claim for the first two terms. As to Q\Q h (v h − u)ϕ dL 3 , from Hölder inequality we deduce that We now prove that It is enough to prove that χ Q h Dv h Du in L p (Q), from here the claim will follow from the semicontinuity of the norm. Since χ Q h Dv h = χ Q h Dv h , this amounts to prove that Q χ Q h Dv h ϕ dL 3 → Q Duϕ dL 3 for every ϕ ∈ L p (Q).
It holds that The first term vanishes as h → ∞ since Dv h Du in L p (Q). Now we estimate the second term Q\Q h |Dv h ϕ| dL 3 . We have Moreover, the following holds as a consequence of Theorem 3.5 in [14] and Fatou Lemma. We are left to prove that First we point out that, since v h weakly converges to u in W 1,p (Q), it follows that v h strongly converges to u in W s,p (Q) for every s ∈ (0, 1). Hence, from Theorem 3.5, v h | S strongly converges to u| S in B p,p s− 2−D f p (S), so in particular v h | S strongly converges to u| S in L p (S). We now set w h := D y v h ∈ L p (Q). In order to prove (7.4), we preliminary prove that From the density of C ∞ (Q) in W 1,p (Q) (see [38,Theorem 2,page 28] . We want to prove that w n h L p (S) ≤ c. By proceeding as in the proof of Theorem 4.5 in [29], since w n h is continuous on S, we can estimate the above norm in terms of the corresponding Darboux sums, and we get Passing to the upper limit as n → ∞, since w n h strongly converges to w h in L p (S h ), from (7.1) we get lim n→∞ w n h L p (S) ≤ c.
Since w n h is bounded in L p (S), there exists a subsequence (still denoted by w n h ) weakly converging to a function w * h in L p (S) for n → ∞. Moreover, from the lower semicontinuity of the norm, we have The above inequality implies that there exists a subsequence of w * h , again denoted by w * h , weakly converging to a function w * in L p (S). By using again the lower semicontinuity of the norm, we get where in the last inequality we used (7.5). Hence (7.4) follows if we prove that w * = D y u a.e. in L p (S). By using the definition of weak convergence and distributional derivative, we get i.e. the thesis. We conclude the proof taking into account the liminf properties of the sum and Proposition 7.4.
If v h is not continuous on Q, from Theorem 6.6 there exists h . By triangle inequality we easily have that w h tends to u weakly in H. Hence from the previous step we have i.e. the thesis.
Proof of condition b). We have to prove that for every u ∈ H there exists {w h } h∈N strongly converging to u in H such that We can suppose that u ∈ V (Q, S). Indeed, if u / ∈ V (Q, S) then Φ p [u] = +∞ and from Lemma 4.8 it follows that there exists a sequence {v h } h∈N converging to u in H and hence lim Let then u ∈ V (Q, S), i.e. u ∈ W 1,p (Q) and u| F ∈ D(S). For the case p = 2, we refer to [27]. Here we investigate the case p > 2. We have to consider two cases.
Step 1. We suppose that u ∈ C(Q), hence u ∈ H. We extend by continuity u to T and we put u this extension. Following the same approach of [30] and [29], we introduce a quasi uniform triangulation τ h of T made by equilateral tetrahedron T j h such that the vertices of the pre-fractal surface S h are nodes of the triangulation at the h-th level. Let S h be the space of all the functions being continuous on T and affine on the tetrahedrons of τ h . We indicate by M h the nodes of τ h , that is the set of the vertices of all T j h . For a given continuous function u, we denote by I h u the function which is affine on every T j h ∈ τ h and which interpolates u in the nodes P j,i ∈ M h Q h . We set w h = I hû and we prove that {w h } strongly converges to u in H, which is equivalent to prove that (see Lemma 4.5) We know that w h − u W 1,p (T ) → 0 (7.6) as h goes to ∞ (see [13]) and hence w h − u W 1,p (Q) → 0. From Theorem 3.6, there exists a constant c independent of h such that Then we have We now prove condition b) for the sequence w h . We note that from Proposition 7.4 We have that then, by taking the limit for h → ∞, we have the thesis (since D(w h −u) L p (Q) → 0 for h → ∞).
We have only to prove that Since w h = I hû , we have that where l j = (j − 1) 3 −h and y i = (i − 1) 3 −h for j = 1, . . . , 3N , i = 1, . . . , M . Hence we get Passing to the upper limit, we get In the same way one can prove that Taking into account the limsup property of the sum the conclusion of the theorem follows.
Step 2. If u ∈ V (Q, S), but u is not continuous, from Theorem 6.4 there exists ψ n ∈ V (Q, S) C(Q) such that ψ n → u in H and ψ n − u V (Q,S) → 0. Let n ∈ N fixed such that ψ n − u V (Q,S) ≤ 1 n and ψ n − u H ≤ 1 n . Byψ n we denote a continuous extension in T . From Step 1 we have that for every fixed n ∈ N I hψn strongly converges toψ n in H, I hψn converges toψ n in W 1,p (T ) when h → ∞ and Passing to the upper limit for n → ∞ to both sides of the above inequality we obtain We now want to apply Corollary 1.16 in [3] for proving that there exists an increasing mapping h → n(h) such that, denoting by w h = I hψn(h) , we have that w h converges to u in H and lim . To this aim we have to prove that Passing to the upper limit for h → ∞, we obtain Then Corollary 1.16 in [3] provides the thesis.
In the following Theorem we deduce the G-convergence of the associated subdifferentials.
For the proof see Theorem 7.46 in [45]. This result will be crucial for the convergence of the solutions of the nonlinear abstract Cauchy problems. 8. Convergence of the solutions. We now consider the abstract homogeneous Cauchy problem (P ) du dt + Au 0, t ∈ [0, T ] u(0) = u 0 , where A is the subdifferential of Φ p , T is a fixed positive number, and u 0 is a given function. We now recall some results on the properties of nonlinear semigroups generated by the (opposite of) subdifferential of a proper convex lower semicontinuous functional on a real Hilbert space (see Theorem 1 and Remark 2 in [6], see also [5]).
According to [ In addition, −∂ϕ generates a (nonlinear) semigroup {T (t)} t≥0 on H, where for every t ≥ 0,T (t) is the composition of the semigroup T (t) on D(ϕ) with the projection on the convex set D(ϕ). In our case it turns out that, from Theorem 8.1, the subdifferentials ∂Φ p and ∂Φ (h) p are maximal, monotone and m-accretive operators on H and H h respectively. Then, if we denote with T p (t) and T respectively, these semigroups are strongly continuous and contractive on H and H h (see Proposition 2.5 in [28] for the fractal case).
Theorem 2.7 in [28] states the following result.
Theorem 8.2. If u 0 ∈ D(−A), then (P ) has a unique strong solution u ∈ C([0, T ]; H) defined as u = T p (·)u 0 such that u ∈ W 1,2 ((δ, T ); H) for every δ ∈ (0, T ). Moreover u ∈ D(−A) a.e. for t ∈ (0, T ), √ t du dt ∈ L 2 (0, T ; H) and Φ p [u] ∈ L 1 (0, T ). Moreover, from Theorem 2.6 in [28] it can be proved that the solution u of problem (P ) solves the following problem(P ) on Q for t ∈ (0, T ] in the following weak sense: We now come to the pre-fractal case. For each h ∈ N fixed, we consider the abstract homogeneous Cauchy problem where A h is the subdifferential of Φ   = δ h f, ψ L 2 (S h ),L 2 (S h ) for every ψ ∈ W 1,p (S h ), where ∂u h ∂n h denotes the normal derivative across S h and ∆ p,y := div(|D y | p−2 D y ).