Asymptotics for venttsel' problems for operators in non divergence form in irregular domains

We study a Venttsel' problem in a three dimensional fractal domain for an operator in non divergence form. We prove existence, uniqueness and regularity results of the strict solution for both the fractal and prefractal problem, via a semigroup approach. In view of numerical approximations, we study the asymptotic behaviour of the solutions of the prefractal problems and we prove that the prefractal solutions converge in the Mosco-Kuwae-Shioya sense to the (limit) solution of the fractal one.

1. Introduction. In this paper we study a parabolic Venttsel' problem in a threedimensional fractal domain. Venttsel' conditions are the most feasible boundary conditions for an elliptic or parabolic problem, they include Dirichlet, Neumann and general oblique boundary conditions as special cases.
They appeared for the first time in ( [47]) in the framework of probability theory. From the point of view of applications they occur in different contexts such as threedimensional water wave theory, models of heat transfer and hydraulic fracturing (see [44], [25], [7]).
In the framework of heat transfer, Venttsel' boundary conditions appear when considering the asymptotic behaviour of heat flow problems for highly conductive coated structures, see [13] for details. The interest in studying the heat flow across irregular domains with fractal boundaries arises from the fact that a lot of industrial and natural processes lead to the formation of rough surfaces or take place across them; fractal geometries turn out to be useful tools to describe such wild geometries. For example the current flow across rough electrodes in chemistry (see [43]) and the diffusion processes in physiological membranes are transport phenomena taking place across irregular layers/boundaries. The literature on Venttsel' problems in regular domains is large, we refer to [14] and the references listed in, as to Venttsel problems in fractal domains the first results, to our knowledge, can be found in [32], where a second order operator in divergence form in a two dimensional fractal domain is considered (see also [9] and [10] for the numerical approximation). It is to be pointed out that the extension to the three dimensional case will require different tools to study the asymptotic behaviour of the solutions.
More precisely, we consider the following boundary value problem for a second order operator in non divergence form with Venttsel's boundary conditions: here L is an operator in non divergence form, Lu = 3 i,j=1 a ij ∂ ij u + a 0 u, a ij are symmetric, uniformly Lipschitz functions in Q satisfying suitable ellipticity conditions (see condition (H 1 ) in Section 4) and a 0 is a positive L ∞ (Q) function, Q is the three-dimensional domain with lateral boundary S = F × [0, 1], where F is the Koch snowflake; ∆ S is the fractal Laplacian on S (see Theorem 4.5 in Section 4), b is a continuous strictly positive function on S, ∂u ∂n A is the co-normal derivative across ∂Q to be defined in a suitable sense (see Theorem 8.1), f (t, P ) is a given function in C θ ([0, T ]; L 2 (Q, m)), θ ∈ (0, 1) and m is the sum of the three-dimensional Lebesgue measure and of a suitable measure g supported on S (see Section 2).
From the point of view of numerical analysis it is also crucial to study the corresponding approximating (prefractal) problems (P h ). To this aim the asymptotic behaviour, as h → ∞, of the approximating solutions is studied. More precisely, we consider for each h ∈ N, the prefractal problems  1] are the corresponding approximating polyhedral surfaces, where F h is a prefractal curve approximating F (see Section 2); ∆ S h is the piecewise tangential Laplacian defined on S h , ∂u ∂n A h is the co-normal derivative across ∂Q h to be defined in a suitable sense (see Theorem 8.2), f h (t, P ) is a given function in C θ ([0, T ]; L 2 (Q, m h )), θ ∈ (0, 1); m h is the sum of the three-dimensional Lebesgue measure and of the surface measure δ h σ of S h , where δ h is a positive constant (see Section 4).
The natural functional setting for these problems is L 2 (Q, m) and L 2 (Q, m h ), respectively because of the presence of the time derivative in the boundary conditions. In this setting, existence and uniqueness results are proved via a semigroup approach, in Section 7 we consider the abstract Cauchy problems (P ) and (P h ), h ∈ N and we prove the existence and uniqueness of the strict solutions for (P ) and (P h ), for every h ∈ N, (see Theorem 7.1 and Theorem 7.2).
The study of the asymptotic behaviour of the solutions of problems (P h ), as h → ∞ brings us to the framework of varying Hilbert spaces L 2 (Q, m h ) . The convergence of the solutions of problems (P h ) to problem (P ) is thus obtained via the Mosco-Kuwae-Shioya convergence of the approximating energy forms E (h) (see [39] and [26]) in varying Hilbert spaces. Actually the convergence of energies implies the convergence of semigroups in a suitable sense, (see Section 7). Finally, in Theorems 8.1 and 8.2 we prove that problems (P ) and (P h ) are the strong formulations of the abstract problems (P h ) and (P ), respectively.
It is to be pointed out that the proof of the convergence of the energies in the three dimensional case, with respect to the two-dimensional one dealt with in [32], is a delicate issue and it relies on new density results, for functions in the domain of the energy form E related to (P ), recently proved in ( [29]).
We point out that from our asymptotic analysis it turns out that the solution of the limit fractal problem (P ) can be indeed approximated by the prefractal solutions of the corresponding approximating prefractal problems (P h )) (see Theorems 7.4 and 7.5). It is still an open problem, even in the bidimensional case and when the operator is the Laplacian, to obtain a quantitative estimate of the rate of convergence in terms of h. Nevertheless our results are a first step towards the study of the numerical approximation of the prefractal problem (P h ) for every fixed h. By proceeding as in [9] and [10] one can perform an a priori error estimate of the numerical approximation by a finite element scheme in space and a finite difference scheme in time. We stress the fact that the domains Q h are not convex polyhedral domains hence H 2 regularity of the solution u h is deteriorated by the presence of reentrant wedges and vertices. A delicate regularity analysis of the solution u h is necessary in order to obtain an optimal rate of convergence. This will object of a forthcoming paper.
The outline of the paper is the following: in Section 2 we introduce the geometry of the Koch snowflake and the domains Q and Q h ; in Section 3 we introduce the main functional spaces; in Section 4 we define the prefractal and fractal energy forms; in Section 5 we introduce the notion of convergence in varying Hilbert spaces; in Section 6 we recall the density Theorems for functions in the domain of the energy form E, we give the definition of Mosco-Kuwae-Shioya convergence of energy forms and we prove the convergence of the approximating energies E (h) to E (see Theorem 6.8); in Section 7 we consider the abstract Cauchy problems in both the fractal and prefractal case and we state the existence and uniqueness Theorem for (P h ) and (P ) respectively; we then prove that the solutions of problems (P h ) do converge to the solution of the problem (P ) in a suitable sense (see Theorem 7.4); in Section 8 we prove that the solutions of the abstract problems (P h ) and (P ) solve in a suitable strong sense problems (P h ) and (P ) respectively.
2. Geometry of Q, S and S h . We denote by |P − P 0 | the Euclidean distance in R n and the Euclidean balls by B(P 0 , r) = {P ∈ R n : |P − P 0 | < r}, P 0 ∈ R n , r > 0. By the snowflake F we denote the union of three complanar Koch curves K i (see [12]).
We assume that the junction points A 1 , A 3 , A 5 are the vertices of a regular triangle with unit side length, that is log3 and where B(P, r) denotes the Euclidean ball in R 2 . K 1 is the uniquely determined self-similar set with respect to four suitable contractions ψ (1) , ..., ψ (4) , with respect to the same ratio 1 3 (see Section 3.2 in [15]). Let V (1) 0 h . It holds that K 1 = V (1) . Let K (0) 1 denote the unit segment whose endpoints are A 1 and A 3 and K We set K h+1 the set of its vertices.
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) , and the corresponding polygonal . In order to approximate F , we define the increasing sequence of finite sets of points In the following we denote by the closed polygonal curve approximating F at the (h + 1)−th step. By S h we denote F h × I, where F h is the prefractal approximation of F at the step h, I = [0, 1]. S h is a surface of polyhedral type. We give a point P ∈ S h the Cartesian coordinates P = (x, x 3 ), where x = (x 1 , x 2 ) are the coordinates of the orthogonal projection of P on the plane containing F h and x 3 is the coordinate of the orthogonal projection of P on the x 3 -line containing the interval I.
By Ω h we denote the open bounded two-dimensional domain with boundary F h . By Q h we denote the domain with S h as lateral surface andΩ h : where dl is the arc-length measure on F h and dx 3 is the one-dimensional Lebesgue measure on I. We introduce the fractal surface S = F × I given by the Cartesian product between F and I. It can be defined on S the finite Borel measure supported on S. The measure g is a d-measure (see Definition 3.1), that is there exist two positive constants c 1 , c 2 where d = d f + 1 = log12 log3 and B(P, r) denotes the Euclidean ball in R 3 . By Ω we denote the two-dimensional domain whose boundary is F . By 3. Functional spaces. By L 2 (·) we denote the Lebesgue space with respect to the Lebesgue measure L 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 H s (G), s ∈ R + we denote the Sobolev spaces, possibly fractional (see [40]). D(G) is the space of infinitely differentiable functions with compact support on G.
if there exists a Borel measure µ with suppµ = T such that for some c 1 , c 2 > 0 Remark 3.3. It is known that the limit exists at quasi every P ∈ G with respect to the (s, 2)−capacity (see [1] . From now on we denote by u| S h the trace operator, that is u| S h = γ 0 u. The following Theorem characterizes the trace on the polyhedral set S h of a function belonging to the Sobolev space H β (R 3 ).
Proof. We adapt the proof from the two dimensional case treated in [8]. Any u ∈ H β (R n ) can be written in terms of Bessel kernels G β , of order β, that is u = G β * g, g ∈ L 2 (R 3 ), (see [45]). Then where 0 < a < 1 will be chosen later. By using the estimates for the Bessel kernels and Lemma 1 on page 104 in [22], we get where C 1 is independent of h. Moreover, since S h is a 2−set, with c 2 = Cδ −1 h , (according to Definition 3.1 we get again from Lemma 1 on page 104 in [22] By choosing a in order to satisfy (3.2) and (3.3), we get The following theorem that characterizes the trace on the set S of a function belonging to Sobolev spaces H β (R 3 ) is a consequence of Theorem 1 in Chapter 5 of [22] as the fractal S is a d−set.
It is possible to prove that the domains Q h are ( , δ) domains with parameter independent of the increasing number of sides of S h . We recall an extension Theorem for ( , δ) domains (see [20]) adapted to the present case: There exists a bounded linear extension operator Ext J : such that, for any β > 0, with C β depending on β.

Besov spaces.
The following trace Theorem is due to Jonsson-Wallin, for the proof see Theorem 1 of Chapter VII in [22]: 1. There exists a linear and continuous operator, (trace operator), γ 0 : 2. There exists a linear and continuous operator, (extension operator), Ext : From now on we denote by u| S the trace operator, that is u| S = γ 0 u.

Energy forms and semigroups.
4.1. The energy forms E (h) . Let Q, S, Q h and S h be defined as in Section 2. We consider the energy forms E S h on S h = F h ×I, h ∈ N. By l we denote the arc-length coordinate on each edge F h and we introduce the coordinate h of S h . By dl we denote the 1-dimensional measure given by the arc-length l, and by dσ the surface measure on S where σ 1 h , σ 2 h are positive costants, D l denotes the tangential derivative along the prefractal F h , and u ∈ H 1 (S h ). By the Fubini theorem, E S h can be written in the form We denote by E S h (u, v) the corresponding bilinear form. Let us consider now the function space Let us consider the energy forms ( We recast L h u as the sum of an operator in divergence form and a lower order operator, by Leibniz formula: From now on we consider the following energy forms

The corresponding bilinear form is
In the following we consider also the space (4.7)

Moreover from assumption (H
We note that in assumption (H 1 h ) the constant λ could depend on h, in the asymptotic analysis they cannot degenerate (i.e. they cannot tend to zero ) and it has to be assumed that inf h λ h > 0.

4.2.
The energy form E. By proceeding as in [15] we construct an energy form on F , by defining a Lagrangian measure L F on F , which has the role of the Euclidean Lagrangian dL(u, v) = Du Dv dx. The corresponding energy form on F is given by We now define the energy form on S and the fractal Laplacian ∆ S .
The form E S is defined for u ∈ D(S), . Now we introduce the energy form on Q. Let us consider the space Let us consider the energy form , is a non divergence operator, a ij are uniformly Lipschitz continuous functions in Q and a 0 is a positive essentially bounded function in Q, ∂ i denotes the partial derivative with respect to x i , i = 1, 2, 3 and ∂ ij = ∂ 2 ∂xi∂xj ; ( From now on we consider the energy form: where A = [a ij ]. We denote by L 2 (Q, m) the Lebesgue space with respect to the measure dm = dL 3 + dg. ByẼ(u, v) we denote the bilinear form  Moreover from assumption (H 2 ) we deduce thatẼ[u] is a positive definite nonsymmetric form. We decomposeẼ(u (4.11) A is the infinitesimal generator of a strongly continuous semigroup {T (t)} t≥0 . Proof. The proof follows from Chapter 17, Section 6 in [11], sinceẼ[·] is weakly coercive that is there exist positive constants λ 0 and α 0 :  Proof. The proof follows from Chapter 17, Section 6 in [11], sinceẼ h [·] is weakly coercive that is there exist positive constants λ 0 and α 0 depending on h such that: In the following, for every α > 0, we denote byG α ,G h α ,Ĝ α andĜ h α the resolvents and coresolvents associated withÃ and its adjointÂ. From Thoerem 2.8 in [38] we have that the following equalities hold

5.
Varying Hilbert spaces. We introduce the notion of convergence in varying Hilbert spaces; for more details, see [26].
where m h is the measure defined in (4.7), with norms We note that where M j denotes a segment of h-generation.
Since u(·, x 3 ) is continuous on F h for each x 3 ∈ [0, 1], by the mean value Theorem, there exists ξ j ∈ M j such that We can write where P j is one of the endpoints of M j . The first term of right-hand side of the inequality tends to zero as h → ∞ from the Corollary 3.4 in [34], while the second vanishes since |u| 2 is uniformly continuous in every M j . Since the thesis follows from dominated convergence theorem.
6. M-convergence of the energy forms and density theorems. In this section we prove the main theorem of this paper, the M-K-S-convergence of the energy forms. The proof relies on some crucial density results for the functions in the domain of the energy form E specialized to the case of interest, for further details we refer the reader to [29].
(See Theorem 5.4 in [29]). 6.3. M-convergence of the symmetric energy forms. We now give the definition of M -convergence of symmetric forms in the case of varying Hilbert space, by using the definition of Kuwae and Shioya in [26].
We put the form and For every w ∈ C we set ϕ h = wχ Q h and ϕ = wχ Q : ϕ h ∈ H h and ϕ ∈ H. We prove that ϕ h strongly converges to ϕ in H. This result follows from Lemma 5.7, in fact the first claim holds since and Q h is a family of sets invading Q. By the same argument it follows that From (6.13) and the choice of ϕ h and ϕ 14) The constant sequence {w} strongly converges to w in H; choosing ϕ h = w in (6.13) and taking into account (6.14), by difference we obtain We now prove the weak convergence of v h to u in L 2 (Q). We first prove the convergence for every φ ∈ C(Q), then the claim will follow by density.
since φχ Q\Q h strongly tends to zero in H and φχ Q h strongly converges to φχ Q in H.
The first term in the right hand side of (6.15) can be estimated as follows: Since v h weakly converges in H 1 (Q) to u, then v h strongly converges to u in H α (Q) for every α ∈ (0, 1). Considering the extension of (v h − u) to H α (R 3 ), it follows from Theorems 3.5 and 3.8 From these inequalities it follows that Proposition 4.4 in [20]). Now we estimate the second term on the right hand side of (6.15).
It is possible to estimate from above the first and the third term of the right hand side of this inequality with g n − u H 1 (Q) , and hence we conclude that for every ε > 0, there exists n ε ∈ N such that these two terms are less than cε.
If we choose n > n ε , the second term in the right-hand side goes to 0 for h tending to +∞, since H h converges to H. Now we state and proof the main theorem of this Section. 1, 2, 3 and that a h ij and a h 0 converge a.e. in Q to a ij and a 0 respectively, then the sequence E (h) converges in the sense of Mosco, Kuwae, Shioya to the form E.
Proof. Condition 1. Let For every h ∈ N from Theorem 3.7 there exists a continuous linear operator Ext : and v h H 1 (Q) ≤ cC, thus there exists a subsequence, still denoted by v h weakly converging to v in H 1 (Q) and hence strongly in L 2 (Q). By Proposition 6.6 it follows that v h weakly converges to u in L 2 (Q).
We want to prove that v = u a.e. that is Q ( v − u)ϕdL 3 = 0 for each ϕ ∈ L 2 (Q).
Since v h → v in L 2 (Q) and v h weakly converges to u in L 2 (Q), it follows that the first two terms of right hand side vanish. Moreover, from Holder inequality and . From the assumptions it follows that From Severini-Egorov Theorem it follows that follows from Remark 5.1 in [34]. Thesis follows from the liminf properties of the sum. Condition 2.We suppose that u ∈ V (Q, S).
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 [31], we introduce a quasi uniform triangulation τ h of T made by equilateral tetrahedron T j h such that the vertices of the prefractal 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 put w h = I h u, and we prove that {w h } strongly converges in H,using the Lemma 5.6: we have to for h tending to ∞ (see [17]) and hence w h − u H 1 (Q) → 0. From Theorem 3.5, there exists c independent of h such that The thesis follows from the weak convergence of v h to v in H and from the fact that We now show that the above sequence {w h } satisfies condition 2 of M -convergence. In fact lim h→∞ δ h S h b|w h | 2 dσ = S b|u| 2 dg. From [30] we have lim h→∞ We prove that The thesis follows since and, from the assumptions on c h ij , a h 0 and on w h , we deduce that . Then the thesis follows from the limsup properties of the sum.
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, 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 H 1 (T) when h → ∞ and Applying the upper limit for n → ∞ to both sides of the above inequality we obtain n . Passing to the upper limit for h → ∞, we obtain Then Corollary 1.16 in [2] provides the thesis.
From the previous Theorem we deduce that Proposition 6.9. For every u ∈ V (Q, S) there exists a sequence w h ∈ H h strongly converging to u ∈ H such that 6.4. Convergence of the total energy formsẼ h . We now study the convergence of the energy formsẼ h (u, v) toẼ(u, v).
Since we are now in the framework of non symmetric forms we make use of the results in [38], [18], [42], [46] and [37] adapted to the present case. Under the hypothesis of Theorem 6.8 the following Theorem holds.
We now prove 2. From Proposition 6.9 and Remark 2.47 in [46] we have that the sequence E h [u] satisfies (6.16). It remains to prove 2. for the term By proceeding as in step 2 of condition 2. in Theorem 6.8 we consider the sequence w h , it turns out that w h ∈ V (Q, S h ) and w h → w in H 1 (Q) Where the limit follows taking into account that b h i L ∞ ≤ M for every i, and that b h i → b i q. uniformly on Q and that the Lebesgue measure of Q − Q h vanishes. Remark 6.11. We point out that from the hypothesis on b h i in Theorem 6.10 the constants in (4.12) do not depend on h. Now we state a Theorem that follows from Theorem 6.10, which is a generalization of Theorem 2.4 in [26]. Theorem 6.12. LetẼ (h) andẼ be the energy forms defined in 4.5 and in 4.8, respectively; under the assumptions of Theorem 6.10 the semigroups {T h (t)} associated with the formẼ h converge, for every t ≥ 0, to the semigroup T (t) associated with the formẼ, in the sense of Definition 5.10.
Proof. In order to prove the convergence of the semigroups we prove that for every α > 0 G h α strongly converges to G α in the sense of Definition 5.10. Taking into account proposition 5.12 we prove thatĜ h α weakly converges toĜ α . We follow the arguments of [42] adapted to the framework of varying Hilbert spaces in [46] (see Theorem 2.41). Let u h ∈ H h be a sequence weakly converging to u ∈ H in H. Let We now prove thatw ∈ V (Q, S) by proving that condition 1. in Theorem 6.10 is satisfied that is sup h w h V (Q,S h ) < +∞ From (4.12) and Remark 6.11 we deduce that (6.17)) and v =w we havẽ From Theorem 2.21 in [46] the convergence of semigroups follows.

7.
Existence results for the abstract fractal and prefractal problems and convergence of the solutions. Let us consider 0 ≤ t ≤ T u(0) = 0 (7.18) and for every h ∈ N  [36] we deduce the following existence results.
Theorem 7.1. Let 0 < θ < 1, f ∈ C θ ([0, T ]; L 2 (Q, m)) and let where T (t) is the analytic semigroup generated byÃ. Then u is the unique strict solution of (7.18) , that is and there exists c such that the following inequality holds: where T h (t) is the analytic semigroup generated byÃ h . Then u h is the unique strict solution of (7.19), that is and u h (0) = 0, and there exists C, independent from h, such that the following inequality holds: This Section is devoted to the study of the behavior of u h when h → ∞. We denote K h = L 2 ([0, T ]; H h ) and K = L 2 ([0, T ]; H). It holds that K h converges to K in the sense of definition 5.1, where the set C = C([0, T ]×Q) and Φ h is the identical operator on C. We denote K = ( Moreover, since v h weakly converges to v in H for every t ∈ [0, T ], it follows that From Lemma 5.5, the contraction property of T h and the assumption (7.24) f h C θ ([0,T ];H h ) < c, we have that there exists a constant c independent from h such that The claim follows from dominated convergence Theorem. Now we prove 2). We note that where the last inequality follows from (7.23) and (7.24). Thus the sequence {u h } is equibounded in [0, T ] and from 1) By applying dominated convergence Theorem we obtain that From 1) it follows in particular that for every t ∈ [0, T ] (u h (t), ψ(t)) H h → (u(t), ψ(t)) H , ∀ψ ∈ C([0, T ] × Q).
From Proposition 7.3 we proved 2). We have to prove that v = du dt . From definition of weak convergence we can write where ∂u ∂n A is the co-normal derivative, defined as an element of (B 2,2 β (S)) . Moreover ∂u ∂n A ∈ C([0, T ]; B 2,2 β (S)) ).
Here the derivatives are intended in the distributional sense. By proceeding as in [28] one can prove a Green formula for the domain Q. The boundary ∂Q is an arbitrary closed set in the sense of [21], that is ∂Q = S ∪Ω, and it supports w the measurem = χ S g + χΩσ. Hence we have: ∂ i (a ij (P ) ∂ j u(t, P )) ϕdL 3 (8.28) for every t ∈ [0, T ] and for every ϕ ∈ H 1 (Q). By following [32] it holds ∂u ∂n A ∈ C([0, T ]; (B 2,2 β (S)) ). Now let ψ be an arbitrary function in V (Q, S) for every fixed t in [0, T ]. Multiplying (7.18) and integrating over Q, we obtain