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International Journal of Partial Differential Equations
Volume 2014 (2014), Article ID 461046, 13 pages
Semilinear Evolution Problems with Ventcel-Type Conditions on Fractal Boundaries
1Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Università degli Studi di Roma “La Sapienza”, Via A. Scarpa 16, 00161 Roma, Italy
2Dipartimento di Matematica, Università degli Studi di Roma “La Sapienza”, Piazzle Aldo Moro 2, 00185 Roma, Italy
Received 30 April 2013; Accepted 17 October 2013; Published 22 January 2014
Academic Editor: William E. Fitzgibbon
Copyright © 2014 Maria Rosaria Lancia and Paola Vernole. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
In this paper we study the parabolic semilinear second-order transmission problem which we formally state as where is the bounded open set , and is a “cylindrical” layer dividing the set into two subsets and (see Figure 2). When is the Koch-type surface , where is the snowflake and (see Section 2), is the energy functional introduced in (12); when is the prefractal surface , is the energy functional introduced in (24). is a nonlinear function from a subset of into . denotes the restriction of to , denotes the jump of across , denotes the Laplace operator defined on the layer (see (12) in Section 3), and denotes the jump of the normal derivatives across , to be intended in a suitable sense.
More precisely, we assume that is a nonlinear mapping from to for any fixed , locally Lipschitz; that is, Lipschitz on bounded sets in with Lipschitz constant when restricted to , satisfying suitable growth conditions (see conditions (i) and (ii) in Section 4). Examples of this type of nonlinearity include, for example, which occur in combustion theory (see ) and in the Navier-Stokes system (see ).
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  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 ); 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 domain and the cylindrical layer ; this fact has a counterpart in the structure of the energy functional and hence on problem . In  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 ,;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 of the solution to 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 . 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 (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  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 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.
2. Geometry of the Fractal Layers and
In the paper by we denote the Euclidean distance in and the Euclidean balls by , , . By the Koch snowflake , we will denote the union of three coplanar Koch curves (see ) , , and as shown in Figure 1. We assume that the junction points , , and 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 .
The Hausdorff dimension of the Koch snowflake is given by . 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 , .
The measure has the property that there exist two positive constants and : where and denotes the Euclidean ball in . 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  Section 2.2).
Let denote a bounded open set in ; in our basic model, denotes the parallelepiped and denotes a “cylindrical” layer in of the type , where and is the Koch snowflake. We assume that is located in a median position inside and divides in two subsets and (see Figure 2).
We give a point the Cartesian coordinates , where 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 : , , .
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 and : where and denotes the Euclidean ball in . As is supported on , it is not ambiguous to write in (5) in place of . Thus turns out to be a -set with (see Appendices A and B).
By , we denote the prefractal layer of the type , is the piecewise linear prefractal approximation of at the step . is a surface of polyhedral type. divides in two subsets , .
We give a point the Cartesian coordinates , where are the coordinates of the orthogonal projection of on the plane containing and is the coordinate of the orthogonal projection on the -line containing the interval .
3. 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 , , is a measure-valued map on which is bilinear symmetric and positive ( is a positive measure.) The measure-valued Lagrangian takes on the fractal the role of the Euclidean Lagrangian . 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 ). In  the Lagrangian 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 and we refer to Section 4 in  for details; in particular in Definition 4.5 a Lagrangian measure on and the corresponding energy form as with domain have been introduced. The domain , which is a Hilbert space with norm has been characterized in terms of the domains of the energy forms on (see  Theorem 4.6).
In the following, we will omit the subscript , the Lagrangian measure will be simply denoted by , and we will set ; an analogous notation will be adopted for the energies.
We define the energy forms on the fractal layer by setting where and are positive constants. Here denotes the measure-valued Lagrangian (of the energy form of with domain ) now acting on and as function of for a.e. ; is the -Hausdorff measure acting on each section of for a.e. with ; denotes the derivative in the direction.
The form is defined for , where is the closure in the intrinsic norm of the set where .
In the following, we will also use the form which is obtained from by the polarization identity:
Proposition 2. In the previous notations and assumptions, the form with domain is a regular Dirichlet form in and the space is a Hilbert space under the intrinsic norm (9).
The proof can be carried on as in Proposition 3.1 of . For the definition and properties of regular Dirichlet forms, we refer to . We now define the Laplace operator on . As is a closed, bilinear form on , there exists (see Chapter 6, Theorem 2.1 in ) a unique self-adjoint, nonpositive operator on —with domain dense in —such that Let denote the dual of the space . We now introduce the Laplace operator on the fractal as a variational operator from by for and for all , where is the duality pairing between and . We use the same symbol 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 dimension of . We find that if is the number of eigenvalues associated with smaller than , then . It can be shown that in our case (see [28, 29]). We stress the fact that in the fractal case , while in the Euclidean setting .
The space is nontrivial; see Proposition 3.3 of . We now introduce the energy form defined on the domain . Here and in the following, denotes the 3-dimensional Lesbesgue measure and denotes the corresponding bilinear form defined on .
As in Theorem 3.2 of , the following result can be proved.
Proposition 3. The form defined in (15) is a regular Dirichlet form in and the space is a Hilbert space equipped with the scalar product
Proposition 4. The space is embedded in .
Proposition 5. The space is embedded in , .
As is a closed bilinear form on with domain dense in , there exists (see Chapter 6 Theorem 2.1 in ) a unique self-adjoint nonpositive operator on with domain dense in such that Moreover in Theorem 13.1 of  it is proved that to each closed symmetric form a family of linear operators can be associated with the property and this family is a strongly continuous resolvent with generator , which also generates a strongly continuous semigroup .
For the reader’s convenience, we recall here the main properties of the semigroup ; the reader is referred to Proposition 3.5 in  for the proof.
Proposition 6. Let be the semigroup generated by the operator A associated with the energy form in (19). Then is an analytic contraction positive preserving semigroup in .
Remark 7. It is well known that the symmetric and contraction analytic semigroup uniquely determines analytic semigroups on the space (see Theorem ) which we still denote by and by its infinitesimal generator.
From Theorem 2.11 in , 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 and ; here is the spectral dimension of .
From interpolation theory results, it can be proved (see Section 3.1 in ) that
3.2. The Energy Forms
By we denote the parallelepiped as defined in Section 3 and by we denote the prefractal layer of the type , , is the prefractal approximation of at the step (see Section 2). divides in two subsets , .
We first construct the energy forms on the prefractal layers , . By we denote the natural arc-length coordinate on each edge of and we introduce the coordinates , , and on every affine “face” of . By we denote the one-dimensional measure given by the arc-length and by are denote the surface measure on each face of ; that is, . We define by setting where and are positive constants and , the Sobolev space of functions on the piecewise affine set (see Appendices A and B). By Fubini theorem, we can write this functional in the form We denote the corresponding bilinear form by . In the sequel we denote by the symbol the trace to .
Consider now the space of functions as it is not trivial as it contains .
Consider now the energy form defined on the domain .
By we will denote the corresponding bilinear form defined on .
Theorem 9. The form , defined in (26), with domain is a regular Dirichlet form in and the space is a Hilbert space equipped with the scalar product
For the proof, see Theorem 4.1 in .
We denote by the corresponding energy norm in ; that is, Proceeding as in Section 3.1 we denote by , , and the resolvents, the generators, and the semigroups associated to , for every , respectively.
As in Proposition 6, the following result can be proved.
Proposition 10. Let be the semigroup generated by the operator associated with the energy form in (27). Then is an analytic contraction positive preserving semigroup in .
By proceeding as in Remark 7, one can show that for every the symmetric and contraction analytic semigroup uniquely determines analytic semigroups on the space (see Theorem ) which we still denote by and by its infinitesimal generator.
The following estimate on the decay of the heat semigroup holds (see e.g., ).
Proposition 11. There exists a positive constant such that where does not depend on h. One considers the cases and ; 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  on the convergence of the approximating energy forms to the fractal energy . In this asymptotic behaviour, the factors and 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 . The convergence of functional is here intended in the sense of the -convergence which we define below.
3.3.1. The -Convergence of Forms
We recall, for the sake of completeness, the definition of -convergence of forms introduced by Mosco in .
Definition 12. A sequence of form -converges to a form in if (a) converging weakly to in (b) there exists converging strongly to in such that
Definition 13. The sequence of forms is asymptotically compact in if every sequence with has a subsequence strongly convergent in .
Proposition 14. The sequence of forms (26) is asymptotically compact in .
Remark 15. We point out that, as the sequence of forms (26) is asymptotically compact in , -convergence is equivalent to the -convergence (see Lemma 2.3.2 in ) and thus we can take in (a) strongly converging to in .
Theorem 16. Let and ; then the sequence of forms defined in (26) -converges in the space to the form defined in (15). The sequence of semigroups associated with the form converges to the semigroup associated with the form in the strong operator topology of uniformly on every interval .
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 and (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 .
We consider the abstract Cauchy problems as and for every where and are the generators associated, respectively, to the energy form and the energy forms introduced in (15) and (26), is a fixed positive real number, and and are given functions in . We assume that is a mapping from , locally Lipschitz, that is, Lipschitz on bounded sets in ; we let denote the Lipschitz constant of : where . We also assume that . This assumption is not necessary in all that follows, but it simplifies the calculations (see ). In order to prove the local existence theorem, we make the following assumptions on the growth of when .
We set for brevity ; we note that , for , and .(i)There exists such that .(ii)Consider for every .
We note that (ii) implies (i) for all since is nondecreasing and Thus is bounded as which implies (i) for .
In Theorem 5.1 of , the following local existence theorem has been proved.
Theorem 17. Let condition (i) hold. Let be sufficiently small if and
There is a and a
with satisfying(1), and ;(2)for every ,
with the integral being both an -valued and -valued Bochner integral;(3)if is strongly measurable with and also satisfies (43), then , for every .
Let condition (ii) hold; there exist a and a unique with satisfying(1)(2)for every , satisfies (43) with the integral being both an -valued and -valued Bochner integral;(3)if is strongly measurable with bounded and also satisfies (43), then , for every .
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 semigroup both on and , the following regularity result for the maximal solution holds (see Theorem 5.3 ).
Theorem 18. Under the assumptions of Theorem 17, one has that the solution can be continuously extended to a maximal interval as a solution of (43), until as , and it is a classical solution; that is, and satisfies
We now recall the convergence results of the sequence of the approximating solutions when h goes to infinity (see Theorem 6.2 in ).
Theorem 19. Let and be the mild solutions of problems and ; let and be as in Theorem 16. In the notations and assumptions of Theorem 17, one has the following;(a)let assumption (i) hold; let and belong to with and in L; then (b)if assumption (ii) holds and in , then with .
5. Strong Formulation of the Transmission Problems
5.1. The Fractal Layer
Theorem 20. Let be the solution of problem . Then one has, for every fixed , where is the restriction of to , , is the inward “normal derivative,” to be defined in a suitable sense, = is the jump of the normal derivative, and is the fractal Laplacian. Moreover .
Proof. Let be an arbitrary function in such that ; by multiplying for (36) in and integrating over we have
From (19) and taking into account that , we have
From the arbitrariness of , we have that, for fixed ,
From the density of in and since , we obtain the first assertion in (49). From this equality, we obtain and since the right-hand side belongs to we deduce that ; hence , where
here the Laplacian is intended in the distributional sense. By proceeding as in (3.26) of , we prove that, for every fixed , the normal derivative is in the dual of the space , where and
for every and every . By proceeding as in Section 6.1 of , we can prove that .
From Proposition 4 and proceeding as in Section 6 of , it can be proved that the transmission condition That is, for every , As a consequence of Theorem 20, the solution of problem is the solution of the following transmission problem. For every , (j) (jj) (jjj) (jv) (v)
Remark 21. Actually from Proposition 6, one deduces that equalities (jv) and (v), respectively, hold in and in with .
5.2. The Prefractal Layer
Theorem 22. Let be the solution of problem . Then one has, for every fixed , where is the restriction of to , is the jump of the normal derivatives across , , , is the inward normal vector, and is the piecewise tangential Laplacian associated to the Dirichlet form . Moreover .
Proof. The first equality in (62) easily follows by proceeding as in Theorem 20. From this, it follows that, for every ,
For every fixed , let denote the restriction of the solution to . By usual duality arguments (see Appendix 4 in ), the normal derivatives , belong to the dual space of . By proceeding as in Section 6.2 of , it is possible to prove that .
Then, by the Green formula for Lipschitz domains, one can prove that That is, the transmission condition holds in the dual of (see Proposition 2.2 in  for details). In order to prove that , we proceed as in Section 4.2 of . Let us consider, for each fixed , the weak solutions and in of the following auxiliary problems: The regularity of follows from the regularity of and since From a regularity result of Jerison and Kenig (see Theorems 2 and 3 of ), we deduce that and .
As to the solution of (67), we preliminary observe that the right-hand side in the first equation of (67) belongs to . From Proposition 4.5 in , it follows that where and ; hence for every ; then by trace results (see Proposition A.1), we obtain, for , and . It follows from (67), (68), and (69) that , ; hence the jump belongs to . As is dense in (see e.g., ), we deduce that the transmission condition (64) actually holds in the -sense and in particular . The proof that easily follows from (69), (72), and the fact that , , , and belong to .
From Theorem 22, it follows that the solution of problem is the solution of the following transmission problem. For every , (j) (jj) (jjj) (jv) (v)
6. Convergence Results
Now we are interested in the behavior of the sequence when goes to .
Theorem 23. Let and be the solutions of problems and according to Theorem 19. Let and be as in Theorem 16. For every fixed positive , one has(i) converges to in ;(ii) weakly converges to in ;(iii) weakly converges to in ;(iv) converges to in .
Proof. We prove condition (i), that is,
From (38), we have
From Theorem 19 (a), we have
And hence, for every fixed ,
This concludes the proof of condition (i).
We now prove condition (ii). From the local Lipschitz continuity of and the Hölder continuity of in into , one can prove that is bounded by a constant which does not depend on ; actually the constants depend only on the constants of the semigroups which in turn do not depend on . From this, together with Theorem 18, we have that there exists a constant independent of such that
Thus in particular it holds ; thus, for every fixed , .
From (82), it follows that for each , belongs to and .
From the boundedness of the sequence in , it follows that there exists a subsequence, which we denote with and a function such that weakly converges to in as goes to .
In order to prove (ii), it is enough to prove that .
Since is dense in , for every , we have
Integrating by parts the left-hand side, we get From (47) or (48), we have From the uniqueness of weak limit, we get a.e.. From the convergence of the sequence to in and the weak convergence of the subsequence to in , we deduce that the whole sequence weakly converges to in .
We now prove condition (iii). It is an easy consequence of (i) and (ii). In fact ; taking the weak limit in , we get the thesis.
We now prove condition (iv). From (i), (iii), and the property of the scalar product in , 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 . We now prove that From Theorem 19, it follows in particular that converges to in ; hence and in ; in particular (91) holds. We now prove assertion (iv) as Taking the upper limit as , we have where the last inequality follows from in . Hence the sequence converges to in and therefore converges to in .
Proposition 24. Let and be the solutions of problems and , respectively. Then and .
Proof. We prove the thesis for . From Theorem 18, it follows that and . Since , we obtain ; hence . The thesis follows as . The result for can be proved analogously.
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 , let be the prefractal surface approximating the Koch-type surface and let us denote every affine “face” of by ; divides into two subsets and .
By we denote the Lebesgue space with respect to the Lebesgue measure on subsets of , which will be left to the context whenever that does not create ambiguity. Let be a closed set of ; by we denote the space of continuous functions on ; by we denote the space of continuous functions vanishing on