1. Introduction
In this paper we study a semilinear problem in a fractal domain with semilinear dynamical boundary conditions.
The model problem, we consider can be formally stated as follows:
![](//html.scirp.org/file/11-7402218x5.png)
where
is the (open) snowflake domain and
is the union of three Koch curves (see Section 2).
is a non linear function from a subset of
into
; m is the sum of the 2-dimensional Lebesgue measure and of the Hausdorff measure of
(see Section 2.1).
denotes the Laplace operator defined on
(see (3.4) in Section 3),
is a positive constant,
is a strictly positive continuous function
in
is the normal derivative across
intended in a suitable sense.
More precisely, we assume that
is a non linear mapping from
to
for any fixed
locally Lipschitz i.e. Lipschitz on bounded sets in
with Lipschitz constant
restricted to
satisfying a suitable growth condition (see condition (g)) in Section 4). Examples of this type of non linearity include e.g.
which occurrs in combustion theory (see [1] ) and in the Navier Stokes system (see [2] ).
Problem
presents a non linear dynamical boundary condition (known also as Venttsel’ boundary condition [3] ). Problem
models a fluid diffusion within a semipermeable membrane and heat flow subject to non linear cooling on the boundary (see [4] [5] ). The literature on boundary value problems with dynamical conditions is huge, we refer to [6] for a derivation of such boundary conditions and to [7] and the references listed in. All these papers deal with smooth domains. The case of irregular domains is studied in [8] - [12] .
In the present case we consider the case in which the non linearity appears both in bulk and on the boundary. We study the problem by a semigroup approach. More precisely we consider the corresponding abstract Cauchy problem:
(1.1)
where
is the generator associated to the energy form
introduced in (3.8),
is a fixed positive real number,
is a given function in
. We assume that
is a mapping from
locally Lipschitz i.e. Lipschitz on bounded sets in
; we let
denote the Lipschitz constant of
:
(1.2)
whenever
.
A is the generator of the analytic contraction positivity preserving semigroup
from
into
associated to
. We study problem
via the corresponding integral equation
(1.3)
In order to prove the existence of the solutions to (1.3) the usual way is to use a contraction argument in suitable Banach spaces see e.g. [13] . Usually the functional setting is that of an interpolation space between the domain of the generator
and
or the domain of a fractional power of
, we refer the reader to [13] - [17] . In our fractal case we do not know the domain of
We stress the fact that it is not neither known a characterization of the domain of the fractal Laplacian
To overcome this difficulty we adapt the abstract approach in [18] to prove local existence and uniqueness results for the mild solution. The key tool in [18] is an assumption on the estimate of the semigroup
as a bounded operator from
to
(see (2.1) in [18] ). In the present case we take into account that our problem has a probabilistic interpretation [19] ; this, in turn, allows us to deduce an analogue estimate of
as a bounded map from
to
see (3.15). We then deal with the strong formulation of the B.V.P. satisfied by the mild solution, which is of course of great interest in the applications, actually we prove that the solution of problem
solves in a suitable sense Problem
see Theorems 5.1 and 5.2.
The layout of the paper is the following in Section 2 we recall the preliminaries on the geometry and the functional spaces. In Section 3 we consider the energy forms and the associated semigroups. In Section 4 we consider the abstract Cauchy problem
and we prove local and global existence results. Finally in Section 5 we prove that the solution of the abstract Cauchy problem
solves problem
in a suitable sense.
2. Preliminaries
2.1. Geometry
In the paper we denote by
points in
, by
the Euclidean distance and by
the Euclidean balls. By the Koch snowflake F, we will denote the union of three coplanar Koch curves (see [20] )
,
and
as shown in Figure 1. We assume that the junction points
,
and
are the vertices of a regular triangle with unit side length, i.e.
. From now on we assume that a clockwise orientation is given on
.
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
(2.1)
where
denotes the normalized
-dimensional Hausdorff measure, restricted to
,
.
The measure
has the property that there exist two positive constants
,
such that
(2.2)
where
and where
denotes the Euclidean ball in
. As
is supported on
, it
is not ambiguous to write in (2.2)
in place of
. In the terminology of the following section we say that
is a d-set with
according to [21] .
Remark 2.1. The Koch snowflake can be also regarded as a fractal manifold (see [22] ).
We denote by
the (open) snowflake domain.
2.2. Functional Spaces
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. By
we denote the Hilbert space of square summable functions on
with respect to the invariant measure
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
. Let
be an open set of
, by
, where
we denote the usual (possibly fractional) Sobolev spaces (see [23] );
is the closure of
, (the infinitely differentiable functions with compact support on
), with respect to the
-norm.
We now recall a trace theorem.
For
in
, we put
(2.3)
at every point
where the limit exists. It is known that the limit (2.3) exists at quasi every
with respect to the
-capacity [24] .
Definition 2.2. Let
be a closed non-empty subset. It is a d-set
if there exists a Borel
measure
with
such that for some constants
and ![]()
(2.4)
Such a
is called a d-measure on
.
Proposition 2.3. The set
is a d-set with
. The measure
is a d-measure.
See [22] and [25] .
Throughout the paper
will denote possibly different constants.
We now come to the definition of the Besov spaces.
Actually there are many equivalent definitions of these spaces see for instance [21] and [26] . We recall here the one which best fits our aims and we will restrict ourselves to the case
,
; the general setting being much more involved see [18] . By
we denote the space of functions
![]()
where
![]()
Theorem 2.4. Let
then
is the trace space to F of
in the following sense:
1)
is a continuous linear operator from
to
,
2) there is a continuous linear operator
from
to
such that
is the identity operator in
.
For the proof we refer to Theorem 1 of Chapter VII in [21] , see also [26] .
From now on we denote
by
.
3. Energy Forms and Semigroups Associated
3.1. The Energy Form E
In Definition 4.5 of [22] a Lagrangian measure
on
and the corresponding energy form
as
(3.1)
with domain
have been introduced. The domain
, which is a Hilbert space with norm
(3.2)
has been characterized in terms of the domains of the energy forms on
(see [22] 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.
In the following we shall also use the form
which is obtained from
by the polarization identity:
(3.3)
It can be proved as in Proposition 3.1 of [22] , that:
Proposition 3.1. 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 (3.2).
For the definition and properties of regular Dirichlet forms we refer to [27] . We now define the Laplace operator on
. As
is a regular Dirichlet form on
, with domain
dense in
, there exists (see Chap. 6, Theorem 2.1 in [28] ) a unique self-adjoint, non positive operator
on
―with domain
dense in
―such that
(3.4)
Let
denote the dual of the space
. We now introduce the Laplace operator on the fractal
as a variational operator from
by
(3.5)
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 (3.4) and as a variational operator in (3.5). It will be clear from the context to which case we refer.
In the following we denote by
(3.6)
defined in
where
denotes a strictly positive continuous function in
is also a Dirichlet form in ![]()
Consider now the space of functions ![]()
(3.7)
The space
is non trivial. We now introduce the energy form
(3.8)
defined on the domain
. In the following we denote by
the Lesbegue space with respect to the measure
with
(3.9)
By
, we will denote the corresponding bilinear form
(3.10)
defined on
.
Proposition 3.2. The form
defined in (3.8) is a Dirichlet form in
and the space
is a Hilbert space equipped with the scalar product
(3.11)
We denote by
the norm in
associated with (3.11) , that is
(3.12)
Resolvents and Semigroups Associated to Energy Forms
As
is a closed bilinear form on
, with domain
dense in
, there exists (see chap. 6 Theorem 2.1 in [28] ) a unique self-adjoint non positive operator
on
, with domain
dense in
, such that
(3.13)
Moreover in Theorem 13.1 of [27] 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 A, which also generates a strongly continuous semigroup ![]()
With similar arguments it can be proved that there exists a nonnegative self-adjoint operator
with
domain
such that
we denote by
the
strongly continuous semigroup associated to
on ![]()
Proposition 3.3. Let
and
be the semigroups generated by the operator A and
respectively, associated to the energy form in (3.13) and in (3.6). Then
and
are analytic contraction positive preserving semigroups in
and
respectively.
Proof. The contraction property follows from Lumer Phillips Theorem on dissipative operators (Chapter 1 Theorem 4.3 in [16] ). In order to prove the analyticity it will be enough to prove that there exists a positive ![]()
such that
(see Proposition 3 Section 6 in Chapter XVII in [29] ). Moreover since
the semigroup is Markovian it is positive preserving. □
Remark 3.4. It is well known that the symmetric and contraction analytic semigroup
uniquely determines analytic semigroups on the space
see (Theorem 1.4.1 [30] ) which we still denote by
and by
its infinitesimal generator.
Let
denote the spectral dimension of
[31] [32] . By Theorem B3.7 in [33] one can prove
Proposition 3.5. For any
is a bounded operator and
(3.14)
Proof. The result follows by using the equivalence between (3.14) and Nash inequality. Actually it holds that for any ![]()
![]()
(see [34] ). □
From Theorem 2.11 in [19] the following estimate on the decay of the heat semigroup holds.
Proposition 3.6. There exists a positive constant
such that
![]()
We will consider the case
and
.
We remark that this property is called supercontractivity ( see e.g. [30] ).
From now on we set
for ![]()
We recall that for every
, and
![]()
From interpolation result theory (see e.g. [35] ), it can be proved that for every ![]()
![]()
with
(3.15)
where
and ![]()
In particular we will often use that
is bounded from
with
![]()
with
and ![]()
Taking into account 2.6 and
we obtain
![]()
We study the solvability of the Cauchy problem:
(4.1)
where
is the generator associated to the energy form
introduced in (3.8),
is a fixed positive real number,
is a given function in
. We assume that
is a mapping from
locally Lipschitz i.e. Lipschitz on bounded sets in
; we let
denote the Lipschitz constant of
:
(4.2)
whenever
. We also assume that
. This assumption is not necessary in all that follows but it simplifies the calculations (see [18] ). In order to prove the local existence theorem we make the following assumption on the growth of
when ![]()
![]()
we note that
for
and ![]()
Let
. Following the approach in Theorem 2 in [18] and adapting the proof of Theorem 5.1 in [8] we have:
Theorem 4.1. Let condition (g) hold. Let
be sufficiently small, if
and
(4.3)
There is a
and a unique
![]()
with
and
satisfying for every
:
(4.4)
with the integral being both an
-valued and
-valued Bochner integral.
The claim of the Theorem is proved by a contraction mapping argument on suitable spaces of continuous functions with values in Banach spaces. We adapt the proof of Theorem 5.1 in [8] to the new functional setting and for the reader’s convenience we recall it.
Proof. Let
be the complete metric space defined as follows
(4.5)
equipped with the metric
![]()
Since condition (g) holds we choose
such that
for ![]()
For
, let
. By using arguments similar to those used in the proof of Lemma 2.1 of [36] we can prove that
and of course
. We now prove that
(4.6)
Taking into account (4.3) there exists
such that
for all
.
![]()
from (4.5) we have that
![]()
where
thus choosing
(4.6) is proved. It remains to prove
that, for a suitable choice of
is a contraction.
![]()
Therefore we have
![]()
We consider now
It holds
![]()
![]()
In order to prove that it is a contraction it’s enough to choose
such that
and
. □
Remark 4.2. If
then
Thus condition (g) is satisfied for ![]()
with
.
Since
is an analytic semigroup on both
and
from Corollary 2.1 in [18] , the following regularity result holds (see also Theorem 5.3 in [8] ).
Theorem 4.3. Under the assumptions of Theorem 4.1 we have.
a) The solution
can be continuously extended to a maximal interval
as a solution of (4.4), until
.
b)
![]()
and satisfies
![]()
i.e. it is a classical solution.
Proof. As to the proof of condition a), we follow Theorem 4.2 in [18] . From the proof of Theorem 4.1 it turns out that the minimum existence time for the solution to the integral equation is as long as
(see also Corollary 2.1. in [18] ).
To prove that the mild solution is classical we use the classical regularity results for linear equations (see e.g. Theorem 4.3.4. in [13] ) by proving that
is Hölder continuous on
into
for any fixed
Taking into account the local Lipschitz continuity of
it is enough to show that
is H
continuous on
into
. Let
we set
if we prove that
![]()
then, as
due to the uniqueness of the solution of (4), then
![]()
for every
hence
is a classical solution (see claim b). Let
Since
is an analytic semigroup,
is continuosly differentiable on
, hence Hölder continuous with any exponent
. It is enough to show that
is Hölder continuous.
For
is a bounded operator in
and from Theorem 11.3 and 12.1 in [37] there exists a constant c such that
![]()
![]()
Now let
then
![]()
![]()
Hence,
![]()
If we choose
it follows
As to the function
it holds
![]()
Hence
Therefore if
is Hölder continuous on
with exponent
. □
We now give a sufficient condition on the initial datum in order to obtain a global solution adapting Theorem 3 (b) in [38] see also Theorem 5.4 in [8] .
Theorem 4.4. Let condition (g) hold. Let
a.e. and
is sufficiently small, then there exists a nonnegative
which is a global solution of (4.4).
Proof. Since
, from (3.15) it follows that
is a bounded operator from
into
with
![]()
hence
![]()
by choosing
sufficiently small from Theorem 4.1 there exists a local solution of (4.4),
. Furthermore from Theorem 4.1
and
. From Theorem 4.3 (a) to show that
is a global solution it is enough to show that
is bounded for every
We will prove that
is bounded for every ![]()
and we will use the notations of the proof in Theorem 4.1.
![]()
Let
is a continuous non decreasing function with
which satisfies
![]()
if
and
then
can never equal
If it did we would have
i.e.
which is false. This proves that for
sufficiently small
must remain bounded. □
5. Strong Interpretation and Regularity Results
Theorem 5.1. Let
be the solution of problem
. Then we have for every fixed ![]()
![]()
and for every ![]()
(5.7)
where
, is the inward “normal derivative”, to be defined in a suitable sense. Moreover
![]()
Proof. By proceeding as in Theorem 6.1 of [39] and taking into account that
we obtain for each ![]()
(5.8)
from this we deduce
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 [40] [41] we prove that,
for every fixed
, the normal derivative
is in
the dual of the space
, where
and
(5.9)
for every
and every
and by proceeding as in 6.1 of [39] we prove that
.
Let
be an arbitrary function in
, for every fixed
we multiply Equation (4.1) in
and we integrate over ![]()
(5.10)
the left hand-side of (5.10) can be written as:
![]()
from (3.13) we deduce
(5.11)
(5.12)
taking into account that
from (5.9), we have
![]()
from (5.11) we have
![]()
by proceeding as in Section 6.1 of [39] it can be proved that
![]()
and the boundary condition holds in
that is
(5.13)
As a consequence of Theorem (5.1) the solution of problem
is the solution of the following problem. For every
,
![]()
Theorem 5.2. Let
be the strict solution of problem
Then for every
![]()
Proof. For every
we consider the weak solutions
and
of the following auxiliary problems
(5.14)
(5.15)
The regularity of
follows from the regularity of
and
since
(5.16)
We note that for every
(see Corollary 3.3 in [42] ) thus in particular
Since
is a quasicircle from Theorem 2.7 in [43] it is also a non-tangentially accessible domain (N.T.A.), this implies that it is regular for the Dirichlet problem (5.14) in the sense of Jerison and Kenig (see Definition 2.12 in [43] ); this yields in particular that
As to the regularity of
taking into
account that
from Theorem 1.3 in [44] part B, it follows that
this concludes
the proof.
Acknowledgements
The authors have been supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilit e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).