Binormal Measures
EMMANUEL P. SMYRNELIS, PANAYOTIS SMYRNELIS
Department of Mathematics,
University of Athens,
Panepistimiopolis 15784, Athens,
GREECE
Abstract: Our starting point is the measure
, where
is the harmonic measure relative
to ⊂⊂
and are concentric balls of ; , are functions depending on and the radii of
(1,2). Generalizing the above measure, we introduce and study the binormal measures as well as their
relation to biharmonic functions.
Key-Words: - Normal measures, binormal measures, biharmonic functions, mean value properties, applications
to PDE (MSC 2020: 31B30, 31D05, 35B05)
Received: September 17, 2022. Revised: April 7, 2023. Accepted: May 1, 2023. Published: May 17, 2023.
1 Introduction
The characteristic mean value property of harmonic
(respectively parabolic) functions involves the
measures
, where is the Dirac
measure at and
is the harmonic
(respectively parabolic) measure relative to
and, supported by the sphere (respectively
by the level surface of the heat kernel). The
adjoint potential of these measures is equal to zero
on ∁ (the complement of ), or equivalently, their
swept measures satisfy∁
0.
In 1944, G. Choquet and J. Deny generalized the
measure
, and introduced the normal
distribution. Moreover, they proved some
characteristic properties of solutions of the
equations 0, and 0 in. Next, in
1967, de La Pradelle following an idea of [8],
extended the notion of normal measure to the setting
of Brelot's theory, [3]. Finally, in 1971, E.
Smyrnelis, using the extended notion of normal
measure, proved several characteristic properties of
normal measures and harmonic functions in Brelot
spaces, applicable to solutions of 0, where is
a second-order linear elliptic operator in.
On the other hand, biharmonic functions (that is,
solutions of 0) satisfy a mean value property
which involves the measures
,
where , are functions of ⊂⊂
and
of the radii , of the concentric spheres ,
(cf., [17]). The scope of this article is to
generalize this property and study some related
issues, for the solutions of the equation
0, where 1,2 is a second-order linear
elliptic differential operator. The idea is to work in a
biharmonic elliptic space, and use special general
measures, applicable to the above equation, in
particular; note that to this biharmonic elliptic space,
we associate a 1-harmonic and a 2-harmonic space
that in the applications correspond respectively to
the solutions of the equations0, and
0.
To this end, we first introduce in Section 2, the
binormal pair of measures , supported by
the compact set , as the pair such that the swept
measures on ∁ of ≔,0 and ≔0,
vanish. Since , it follows that ∁
∁ ∁ or ,∁ ,0
∁ 0,
∁ (cf.,
[15]).
The pair ,0 is called a pure biharmonic pair if
,0∁ 0,0 or equivalently if the pure adjoint
potential pair vanishes on ∁.
The pair 0, is called 2-normal if0,∁
0,0) or equivalently if the 2-adjoint potential
vanishes on ∁ (see, [14]).
Several examples of the aforementioned pairs of
measures are given in Section 3.
In Section 4, we prove the characteristic mean value
properties of biharmonic pairs in relation to
biharmonic pairs of measures.
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Section 5 is devoted to the study of the properties of
binormal pairs of measures. Furthermore, we show
that the linear combinations of the pairs
∁,
∁ are dense for the vague topology, in the
space of the pure binormal pairs of measures, where
(∁,∁
,0∁. Analogous results hold
for the measures
∁ (respectively
∁) in the space of 1-normal (respectively 2-
normal) measures, where ∁ is the swept
nonnegative measure of in the 1-harmonic space
(respectively ∁ is the swept nonnegative
measure of in the 2-harmonic space). Finally, we
examine the relation between binormal and normal
measures.
Note. In this work, we use the term `measure' for
`signed measure'.
2 Reminders, Definitions, and
Preliminary Results
Let us first point out there are equivalent views of
potential theory. We refer for instance to [1], [11].
In this paper, our setting is a general biharmonic
space, as the space of solutions of the system
, 0, where 1,2 is a
second-order linear elliptic or parabolic differential
operator (cf., [15]). From this space, one can
construct using Green’s pairs, the associated adjoint
space corresponding to the system
∗
,
∗0, which is in duality with the initial space
(cf., [20]). In this context, the potential theory of the
harmonic case can be extended, and appropriate
tools are provided to study boundary value
problems. We also point out in [21], [22], two
different approaches to the study of the biharmonic
boundary value problem.
In what follows, we briefly present the main facts
about biharmonic spaces. These spaces have been
inspired by the classical biharmonic equation
0, and we point out that the polyharmonic
case can be studied with the same approach. For
more details, we refer to [12].
We consider a locally compact, connected space
with a countable basis. We denote by
(respectively) the set of all nonempty open sets
(respectively the set of all nonempty relatively
compact open sets) in .
Let be a map that associates to each ∈ a
linear subspace of which is composed
of compatible pairs , in the sense that if
0 on an open set, then also vanishes there. The
pairs of are called biharmonic on.
On the other hand, a set ∈
with ∂ ⌀ is
called -regular if the following conditions hold:
• The Riquier boundary value problem has
only one solution
,,
, associated
to the pair ,
∈
∂∂.
• The inequalities 0 1,2
imply that
, 0, while the inequality 0
implies that
, 0. Hence, for every ∈
, there exists a unique system
,
,
of Radon nonnegative
measures on ∂, such that
,
, while
,
.
Next, we recall that a pair of functions ,
defined on ∈ , is called hyperharmonic if
• :→∞,∞,
• is lower semi-continuous,
• and the inequalities
, as well as
,
hold for every regular set ⊂⊂, and
every ∈.
Let us also mention that if the function is finite
on a dense subset of, then the hyperharmonic pair
, is called superharmonic on. Finally, a
nonnegative superharmonic pair , will
be called potential pair (on), if ,0,0 is
the only biharmonic pair satisfying0
1,2
.
The space , with the axioms I, II, III, and IV
introduced in [15], is called biharmonic. A
biharmonic space is called elliptic if, for every ∈
and every regular set ∋, we have
supp
supp
supp
∂; it will
be called strong if there exists a strictly positive
potential pair on . In a biharmonic space, we
associate the underlying harmonic spaces ,
and,, which correspond respectively to the
solutions of the equations0, and 0
in the classical case. We use respectively the
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prefixes 1 or2, to refer to the harmonic spaces
defined previously.
We shall say that the hyperharmonic (respectively
superharmonic/potential) pair , is pure, if
given a nonnegative 2-hyperharmonic function
on , is the smallest nonnegative function such
that , is a nonnegative hyperharmonic
(respectively superharmonic/potential) pair on .
The -harmonic (respectively biharmonic) support
of a -hyperharmonic function (respectively
hyperharmonic pair) is defined as the smallest
closed set such that the function (respectively the
pair) is -harmonic (respectively biharmonic) in its
complement (1,2). We call Green’s pair, a pure
potential pair with punctual biharmonic support. We
also recall that if is a numerical function on an
open set , the function is defined as follows:
lim
→
∈ inf.
In [20], we define and study the adjoint biharmonic
spaces corresponding to the adjoint
equation∗0, that is, to the system:
∗
,
∗0.
The asterisk symbol is used in the sequel to refer to
adjoint spaces.
Our setting will be a strong biharmonic elliptic
connected space. We assume the proportionality of
-Green’s potentials and -adjoint Green’s
potentials, and also the existence of a topological
basis of completely determining domains for the
associated -harmonic spaces1,2
. For the
notions and notations not explained in this work, we
refer to [15], [9].
Definition 1. Let , be Radon measures supported
by a compact set ⊂ , and let
,
, with 0, 0, 1,2
.
• The pair , is called binormal for if
,0∁ 0,0 and0,∁ 0,0.
• The pair ,0 is called pure binormal for
if,0∁ 0,0.
• The pair 0, is called 2-normal for
if0,∁ 0,0.
Let us consider the open subset ⊂ , the
points, ∈ , the Green’s pair ,
of
biharmonic support and the adjoint Green’s pair
∗,
∗ of support (cf., [19], [20]). We denote
by
,
, the swept pair on of the former
pair, and by
∗,,
∗, the swept pair on of
the latter pair. We also consider the adjoint pure
potential pair
∗
∗,
∗ with associated
nonnegative measure, where
•
∗
∗,
•
∗
∗,
and
∗,,
∗, the swept pair corresponding to
the open set .
Lemma 2. We assert that
∗,
∗,
.
Proof. If
,
is the adjoint swept pair of
,0 on, then it holds:
∗,
∗
∗
∗
∗
∗
∗
.
On the other hand, since we have
∗,
∗
∗
, using, [20],
Lemma 4, and a remark after the proof of [20],
Proposition 4.2, we obtain
∗,,
which completes the proof.
Theorem 3. Let ,0 be a pair of measures
supported by the compact set . Then, the following
properties are equivalent:
(i) ,0 is pure binormal relative to .
(ii) The adjoint pure potential pair
∗,
∗
vanishes on ∁.
Proof. First, we notice that as the pair
∗,
∗ is
adjoint biharmonic on∁, and therefore compatible,
if
∗ =
∗holds on∁, then
∗ =
∗ also holds
on∁. In other words, if ,0 is pure binormal,
then is 1-normal.
⇒. The equality
∗
∗ on ∁ implies
that the respective reduced functions satisfy
∗∁ =
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∗∁
in, and it follows from Lemma 2 that
∁
=
∁
. In view of
[15], Theorem 7.11, we have
∁
,
∁
,
∁
,
where : ,0,
:
,0 and
∁
∁
,
∁
,
∁
,
∁
,
,
∁
, are the respective
swept pairs on ∁; therefore
∁
=
,
∁
,
∁
,
1,2
. Finally, using [15], Theorem 7.1, (cf.
also, [14]), we deduce that
,
∁
,
∁
, and
,
∁
=
,
∁
or equivalently
,
∁
∗
=
,
∁
∗
in ; it follows that
,
∁
=
,
∁
, hence
∁
=
∁
= 0.
⇒. The previous arguments can be reversed
to prove the converse implication.
Remark 4. The case of the pair 0, with
0,
∁
0,0 was studied in [14], and it was
established that 0,
∁
= 0,0 ⇔
∗
= 0 on∁.
Corollary 5. We suppose that
(that is,
L
L
in the classical case). Let λ,0 be a pure
binormal pair for the compact setK. Then, λ is 1-
and 2-normal, while λ,λ is binormal forK.
Proof. It follows from Theorem 3 that
∗
∗
; as
the pair
∗
,
∗
is adjoint biharmonic on ∁, and
therefore compatible, we have
∗
0 on ∁, and
by assumption,
∗
∗
. We also know
that,
∁
,0
∁
0,
∁
. Consequently,
is 1- and 2-normal, while in view of Definition 1,
, is binormal.
3 Some Examples
The functions such that
0 on an open set
of
satisfy a characteristic mean value property
(see, [17]):
,
where
,
, 1,2
, are concentric balls
with 0
,
⊂,
=
,
=
,
= ‖
‖, ∈
and
, 1,2
, are the
respective harmonic measures.
Let
,
be the Green’s pair in
(cf., [19]); it
is biharmonic on the open set
\. If
⊂
and ∈
, then we have
or equivalently
∗
∗
∗
Example 6. We consider the compact set
;
the pair of measures ,0 with
, where
,
is a pure binormal
pair of measures. We can also take the
decomposition
, where
,
. Moreover, we observe
that the pair , is a binormal pair for K.
Example 7. Let be a measure with compact
support in
. If ∈ ∁
, we obtain:
∗
∗
∗
∗
∗
.
The pair,0, where is pure
biharmonic, while the pair , is a binormal pair.
Note. Obviously, since every compact set is
contained in a ball, we can construct pure binormal
(respectively binormal) pairs from a given measure.
Example 8. Starting from a measure supported by
a compact set ⊂
, G. Choquet and J. Deny (cf.,
[6]) have constructed another measure′ such that
′
on
∪∪
, where the sets
are the connected components of ∁,
is compact,
is the potential generated by , and is the
volume element (and so on for the polyharmonic
case). The potential
´
is defined as follows:
´
,′
,
,
,
,,
where
is the Newtonian kernel, and
,
,
, is the iterated kernel (see,
[12]). If
´
,0, on∁, then
,
,0on∁.
Therefore, the pair , is binormal.
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Example 9. Let
∗,1 be a strictly positive adjoint
biharmonic pair and let
∗ be the associated kernel
of the potential part of
∗. If
∗ is a nonnegative
adjoint 1-hyperharmonic function, the adjoint pair
∗
∗,
∗ is a pure hyperharmonic pair; it will be
an adjoint pure potential pair, if
∗ is an adjoint 1-
potential, continuous with a compact harmonic*
support. Let be a measure supported by a compact
set ⊂ ; we have
∗
∗, as
well as
∗1
∗, where is the
nonnegative measure associated with the adjoint
potential
∗1. Now, let ′ be another measure with
density
∗ relative to; we consider the following
function:
∗
∗
∗
∗
∗
∗
∗
∗
∗.
Therefore, if
∗
∗ 0 on∁, we also have that
∗ 0. Consequently, the pair ,0 is pure
binormal for. On the other hand, if is a 2-
normal measure for, then the pair , will be
binormal for.
4 Some Mean Values Properties of
Biharmonic Pairs
Let us recall some further results on harmonic and
biharmonic spaces (cf., [15], parts X, XI). In a
harmonic space, we consider a potential on ,
which is finite, continuous, and strictly
superharmonic. Let be its associated nonnegative
measure. We define Dynkin’s operators, and ′
relative to, as
limsup
↘
1
′ liminf
↘
2
where ∈ , is an open set with ‾ compact, is
a numerical function on such that the numerator
in (1) and (2) is defined, and
is the harmonic
measure. We can see that
on the
harmonic space, where
,
and ∈ . Moreover, if is the kernel associated
with, then we have , for ∈
. The following inequality 0
(or′0) on an open set ⊂ is also
characteristic of hyperharmonic functions on.
Let, ´ be the operators in (1)-(2) associated to
the space , (1,2). We say that the pair
,
of finite and continuous functions in the
open set ⊂ , is regular if and (or
equivalently ´ and´) are finite and
continuous in.
Next, we define the operators:
limsup
↘
,
′ liminf
↘
.
Since on a relatively compact open set, there exists a
strictly positive biharmonic pair,, we can
assume, without loss of generality, that1. The
Riesz decomposition yields
, where
is a 1-potential and is a 1-harmonic function on
. We have
, and′
′. Moreover, the inequality
(or
′
), at the points where is finite, is a
characteristic property of the hyperharmonic pairs
,.
Proposition 10. , be a binormal pair of
measures supported by a compact set ⊂, where
is an open subset of , and , a biharmonic
pair of functions on . Then, 0, and
0.
Proof. We know that 0 if is a 2-normal
measure relative to a compact set ⊂ (cf., [14],
Proposition 1]). Thus, it remains to prove the other
equality. Let us consider a relatively compact open
set, such that⊂⊂⊂. By [18],
Proposition 1.7, there exist continuous potential
pairs,, and,, which are biharmonic
on, and such that,,,.
We have the decompositions: ,′,
,0, as well as ,′,,0,
where ′,, ′, are pure potential pairs in
, biharmonic on , while and are 1-potentials
in (see, [18], Proposition 2.8 and Proposition 2.2);
moreover, and are 1-harmonic on , since
′
,
′
on , while
′
0,
′
0 on , (cf., [15],
Corollary 11.4). Therefore, we have on:
′
′
, where
is 1-
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harmonic on . Finally, the nonnegative measures
and associated with the pure pairs ′, and
′, (cf., [18], (3.13)), are supported by∁.
As 0, (cf., [14], Proposition 1), we obtain:
′′
0,
where , since
0 holds on
∁ ⊃ ∁.
Next, we shall study the converse of Proposition 10.
Proposition 11. Let be an open subset of and
let , be a pair of regular functions satisfying
0,0 for a family , of
binormal pairs of measures relative to compact sets
⊂
with 0, 0, such that ,
∗
,
∗ ,
,
∗
,
∗ , for all ∈, the open sets forming a
basis of ; then, the pair , is biharmonic on
.
Proof. Let be an open set with ⊂ and
compact. There is a strictly positive biharmonic pair
, on (cf., [15], Theorem 6.9); without loss
of generality, we may assume that1, and we
may replace with . In the associated 1-harmonic
space, the Riesz decomposition implies that
; we consider the kernel
associated with
the potential and the associated operators,
(cf., [15], parts X, XI). The pair
, is
biharmonic since
, and
is a 2-harmonic function (cf., [14], Proposition 2)1.
It follows from Proposition 10 that
0
holds for all satisfying the assumptions of
Proposition 11. At this stage, we consider the
function
on; since the functions
and are continuous on, will also be
continuous on. Therefore, we obtain 0.
In addition, since
∗ 0 on ∁ (see the beginning
of the proof of Theorem 3), is in view of [14],
Proposition 3, a 1-harmonic function, that we
denote by. Therefore,
is the first
1 Analogous notions and results are available in the
adjoint case.
component of a biharmonic pair on, namely, of
the pair
,. Finally, since the pair
, is biharmonic on every open set ⊂ ⊂
, with compact, it will also be biharmonic on .
Corollary 12. Let (1,2) be a second-order
linear elliptic operator with regular coefficients
defined on a domain ⊂
(2). We consider
the biharmonic space of the solutions of the
system
, 0 on . We suppose
that there exists a positive potential pair; therefore,
there exists a positive -potential (1,2) (cf.,
[15], part XI, [9], Chap. VII). Then,
holds for every ∈ ,
where, are concentric balls such that ∈
⊂
⊂ (cf. Section 3). This property is
characteristic of biharmonic2 functions on. We
notice that if
, then we can also write
.
5 Properties of Binormal Pairs of
Measures
Let , be measures,
,
,
with 0, 0, 1,2
, and consider the
pairs :,0, :,0, as well as the pair
:0,. Therefore, we have
∁
,
∁
,
∁
and ∁ 0,∁ (cf. Section 1 and the proof of
Theorem 3).
Theorem 13. The following are equivalent:
(i) The pair ,0 is pure binormal and
the pair 0, is 2-normal.
(ii)
∁ 0, and
∁ 0 1,2
.
(iii)
0 and
0, where ,, , are potential
pairs in with support in ∁.
(iv) The previous potential pairs could be
pure potential pairs.
(v)
0, and 0 hold for
every biharmonic pair of functions
, on an open set ⊃ .
2 The function is called biharmonic on, if it is
the first component of a biharmonic pair on.
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(vi)
∁, and
∁ 0, where
,0∁
∁,
∁ with the part of
supported by the set of points of
where ∁ is 1-thin;
∁, where
0,∁
∁,
∁, with the part of
supported by the set of points where
∁ is 2-thin.
Proof. (i) ⇔ (ii). We have already established the
first part of Theorem 13 in the proof of Theorem 3.
Concerning the second part, we can see that these
implications are well-known in harmonic spaces
(cf., [14]).
(i) ⇒ (v). This is proved in Proposition 10.
(v) ⇒ (i). Suppose there exist points ,∈∁
where
∗
∗,
∗
∗; we
take as , the Green pair ,
and we
have
as well as
, therefore we
get
∗
∗ and
∗
∗,
which contradicts our assumptions (cf. Theorem 3).
(iii) ⇒ (v). By [18], Proposition 1.7, there are two
continuous potential pairs ,, and ,,
which are biharmonic on a relatively compact open
set ′, with ⊂′⊂′⊂, and such that
on ′, (i=1,2).
(v) ⇒ (iii). We choose an open set ⊃ such that
the supports of the potential pairs,, and
, are not contained in ; hence, these pairs
are biharmonic on .
(iii) ⇒ (iv). This is straightforward because (iv) is a
particular case of (iii).
(iv) ⇒ (i). Suppose there exist two points ,∈
∁ such that
∗
∗, and
∗
∗. We take as pure potential pairs supported
on∁, the Green’s pairs ,
, and ,
,
where 0, 1. Therefore, we obtain
,
and1
∗1
∗; clearly, this
contradicts our assumptions (see also Theorem 3).
(ii)⇒(vi). ,0,0,0, with the part of
supported by the set of points where ∁ is not 1-
thin. Setting : ,0, we have,0∁
∁,
∁
∁,
∁
∁,
∁. On the
other hand, we know that
∁ . As
∁ 0, we
deduce that
∁
∁ 0; consequently, it
follows that
∁. Furthermore,
since
∁ 0, we obtain
∁
∁ 0. Finally,
in view of [16], Remark 2.12, we conclude that
∁ 0 (see also, [15], Theorem 7.13).
(vi)⇒(i). We know that
∁
∁
∁, where , is a potential pair; since
∁ 0, it follows that
∁
∁.
Now, if , is the Green’s pair,
, then
we have
∁
∁. That
is,
∗∁
∗
∁, in view of
Lemma 1. As for ∈ , it holds that
∗∁
∗
on∁, so we deduce that
∗
∗
∁; therefore,
∗0 on∁.
Note. We point out that the implication (vi)⇒(ii)
can be established, by reversing the arguments in
the proof of (ii)⇒(vi). We can see in the proof of
[4], Proposition 3, that
∁ holds for every 1-
normal measure.
Theorem 14. Let be a compact subset of. The
following are equivalent:
(i) There exists a pure binormal pair of
measures ,0 for the compact set,
with 0.
(ii) ∁ is1-thin for at least one point of .
Proof. (i)⇒(ii). In view of Theorem 13, we
have
∁ 0, (1,2), and by assumption 0.
If ∁ is not 1-thin at any point of, then we will
obtain
∁ (cf., [14], Proposition 3);
since
∁
,
∁
,
∁ 0, it follows that 0.
This is a contradiction.
(ii)⇒(i). Given a pure binormal pair,0, suppose
that 0. By assumption and in view of Theorem
13, we will obtain
∁
∁ 0 and
0 (since ∁ is 1-thin for at least one point of). As
the measure is supported by the set of unstable
points of, and
∁ is supported by the set of
points where ∁ is not 1-thin, we deduce that
∁ (see, [1], Proposition 4.6, [4], Lemma VIII, 2);
therefore, 0, which is a contradiction.
We denote by the set of measures on. We
endow it with the vague topology, that is, the
topology of the simple convergence on the space of
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continuous functions with compact support.
Similarly, we consider the set with the
respective vague topology. We also denote by
the set of points of, where ∁ is -thin, and by
(resp. , the set of pure binormal pairs of
measures (resp. the set of -normal measures,
1,2) for . Finally, we recall that,0∁
∁,
∁, where
∁ is the swept measure of
on ∁ in the 1-harmonic space, and0,∁
0,
∁, where
∁ is the swept measure of on
∁ in the 2-harmonic space.
Theorem 15.
(i) The pairs
∁,
∁, where ∈
, form a total subset of .
(ii) The measures
∁, where ∈ ,
form a total subset of
.
(iii) The measures
∁, where ∈ ,
form a total subset of.
Proof. (i) First, it is well known that the swept pair
∁,
∁ of ,0 on∁, is expressed by
∁=
∁,
∁ =
∁,
where is any continuous function with compact
support. Next, we recall that by definition of the
integral, there exist points of such that
∣
∁∑
∁∣ ′ (3)
with∑
. Note that by considering a
suitable partition of, we can choose the (same)
coefficients, such that relations (3) and (4) are
satisfied (cf. [5, p. 109-109], [2] and [7, p. 126-127].
Moreover, according to [2], Theorem 1, chap. III,
§2, No. 4, there exists a linear combination ∑
such that
∣∑
∣ ″and ∑
. (4)
Consequently, by combining (3) and (4), we can
write
∑
∁
∁
∑
∁,
and ∑
∁ with ∑
.
Since, by Theorem 13, ,0,0
∁,
∁,
the result follows. Assertions (ii) and (iii) can be
proved in the same way.
Finally, we shall examine how normal and binormal
measures are connected.
Proposition 16.
(i) If ,0 is a pure binormal pair for the
compact set, then the measure is 1-
normal for .
(ii) Conversely, suppose that is a 1-normal
measure for. Then the pair ,0 is not
necessarily a pure binormal pair, even if
the -harmonic spaces coincide (1,2).
Proof. (i) Since the pair
∗,
∗ is biharmonic
adjoint on∁, and therefore compatible, then the
equality
∗
∗ on ∁ implies that
∗
∗
holds there. Consequently, is 1-normal for.
(ii) If
∗
∗ on∁, then we assert that
∗
∗
∗ on∁, where
∗ is an adjoint 2-harmonic
function on ∁. Indeed, since the pure potential
pairs satisfy the relations
∗
∗
∗ and
∗
∗
∗ on∁, we obtain
∗
∗
∗0 on ∁,
that is,
∗
∗
∗, where
∗ is an adjoint 2-
harmonic function on the complement of .
Let us now take for the elliptic operator, the
Laplacian. We have the following inclusion:
:,0ispurebinormal
⊂ :isa1 normalmeasure.
For instance, the measure
, where is
a ball, is normal for , but the pair ,0 is not
a pure binormal pair. On the other hand, the
measure
is 1-normal, and
the pair ,0 is also pure binormal (, are
concentric balls, and⊂
⊂
).
Nevertheless, in general, we have:
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Proposition 17. Let be a 1-normal measure for.
Then, ,0 is a pure binormal pair if and only
if
∁ 0.
Proof. This follows immediately from Theorem 13.
6 Conclusion and Future Work
In the present paper, we consider a biharmonic
elliptic space, corresponding in to the solutions
of the system
, 0, where the
second-order linear elliptic differential operators
1,2 (cf., [15]), have adjoint operators
∗
1,2 satisfying
∗
,
∗0 (cf., [20]).
By introducing binormal pairs of measures, we have
extended the normal measures from the harmonic
case (cf., [6], [10], [14]) to the biharmonic context.
More specifically, by using pure adjoint potential
pairs, we have studied the binormal pairs of
measures satisfying the equivalent properties of
Theorem 3. We have also established some
characteristic properties of biharmonic pairs and
binormal pairs of measures. In addition, we have
pointed out in Theorems 13 and 14, the connection
between binormal pairs of measures and the fine
topologies of the associated harmonic spaces
(corresponding in to the solutions of equations
0 and 0 respectively). On the other
hand, Theorem 15 provides an approximation of
pure binormal pairs of measures by normal
measures.
Finally, Example 6 generalizes a characteristic
property of the classical biharmonic case for the
equation 0 (cf., [17]). It is an important
result that may be useful to study some boundary
value problems, for the above systems. For instance,
it would be interesting to examine the following
biharmonic problem in . Can we determine a
biharmonic function in the interior of a smooth
domain if its values and the values of its normal
derivative are known on the boundary? Here, a
biharmonic function is the first component of a
biharmonic pair. Another interesting open problem
would be the extension of the results in [13], in the
context of the heat equation, to more general
parabolic operators.
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