From Laplacian Transport to Dirichlet-to-Neumann Gibbs) Semigroups
The paper gives a short account of some basic properties of Dirichlet-to-Neumann operators Λγ∂Ω including the corresponding semigroups motivated by the Laplacian transport in anisotropic media (γ ≠ I) and by elliptic systems with dynamical boundary conditions.
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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| Cite this: | From Laplacian Transport to Dirichlet-to-Neumann Gibbs) Semigroups / V.A. Zagrebnov // Журнал математической физики, анализа, геометрии. — 2008. — Т. 4, № 4. — С. 551-568. — Бібліогр.: 23 назв. — англ. |
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| citation_txt | From Laplacian Transport to Dirichlet-to-Neumann Gibbs) Semigroups / V.A. Zagrebnov // Журнал математической физики, анализа, геометрии. — 2008. — Т. 4, № 4. — С. 551-568. — Бібліогр.: 23 назв. — англ. |
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| description | The paper gives a short account of some basic properties of Dirichlet-to-Neumann operators Λγ∂Ω including the corresponding semigroups motivated by the Laplacian transport in anisotropic media (γ ≠ I) and by elliptic systems with dynamical boundary conditions.
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Journal of Mathematical Physics, Analysis, Geometry
2008, vol. 4, No. 4, pp. 551�568
From Laplacian Transport to Dirichlet-to-Neumann
(Gibbs) Semigroups
V.A. Zagrebnov
Centre de Physique Th�eorique - UMR 6207, Universit�e de la M�editerran�ee
Luminy - Case 907, 13288 Marseille, Cedex 09, France
E-mail:Valentin.Zagrebnov@cpt.univ-mrs.fr
Received January 9, 2008
The paper gives a short account of some basic properties of Dirichlet-
to-Neumann operators �
;@
including the corresponding semigroups moti-
vated by the Laplacian transport in anisotropic media (
6= I) and by elliptic
systems with dynamical boundary conditions. To illustrate these notions and
the properties we use the explicitly constructed Lax semigroups. We demon-
strate that for a general smooth bounded convex domain
� R
d the cor-
responding Dirichlet-to-Neumann semigroup
�
U(t) := e
�t�
;@
t�0
in the
Hilbert space L
2(@
) belongs to the trace-norm von Neumann�Schatten
ideal for any t > 0. This means that it is in fact an immediate Gibbs semi-
group. Recently H. Emamirad and I. Laadnani have constructed a Trotter�
Kato�Cherno� product-type approximating family f(V
;@
(t=n))
ng
n�1
strongly converging to the semigroup U(t) for n ! 1. We conclude the
paper by discussion of a conjecture about convergence of the Emamirad�
Laadnani approximantes in the trace-norm topology.
Key words: Laplacian transport, Dirichlet-to-Neumann operators, Lax
semigroups, Dirichlet-to-Neumann semigroups, Gibbs semigroups.
Mathematics Subject Classi�cation 2000: 47A55, 47D03, 81Q10.
1. Laplacian Transport and Dirichlet-to-Neumann Operators
E x a m p l e 1.1. It is well known (see, e.g., [LeUl]) that the problem of
determining a conductivity matrix �eld
(x) = [
i;j(x)]
d
i;j=1, for x in a bounded
open domain
� R
d , is related to "measuring" the elliptic Dirichlet-to-Neumann
An extended version of the author's talk presented on the Lyapunov Memorial Conference,
June 24-30, 2007 (Kharkiv National University, Ukraine), which is based on the common project
with Prof. Hassan Emamirad (Laboratoire de Math�ematiques, Universit�e de Poitiers).
c
V.A. Zagrebnov, 2008
V.A. Zagrebnov
map for associated conductivity equation. Notice that the solution of this problem
has a lot of practical applications in various domains: geophysics, electrochemistry
etc. It is also an important diagnostic tool in medicine, e.g., in the electrical
impedance tomography ; the tissue in the human body is an example of highly
anisotropic conductor [BaBr].
Under the assumption that there is no sources or sinks of current the potential
v(x); x 2
; for a given voltage f(!); ! 2 @
; on the (smooth) boundary @
of
is a solution of the Dirichlet problem:(
div(
rv) = 0 in
;
vj@
= f on @
:
(P1)
Then the corresponding to (P1) Dirichlet-to-Neumann map (operator) �
;@
is
de�ned by
�
;@
: f 7! @vf=@�
:= � �
rvf j@
: (1.1)
Here � is the unit outer-normal vector to the boundary at ! 2 @
and the function
u := uf is the solution of the Dirichlet problem (P1).
The Dirichlet-to-Neumann operator (1.1) is also called the voltage-to-current
map, since the function �
;@
f gives the induced current �ux trough the boundary
@
. The key (inverse) problem is whether one can determine the conductivity
matrix
by knowing electrical boundary measurements, i.e., the corresponding
Dirichlet-to-Neumann operator? Unfortunately, this operator does not determine
the matrix
uniquely, see e.g. [GrUl] and references there.
E x a m p l e 1.2. The problem of electrical current �ux in the form (P1)
is an example of the so-called Laplacian transport. Besides the voltage-to-current
problem the motivation to study this kind of transport comes for instance from
the transfer across biological membranes, see e.g. [Sap], [GrFiSap].
Let some "species" of concentration C(x), x 2 R
d , di�use in the isotropic
bulk (
= I) from a (distant) source localized on the closed boundary @
0 to-
wards a semipermeable compact interface @
on which they disappear at a given
rate W . Then the steady concentration �eld (Laplacian transport with a di�usion
coe�cient D) obeys the set of equations8><>:
�C = 0; x 2
0 n
;
C(!0 2 @
0) = C0; at the source;
(�D) @�C(!) =W (C(!)� 0); on the interface ! 2 @
:
(P2)
Let C = C0(1 � u). Then �u = 0, x 2
. If we put � := D=W , then the
boundary conditions on @
take the form: (I + �@�)u j@
(!) = 1 j@
(!), where
(1 j@
)(!) = �@
(!) is a characteristic function of the set @
, and u(!0) = 0,
!0 2 @
0 on the source boundary.
552 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4
From Laplacian Transport to Dirichlet-to-Neumann (Gibbs) Semigroups
Consider now the following auxiliary Laplace�Dirichlet problem
�u = 0; x 2
0 n
; u j@
(!) = f(! 2 @
) and u j@
0 (!) = 0; (1.2)
with solution uf . Then similarly to (1.1), with the problem (1.2) we can associate
a Dirihlet-to-Neumann operator
�
=I;@
: f 7! @�uf j@
(1.3)
with the domain dom(�I;@
), which belongs to a certain Sobolev space (Sect. 2).
The advantage of this approach is that as soon as the operator (1.3) is de�ned
it can be applied for studying the mixed boundary value problem (P2). This
gives in particular the value of the particle �ux due to Laplacian transport across
the membrane @
. Indeed, one obtains that (I + ��I;@
)u j@
= 1 j@
, and that
the local (di�usive) particle �ux is de�ned as:
� j@
:= D C0(@nu) j@
= D C0(�I;@
(I + ��I;@
)
�11) j@
: (1.4)
Then the corresponding total �ux across the membrane @
� := (�; 1)L2(@
) = D C0(�(I + ��I;@
)
�11; 1)L2(@
) (1.5)
is experimentally measurable macroscopic response of the system expressed via
transport parameters D;C0; � and geometry of @
. Here (�; �)L2(@
) is a scalar
product in the Hilbert space @H := L
2(@
).
The aim of this paper is twofold:
(i) to give a short account of some standard results about Dirichlet-to-Neumann
operators and related Dirichlet-to-Neumann semigroups that solve a certain class
of elliptic systems with dynamical boundary conditions;
(ii) to present some recent results concerning the approximation theory and
the Gibbs character of the Dirichlet-to-Neumann semigroups for compact sets
with smooth boundaries @
.
To this end in the next Sect. 2 we recall some fundamental properties of
the Dirichlet-to-Neumann operators and semigroups, we illustrate them by a few
elementary examples, including the Lax semigroups [Lax].
In Section 3 we present the strong Emamirad�Laadnani approximations of
the Dirichlet-to-Neumann semigroups inspired by the Cherno� theory and by its
generalizations in [NeZag, CaZag2].
We show in Sect. 4 that for compact sets
with smooth boundaries @
the Dirichlet-to-Neumann semigroups are in fact (immediate) Gibbs semigroups
[Zag2].
Some recent results and conjectures about approximations of the Dirichlet-to-
Neumann (Gibbs) semigroups in operator and trace-norm topologies are collected
in the last Sect. 5.
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4 553
V.A. Zagrebnov
2. Dirichlet-to-Neumann Operators and Semigroups
2.1. Dirichlet-to-Neumann Operators
Let
be an open bounded domain in Rd with a smooth boundary @
. Let
be a C1(
) matrix-valued function on
, which we call the Laplacian transport
matrix in domain
.
We suppose that the matrix-valued function
(x) := [
i;j(x)]
d
i;j=1 satis�es the
following hypotheses:
(H1) The real coe�cients are symmetric and
i;j(x) =
j;i(x) 2 C
1(
).
(H2) There exist two constants 0 < c1 � c2 <1 such that for all � 2 R
d we
have
c1k�k
2
�
nX
i;j=1
�i�j
i;j(x) � c2k�k
2
: (2.1)
Then the Dirichlet-to-Neumann operator �
;@
associated with the Laplacian
transport in
is de�ned as follows.
Let f 2 C(@
), and denote by vf the unique solution (see, e.g., [GiTr,
Th. 6.25]) of the Dirichlet problem(
A
;@
v := div(
rv) = 0 in
;
v j@
= f on @
;
(P1)
in the Banach space X := C(
). Here the operator A
;@
is de�ned on its
maximal domain
dom(A
;@
) := fu 2 X : A
;@
u 2 Xg: (2.2)
De�nition 2.1. The Dirichlet-to-Neumann operator is the map
�
;@
: f 7! @vf=@�
= � �
rvf j@
; (2.3)
with the domain
dom(�
;@
) = ff 2 @C(
R) : vf 2 Ker(A
;@
) and j(� �
rvf j@
)j <1g:
(2.4)
Here � denotes the unit outer-normal vector at ! 2 @
, and vf is the solution of
Dirichlet problem (P1).
The solution vf := L@
f of the problem (P1) is called the
-harmonic lifting
of f , where L@
: C(@
) 7! C
2(
) \ C(
) is called the lifting operator with
domain dom(L@
) = C(@
). If T@
: C(
) 7! C(@
) denotes the trace operator
on the smooth boundary @
, i.e., v j@
= T@
v, then [Eng]:
L@
= (T@
jKer(A
;@
)
)�1 and dom(�
;@
) = T@
fKer(A
;@
)g: (2.5)
554 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4
From Laplacian Transport to Dirichlet-to-Neumann (Gibbs) Semigroups
Remark 2.2. Let @X := C(@
). Then (2.5) implies
T@
L@
u = u ; u 2 @X and L@
T@
w = w ; w 2 Ker(A
;@
): (2.6)
One also gets that the lifting operator is bounded: L@
2 L(@X;X), whereas the
Dirichlet-to-Neumann operator (2.3) is obviously not.
Now let H be Hilbert space L2(
) and @H := L
2(@
) denote the boundary
space. In order that the problem (P1) admits a unique solution vf , one has to
assume that f 2 W
1=2
2
(@
), and then vf belongs the Sobolev space W 1
2
(
), see
e.g. [Tay, Ch.7]. So, we can de�ne Dirichlet-to-Neumann operator in the Hilbert
space @H by (2.3) with the domain
dom(�
;@
) := ff 2W
1=2
2
(@
) : �
;@
f 2 @H = L
2(@
)g: (2.7)
Proposition 2.3. The Dirichlet-to-Neumann operator (2.3) with domain (2.7)
in the Hilbert space @H is unbounded, nonnegative, selfadjoint, �rst-order elliptic
pseudodi�erential operator with compact resolvent.
The complete proof can be found, e.g., in [Tay, Ch. 7], [Tay1]. Therefore, we
give here only some comments on these properties of the Dirichlet-to-Neumann
operator (2.3) in @H = L
2(@
).
Remark 2.4. (a) By virtue of de�nition (2.3) for any f 2W
1=2
2
(@
) one gets
(f;�
;@
f)@H =
Z
@
d�(!) vf (!) � �
(!)(rvf )(!) (2.8)
=
Z
dx div(vf(x) (
rvf)(x)) =
Z
dx (rvf(x) �
rvf)(x)) � 0;
since the matrix
veri�es (H2). Thus, operator �
;@
is nonnegative.
(b) In fact to ensure the existence of the trace T@
(� �
r(L@
f)) one has ini-
tially to de�ne the operator �
;@
for f 2W
3=2
2
(@
). Then Dirichlet-to-Neumann
operator is a selfadjoint extension with domain (2.7) and moreover it is a bounded
map �
;@
:W
1=2
2
(@
) 7!W
�1=2
2
(@
).
(c) By (2.8) and since derivatives of the �rst-order are involved in (2.3), one
can conclude that this operator should be elliptic and pseudodi�erential. If
(x) =
I, then �I;@
is, roughly, the operator (��@
)
1=2, where �@
is the Laplace�
Beltrami operator on @
with corresponding induced metric [Tay, Ch.7], [Tay1].
(d) Compactness of the imbedding W
1=2
2
(@
) ,! L
2(@
) implies the compact-
ness of the resolvent of �
;@
.
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4 555
V.A. Zagrebnov
By (a) and (d) the spectrum �(�
;@
) of the Dirichlet-to-Neumann operator
is a set of nonnegative increasing eigenvalues f�kg
1
k=1. The rate of increasing is
given by the Weyl asymptotic formula, see, e.g., [Hor, Tay]:
Proposition 2.5. Let �
;@
(x; �), for (x; �) 2 T
�
@
, be the symbol of the
�rst-order elliptic pseudodi�erential Dirichlet-to-Neumann operator �
;@
. Then
the asymptotic behaviour of the corresponding eigenvalues as k !1 has the form
�k �
�
k
C(@
;�
)
�
1=(d�1)
;
where
C(@
;�
) :=
1
(2�)d�1
Z
�
;@
(x;�)�1
dx d�:
Another important result is due to Hislop and Lutzer [HiLu]. It concerns
a localization (rapid decay) of the
-harmonic lifting of the corresponding eigen-
functions.
Proposition 2.6. Let f�kg
1
k=1 be eigenfunctions of the Dirichlet-to-Neumann
operator: �
;@
= �k�k with k�kkL2(@
) = 1. Let v�k := L@
�k be the
-harmonic
lifting of �k to
corresponding to the problem (P1). Then for any compact C �
and x 2 C one gets the representation
jv�k(x)j = (x; p; C)=�k
p (2.9)
with arbitrary large p > 0. Here (x; p; C) is a decreasing function of the distance
dist(x; @
).
Since by the Weyl asymptotic formula we have �k = O(k1=(d�1)), the decay im-
plied by the estimate (2.9) is algebraic.
Conjecture 2.7. [HiLu]. In fact the order of decay instead of (x; p; C)=�k
p
is exponential: O(exp[� k dist(C; @
)]).
2.2. Example of a Dirichlet-to-Neumann Operator
To illustrate the results mentioned above we consider a simple example which
will be useful below for contraction of the Lax semigroups.
Consider a homogeneous isotropic case:
(x) = I, and let
=
R := fx 2
R
d=3 : kxk < Rg. Then A
;@
R = �@
R and for the harmonic lifting of
f(!) =
X
l;m
f
(R)
l;m Yl;m(�; ') 2W
1=2
2
(@
R)
556 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4
From Laplacian Transport to Dirichlet-to-Neumann (Gibbs) Semigroups
we obtain
vf (r; �; ') =
X
l;m
�
r
R
�l
f
(R)
l;m Yl;m(�; '); (2.10)
since the spherical functions fYl;mg
1
l=0;jmj�l
form a complete orthonormal basis
in the Hilbert space @H = L
2(@
R; d� sin � d').
De�nition (2.3) and (2.10) imply that nonnegative, selfadjoint, �rst-order
elliptic pseudodi�erential Dirichlet-to-Neumann operator
(�I;@
Rf)(! = (R; �; ')) =
1X
l=0
m=lX
m=�l
�
l
R
�
f
(R)
l;m Yl;m(�; ') (2.11)
has discrete spectrum �(�I;@
R) := f�l;m = l=Rg1
l=0;jmj�l
with spherical eigen-
functions
(�I;@
RYl;m)(R; �; ') =
�
l
R
�
Yl;m(�; ') (2.12)
and multiplicity m. The operator (2.11) is obviously unbounded and it has a com-
pact resolvent.
Remark 2.8. Since by virtue of (2.10) the
-harmonic lifting of the eigen-
function Yl;m to the ball
R is
vYl;m(r; �; ') =
�
r
R
�l
Yl;m(�; ');
one can check the localization (Prop. 2.6) and Conjecture about the exponential
decay explicitly. For distances 0 < dist(x; @
R) = R � r � R, one obtains
jvYl;m(r; �; ')j = O(e�l(R�r)=R).
2.3. Dirichlet-to-Neumann Semigroups on @X
To de�ne the Dirichlet-to-Neumann semigroups on the boundary Banach space
@X = C(@
) we can follow the line of reasoning of [Esc] or [Eng]. To this end
consider in X = C(
) the following elliptic system with the dynamical boundary
conditions 8><>:
div(
ru(t; �)) = 0 in (0;1) �
;
@u(t; �)=@t + @u(t; �)=@�
= 0 on (0;1) � @
;
u(0; �) = f on @
:
(P2)
Proposition 2.9. The problem (P2) has a unique solution uf (t; x) for any
f 2 C(@
). Its trace on the boundary @
has the form
uf (t; !) := (T@
uf (t; �))(!) = (U(t)f)(!); (2.13)
where the family of operators fU(t) = e
�t�
;@
gt�0 is a C0-semigroup generated
by the Dirichlet-to-Neumann operator of the problem (P1).
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4 557
V.A. Zagrebnov
The following key result about the properties of the Dirichlet-to-Neumann
semigroups on the boundary Banach space @X = C(@
) is due to Escher�Engel
[Esc, Eng] and Emamirad�Laadnani [EmLa]:
Proposition 2.10. The semigroup fU(t) = e
�t�
;@
gt�0 is analytic, compact,
positive, irreducible and Markov C0-semigroup of contractions on C(@
).
Remark 2.11. The complete proof can be found in the papers quoted above.
So, here we make only some comments and hints concerning Prop. 2.10.
2.4. Dirichlet-to-Neumann Semigroups on @H
The Dirichlet-to-Neumann semigroup fU(t) = e
�t�
;@
gt�0 on @H is de�ned
by selfadjoint and nonnegative Dirichlet-to-Neumann generator �
;@
of Prop. 2.3.
Proposition 2.12. The Dirichlet-to-Neumann semigroup fU(t) = e
�t�
;@
gt
on the Hilbert space @H is a holomorphic quasisectorial contraction with values
in the trace-class C1(@H) for Re (t) > 0.
Remark 2.13. The �rst part of the statement follows from Prop. 2.3. Since
the generator �
;@
is selfadjoint and nonnegative, the semigroup fU(t)gt is holo-
morphic and quasisectorial contraction for Re (t) > 0, see, e.g., [CaZag1, Zag1].
The compactness of the resolvent of �
;@
implies the compactness of fU(t)gt>0,
but to prove the last part of the statement we need a supplementary argument
about asymptotic behaviour of its eigenvalues given by the Weyl asymptotic for-
mula (Prop. 2.5).
This behaviour of eigenvalues implies the second part of Prop. 2.12:
Lemma 2.14. The Dirichlet-to-Neumann semigroup U(t) has values in the
trace-class C1(@H) for any t > 0.
P r o f. Since the Dirichlet-to-Neumann operator �
;@
is selfadjoint, we
have to prove that
kU(t)k1 =
X
k�1
e�t�k <1 (2.14)
for t > 0. Here k � k1 denotes the norm in the trace-class C1(@H). Then the Weyl
asymptotic formula implies that there exists a bounded M and a function r(k)
such that X
k�1
e�t�k �
X
k�1
expf�t[(k=c)
1
d�1 + r(k)]g
� etM
X
k�1
expf�t(k=c)
1
d�1 g:
Here c := C(@
;�
) and the last sum converges for any t > 0, which proves the
equation (2.14).
558 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4
From Laplacian Transport to Dirichlet-to-Neumann (Gibbs) Semigroups
2.5. Example: Lax Semigroups
A beautiful example of explicit representation of the Dirichlet-to-Neumann
semigroup (2.13) is due to Lax [Lax, Ch. 36].
Let
(x) = I, and
=
R (see Sect. 2.2). Following [Lax] we de�ne the
mapping
K(t) : v(x) 7! v(e�t=R x) for any u 2 C(
R); (2.15)
which is a semigroup for the parameter t � 0 in the Banach space X = C(
R):
(K(�)K(t)v)(x) = v(e��=R e�t=R x) = v(e�(�+t)=R x) ; �; t � 0 ; x 2
R : (2.16)
Remark 2.15. It is clear that if v(x) is (
= I)-harmonic in C(
R), then the
function: x 7! v(e�t=R x) is also harmonic. Therefore,
uf (t; x) := vf (e
�t=R
x) = (K(t)L@
Rf)(x) = (L@
Rft)(x); x 2
R; (2.17)
is the harmonic lifting of the function ft(!) := vf (e
�t=R
!) ; ! 2 @
R, where vf
solves the problem (P1) for
= I. Since in the spherical coordinates x = (r; �; ')
one has
@uf (t; x)=@t = �@rvf (e
�t=R
r; �; ')e�t=R (r=R)
and
@uf (t; R; �; ')=@�I = @rvf (e
�t=R
r; �; ')e�t=R;
we get that @uf (t; !)=@t+@uf (t; !)=@�I = 0, i.e., the function (2.17) is a solution
of the problem (P2).
Hence, according to (2.13) and (2.17) the operator family
S(t) := T@
RK(t)L@
R ; t � 0; (2.18)
de�nes the Dirichlet-to-Neumann semigroup corresponding to the problem (P2)
for
(x) = I, and
=
R, which is known as the Lax semigroup. By virtue of
(2.17) and (2.18) the action of this semigroup is known explicitly:
(S(t)f)(!) = vf (e
�t=R
!); ! 2 @
R: (2.19)
Notice that the semigroup relation
S(�)S(t) = T@
RK(�)L@
RT@
RK(t)L@
R = S(� + t); (2.20)
follows from the properties of lifting and trace operators (see Remark 2.2), from
identity (2.16) and de�nition (2.18). One �nds the generator �
=I;@
R of this
semigroup from the limit
0 = lim
t!0
sup
!2@
R
j
1
t
(f � S(t)f)(!)� (�
=I;@
Rf)(!)j (2.21)
= lim
t!0
sup
!2@
R
j
1
t
(vf (R; �; ')� vf (e
�t=R
R; �; '))� (�
=I;@
Rf)(R; �; ')j:
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4 559
V.A. Zagrebnov
Then the operator
(�
=I;@
Rf)(R; �; ') = @rvf (r = R; �; ') (2.22)
for any function f from the domain
dom(�I;@
R) = ff 2 @C(
R) : vf 2 Ker(AI;@
R) and j(@rvf ) j@
R j <1g
(2.23)
is identical to (2.4) for the case
= I and @
= @
R. Therefore, the gene-
rator (2.22) of the Lax semigroup is the Dirichlet-to-Neumann operator in this
particular case of the Banach space @X = C(@
R).
Similarly, we can consider the Lax semigroup (2.18) in the Hilbert space @H =
L
2(@
R; d� sin � d'). Since the generator of this semigroup is a particular case
of the Dirichlet-to-Neumann operator (2.11), by (2.12) and (2.10) we again obtain
the corresponding action in the explicit form
(S(t)f)(!) (2.24)
= (e�t�I;@
R f)(!)) =
1X
l=0
m=lX
m=�l
1X
s=0
(�t)s
s!
�
l
R
�s
f
(R)
l;m Yl;m(�; ')
=
1X
l=0
m=lX
m=�l
(e� t=R)l f
(R)
l;m Yl;m(�; ') = vf (e
�t=R
!); ! 2 @
R;
which coincides with (2.19).
Notice that for t > 0 the Lax semigroups have their values in the trace-class
C1(@H). This explicitly follows from (2.12), i.e., from the fact that the spectrum
of the semigroup generator �(�I;@
R) := f�l;m = l=Rg1
l=0;jmj�l
is discrete and
TrS(t) =
1X
l=0
(2l + 1) e�tl=R <1: (2.25)
The last is proven in the whole generality in Th. 2.14.
3. Product Approximations of Dirichlet-to-Neumann
Semigroups
3.1. Approximating Family
Since in contrast to the Lax semigroup (
= I) the action of the general
Dirichlet-to-Neumann semigroup for
6= I is known only implicitly (2.13), it is
useful to construct converging approximations, which are simpler for calculations
and analysis.
560 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4
From Laplacian Transport to Dirichlet-to-Neumann (Gibbs) Semigroups
One of them is the Emamirad�Laadnani approximation [EmLa], which is mo-
tivated by the explicit action (2.19), (2.24) of the Lax semigroup
(S(t)f)(!) = (T@
RKR(t)L@
Rf)(!) = vf (e
�t=R
!); ! 2 @
R; (3.1)
KR(t) : v(x) 7! v(e�t=R x) for any v 2 C(
R) (orH(
R)):
The suggestion of [EmLa] consists in substituting the family fKR(t)gt�0 by the
�deformed operator family
K
;R(t) : v(x) 7! v(e�(t=R)
(x)
x) for any v 2 C(
R) (orH(
R)): (3.2)
De�nition 3.1. For the ball
R the Emamirad�Laadnani approximating
family fV
;R(t) := V
;@
R(t)gt�0 is de�ned by
(V
;R(t)f)(!) := (T@
RK
;R(t)L@
Rf)(!) = vf (e
�(t=R)
(!)
!); ! 2 @
R: (3.3)
Remark 3.2. (a) Notice that the approximating family (3.3) is not a semigroup
(V
;R(t)V
;R(s)f)(!) = (T@
RK
;R(t)L@
R
ef(s))(!) (3.4)
= v
ef(s)
(e�(t=R)
(!)
!) 6= vf (e
�((t+s)=R)
(!)
!) = (V
;R(t+ s)f)(!):
(b) This family is strongly continuous at t = 0:
lim
t&0
V
;R(t)f = f for any f 2 @X (or @H): (3.5)
(c) By de�nition (3.3) this family has the derivative at t = +0:
(@tV
;R(t)f)(!) jt=0= ��(!) �
(!)(rvf )(!) = �(�
;@
Rf)(!); (3.6)
which for any f 2 dom(�
;@
R) coincides with the (minus) Dirichlet-to-Neumann
operator (2.3).
3.2. Strong Approximation of the Dirichlet-to-Neumann Semigroups
By virtue of Remark 3.2 the Emamirad�Laadnani approximation family veri-
�es the conditions of the Cherno� approximation theorem ([Che, Th. 1.1]):
Proposition 3.3. Let f�(s)gs�0 be a family of the linear contractions on
a Banach space B and let X0 be the generator of a C0-contraction semigroup.
De�ne X(s) := s
�1(I � �(s)), s > 0. Then for s ! +0 the family fX(s)gs>0
converges strongly in the resolvent sense to the operator X0 if and only if the
sequence f�(t=n)ngn�1, t > 0, converges strongly to e�tX0 as n ! 1 uniformly
on any compact t-intervals in R1
+
.
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4 561
V.A. Zagrebnov
Notice that fV
;R(t)gt�0 in the Banach space @X is the family of contractions
because of the maximum principle for the
-harmonic functions vf . Since the
Dirichlet-to-Neumann operator (2.3) is densely de�ned and closed, Remark 3.2
(c) implies that the family X(s) := s
�1(I � V
;R(s)) converges for s ! +0 to
X0 = �
;@
R in the strong resolvent sense.
Similar arguments are valid for the case of the Hilbert space @H. By virtue
of Remark 2.4 the Dirichlet-to-Neumann operator �
;@
is nonnegative and self-
adjoint. This implies again that (3.3) is the family of contractions in @H and that
by Remark 3.2 (c) the family X(s) := s
�1(I � V
;R(s)) converges for s! +0 to
X0 = �
;@
R in the strong resolvent sense.
Resuming the above observations we obtain the strong approximation of the
Dirichlet-to-Neumann semigroup U(t)
Corollary 3.4. [EmLa]
lim
n!1
(V
;R(t=n))
n
f = U(t)f; for every f 2 @X or @H; (3.7)
uniformly on any compact t-intervals in (0;1).
The Emamirad�Laadnani approximation theorem (Cor. 3.4) has the following
important extension to more general geometry than the ball [EmLa].
De�nition 3.5. We say that a bounded smooth domain
in Rd has the pro-
perty of the interior ball if for any ! 2 @
there exists a tangent to @
at ! plane
T!, and such that one can construct a ball tangent to T! at !, which is totally
included in
.
If
has this property, then with any point ! 2 @
, one can associate a unique
point x!, which is the center of the biggest ball B(x!; r!) of radius r! included
in
. For any 0 < r � r!, we can construct the approximating family Vr(t)
related to the ball B(xr;!; r) := fx 2
: jx � xr;!j � rg of radius r, which
is centered on the line perpendicular to T! at the point ! 2 @
, i.e., xr;! =
(r=r!)x! + (1� r=r!)!. Then we de�ne
(V
;r(t)f)(!) := T@
vf
�
xr;! + e�(t=r)
(!)(r �!)
�
: (3.8)
Here �! is the outer-normal vector at !, the function vf = L@
f is the
-harmonic
lifting of the boundary condition f on @
, and T@
is the trace operator
T@
: H1(
) 3 v 7�! v j@
2 H
1=2(@
): (3.9)
Remark 3.6. Notice that:
(a) since �! = (!�xr;!)=r, one gets (V
;r(t = 0)f)(!) := (T@
vf )(!) = f(!);
562 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4
From Laplacian Transport to Dirichlet-to-Neumann (Gibbs) Semigroups
(b) by virtue of (3.8) the strong derivative at t = 0 has the form
(@tV
;r(t = 0)f)(!) = �
(!)�! � (rvf )(!) = �(�
;@
f)(!);
see (3.6).
Proposition 3.7. [EmLa]. Let
has the property of interior ball, and let
inf
!2@
fr > 0 : B(x!; r!) �
g > 0;
sup
!2@
fr > 0 : B(x!; r!) �
g <1:
For any 0 < s � 1 we de�ne V
;sr! , i.e.,
V
;sr!f(!) = vf
�
xs;! + e�(t=(sr!))
(!)(sr! �!)
�
; (3.10)
where xs;! = sx! + (1� s)!. Then for any 0 < s � 1
lim
n!1
(V
;sr!(t=n))
n
f = U(t)f; for every f 2 @X or @H; (3.11)
uniformly on any compact t-intervals in (0;1).
Remark 3.8. By De�nition 3.1 for the ball
R and the constant matrix-valued
function
(x) = I one obviously has V
=I;R(t) = S(t) = U(t). On the other
hand, for a general smooth domain
with geometry verifying the conditions of
Prop. 3.7, one is obliged to consider the family of approximations V
;sr! even for
the homogeneous case
= I.
4. Dirichlet-to-Neumann Gibbs Semigroups
4.1. Gibbs Semigroups
Since by Lemma 2.14 for any Dirichlet-to-Neumann semigroup we obtain
U(t > 0) 2 C1(@H), then one can check that it is in fact a Gibbs semigroup.
To this end we recall the main de�nitions and some results that we need for the
proof (see, e.g., [Zag2]).
Let H be a separable, in�nite-dimensional complex Hilbert space. We denote
by L(H) the algebra of all bounded operators on H and by C1(H) � L(H) the
subspace of all compact operators. The C1(H) is a �-ideal in L(H), that is: if A 2
C1(H), then A� 2 C1(H) and if A 2 C1(H) and B 2 L(H), then AB 2 C1(H)
and BA 2 C1(H). We say that a compact operator A 2 C1(H) belongs to the
von Neumann�Schatten �-ideal Cp(H) for a certain 1 � p <1, if the norm
kAkp :=
0@X
n�1
sn(A)
p
1A1=p
<1; (4.1)
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4 563
V.A. Zagrebnov
where sn(A) :=
p
�n(A�A) are the singular values of A de�ned by the eigenvalues
f�n(�)gn�1 of nonnegative selfadjoint operator A�A. Since the norm kAkp is
a nonincreasing function of p > 0, one gets
kAk1 � kAkp � kAkq > kAk1(= kAk) (4.2)
for 1 � p � q <1. Then for the von Neumann�Schatten ideals this implies the
inclusions
C1(H) � Cp(H) � Cq(H) � C1(H): (4.3)
Let p�1 = q
�1+r�1. Then, by virtue of the H�older inequality applied to (4.1),
one gets kABkp � kAkqkBkr, if A 2 Cq(H) and B 2 Cr(H). Consequently, we
obtain
Lemma 4.1. The operator A belongs to the trace-class C1(H) if and only if
there exist two (Hilbert�Schmidt) operators K1, K2 2 C2(H), such that A =
K1 K2. Similarly, if K 2 Cp(H), then K
p 2 C1(H).
Let K be an integral operator in the Hilbert space L2(D;�). It is a Hilbert�
Schmidt operator if and only if its kernel k(x; y) 2 L
2(D �D;� � �), and then
one gets the estimate kKk2 � kkkL2(D�D;���).
The proof is quite straightforward and can be found in, e.g., [Kat, Sim].
De�nition 4.2. [Zag2]. Let fG(t)gt�0 be a C0-semigroup on H with fG(t)gt>0
� C1(H). It is called the immediate Gibbs semigroup if G(t) 2 C1(H) for any
t > 0, and it is called the eventually Gibbs semigroup if there is t0 > 0 such that
G(t) 2 C1(H) for any t � t0.
Remark 4.3. (a) Notice that by Lem. 4.1 any C0-semigroup such that one has
fG(t)gt>0 � Cp(H) for some p <1 is an immediate Gibbs semigroup.
(b) Since compact C0-semigroups are normcontinuous for any t > 0,
the immediate Gibbs semigroups are k � k1-norm continuous for t > 0.
For more details on the Gibbs semigroups properties we refer to the book
[Zag2].
Corollary 4.4. By virtue of Prop. 2.12, Def. 4.2 and Remark 4.3 the
Dirichlet-to-Neumann semigroup fU(t) = e
�t�
;@
gt on the Hilbert space @H
is a k � k1-holomorphic quasisectorial immediate Gibbs for Re (t) > 0.
4.2. Compact and Tr-norm Approximating Family
Proposition 4.5. [EmLa] For the ball
R the Emamirad�Laadnani appro-
ximating family fV
;R(t)gt�0 consists of compact operators on the Banach space
@X = C(@
R) for any t > 0.
564 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4
From Laplacian Transport to Dirichlet-to-Neumann (Gibbs) Semigroups
The proof follows from Def. 3.1 by Arzela�Ascoli criterium of compactness,
since representation (3.3) and conditions on
imply the uniform bound and
equicontinuity of the sets fV
;R(t)(@X)gt for any t > 0.
For the case of Hilbert space we recall the following useful condition for cha-
racterization of the Tr-class operators [Zag2].
Proposition 4.6. If A 2 L(H) and
P
1
j=1 kAejk < 1 for an orthonormal
basis fejg
1
j=1 of H, then A 2 C1(H).
Theorem 4.7. On the Hilbert space @H = L
2(@
R) the approximating family
fV
;R(t)gt>0 � C1(@H).
P r o f. Since the eigenfunctions f�kg
1
k=1 of the selfadjoint Dirichlet-to-
Neumann operator �
;@
R form an orthonormal basis in L
2(@
R), we apply
Prop. 4.6 for this basis.
Let @
t;
;R := fx!(t) := e
�(t=R)
(!)
!g!2@
R . By representation (3.3) and
by estimate (2.9) one obtains
kV
;R(t)�kk
2 =
Z
@
R
d�(!)jv�k (x!)j
2
� j@
Rj sup
!2@
R
(x!; p; @
t;
;R)
2
=k
2p=(d�1)
: (4.4)
Then, by hypothesis (H2) on the matrix
for the norm of the vector x! in R
d
one gets the estimate
kx!k � ke
�(t=R)
k R � e
�c1(t=R) R:
Hence, for any t > 0 the dist(x!; @
R) � (1 � e
�c1(t=R))R > 0, which for the
estimates in (2.9) and in (4.4) implies that
0 < inf
!2@
R
(x!; p; @
t>0;
;R) � sup
!2@
R
(x!; p; @
t>0;
;R):
Then, for 2p=(d � 1) > 1 the estimate (4.4) ensures the convergence of the series
in the inequality
kV
;R(t)k1 �
1X
k=1
kV
;R(t)�kk;
which �nishes the proof.
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4 565
V.A. Zagrebnov
5. Concluding Remarks: Trace-Norm Approximations
The strong Emamirad�Laadnani approximation theorem (Cor. 3.4) and the re-
sults of Sect. 4.2 proving that Dirichlet-to-Neumann semigroup U(t) and appro-
ximants V
;@
(t=n)
n belong to C1(@H), for all n � 1 and t > 0, motivate the
following conjecture:
Conjecture 5.1. [EmZa]. The Emamirad�Laadnani approximation theorem
is valid in the Tr-norm topology of C1(@H).
Remark 5.2. Notice that the strong approximation of the Dirichlet-to-
Neumann Gibbs semigroup U(t) by the Tr-class family (V
;@
(t=n))
n does not lift
automatically the topology of convergence to, e.g., operator-norm approximation
[Zag2].
Therefore, to prove Conjecture 5.1 one needs additional arguments similar
to those of [CaZag2]. To this end we put the di�erence in question �n(t) :=
(V
;@
(t=n))
n � U(t) in the following form:
�n(t) = f(V
;@
(t=n))
kn � (U(t=n))kng(V
;R(t=n))
mn (5.1)
+ (U(t=n))knf(V
;@
(t=n))
mn � (U(t=n))mng:
Here for any n > 1, we de�ne two variables kn = [n=2] and mn = [(n + 1)=2],
where [x] denotes the integer part of x � 0, i.e., n = kn + mn. Then, for the
estimate of �n(t) in the C1(@H)-topology one gets
k�n(t)k1 � k(V
;@
(t=n))
kn � (U(t=n))knk k(V
;@
(t=n))
mnk1 (5.2)
+ k(U(t=n))knk1 k(V
;@
(t=n))
mn � (U(t=n))mnk:
In spite of Remark 5.2, the explicit representation of approximants
f(V
;@
(t=n))
ng
n�1
allows to prove the corresponding operator-norm estimate.
Theorem 5.3. [EmZa]. Let V
;@
R(t) be de�ned by (3.3). Then one gets the
estimate
k(V
;@
R(t=n))
n
� U(t)k � "(n); lim
n!1
"(n) = 0; (5.3)
uniformly for any t-compact in R1
+
.
To establish (5.3) we use the "telescopic" representation
(V
;@
R(t=n))
n
� U(t) (5.4)
=
n�1X
s=0
(V
;@
R(t=n))
(n�s�1)
fV
;@
R(t=n)� U(t=n)g(U(t=n))s;
and the operator-norm estimate of fV
;@
R(t=n)� U(t=n)g for large n.
566 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4
From Laplacian Transport to Dirichlet-to-Neumann (Gibbs) Semigroups
The next auxiliary result establishes a relation between the family of operators
V
;@
R(t) and the Dirichlet-to-Neumann semigroup U(t).
Lemma 5.4. [EmZa]. There exists a bounded operator W
;@
R(t) on @H such
that
V
;@
R(t) =W
;@
R(t)U(t) (5.5)
for any t � 0.
Now we return to the main inequality (5.2). To estimate the �rst term in
the right-hand side of (5.2) we need Th. 5.3 and the Ginibre�Gruber inequality
[CaZag2]
k(V
;@
(t=n))
mnk1 � C U(mnt=n):
To establish the latter we use representation (5.5) given by Lem. 5.4.
To estimate the second term one needs only the result of Th. 5.3. All together
this gives a proof of Conjecture 5.1 at least for the ball
R.
Acknowledgements. This paper is based on the lecture given by the author
at the Lyapunov Memorial Conference, June 24-30, 2007 (Kharkiv National Uni-
versity, Ukraine). I would like to express my gratitude to the organizers and in
particular to Prof. Leonid A. Pastur for invitation and for support. The confe-
rence talk, as well as the present account, is a part of the common project with
Prof. Hassan Emamirad, whom I would like to thank for fruitful and pleasant
collaboration.
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| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Zagrebnov, V.A. 2016-09-29T20:26:09Z 2016-09-29T20:26:09Z 2008 From Laplacian Transport to Dirichlet-to-Neumann Gibbs) Semigroups / V.A. Zagrebnov // Журнал математической физики, анализа, геометрии. — 2008. — Т. 4, № 4. — С. 551-568. — Бібліогр.: 23 назв. — англ. 1812-9471 https://nasplib.isofts.kiev.ua/handle/123456789/106522 The paper gives a short account of some basic properties of Dirichlet-to-Neumann operators Λγ∂Ω including the corresponding semigroups motivated by the Laplacian transport in anisotropic media (γ ≠ I) and by elliptic systems with dynamical boundary conditions. This paper is based on the lecture given by the author at the Lyapunov Memorial Conference, June 24-30, 2007 (Kharkiv National University, Ukraine). I would like to express my gratitude to the organizers and in particular to Prof. Leonid A. Pastur for invitation and for support. The conference talk, as well as the present account, is a part of the common project with Prof. Hassan Emamirad, whom I would like to thank for fruitful and pleasant collaboration. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Журнал математической физики, анализа, геометрии From Laplacian Transport to Dirichlet-to-Neumann Gibbs) Semigroups Article published earlier |
| spellingShingle | From Laplacian Transport to Dirichlet-to-Neumann Gibbs) Semigroups Zagrebnov, V.A. |
| title | From Laplacian Transport to Dirichlet-to-Neumann Gibbs) Semigroups |
| title_full | From Laplacian Transport to Dirichlet-to-Neumann Gibbs) Semigroups |
| title_fullStr | From Laplacian Transport to Dirichlet-to-Neumann Gibbs) Semigroups |
| title_full_unstemmed | From Laplacian Transport to Dirichlet-to-Neumann Gibbs) Semigroups |
| title_short | From Laplacian Transport to Dirichlet-to-Neumann Gibbs) Semigroups |
| title_sort | from laplacian transport to dirichlet-to-neumann gibbs) semigroups |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/106522 |
| work_keys_str_mv | AT zagrebnovva fromlaplaciantransporttodirichlettoneumanngibbssemigroups |