- Email: [email protected]
Spectral theory for isotropic fractal drums
C. R. Acad. Sci. Paris, t. 326, !Xrie I, p. 1269-1274, Analyse matht!matique/Mafhematical Analysis
Spectral David
theory
EDMUNDS
1998
for isotropic
‘, Hans TRIEBEL
fractal
’
” School of Mathematical Sciences, University of Sussex, Brighton BNl E-mail: d.r.edmunds~susspx.a~.uk I’ Fakult&t fiir Mathematik untl Informatik, Friedrich-Schiller-Universit~t E-mail: [email protected] (Rrcu
Ir 15 avril
Abstract.
1998, accept6 If* 27 avril
9QH, IJK jena,
07740 Jrna,
Germany
1998)
We study the behaviour of eigenvalues in problems which correspondto the vibrations of a drum, the whole massof which is concentratedon a fractal subsetof the drum. 0 AcadCmiedes Sciences/Elsevier,Paris
La thkorie RCSUtllC.
drums
spectrale
pour les tambours fiactals
isotropiques
Nous t2udkm.s le comportement des valeurs propres pour drs prohl&mes correspondent aux vibratiorrs d’un tambour dent toute la maLsse est c.oncentr& WI .wwen.semble fractal riu tamhour. 0 Acadhie des ScienceslElsevier, Paris
Version frangaise
qui sur
abr&e
On dit qu’une fonction monotone 9 : (0, l] -+ (0, w) est admissible si X0(2-.‘) x Q(2-“j), j = 0,l: 2,. . _ Par exemple, 9(t) = Ilog,(ct)l” (t’ E Iw, 0 < I: < 1) est admissible. Soit I’ un compact de 03” (r~ _>1) et soit \I/ une fonction admissible. Si 0 < d < u,, alors on dit que I’ est un ensemble (d: $) s’il existe une mesure de Radon 1” dkfinie sur iw” et des nombres cl, c2 > 0 tels que pour toute boule B(y, T-) c Iw” centrke en y E I’ et de rayon ‘r E (U, I).
Si la fonction Q est admissible et dkroissante, avec ,.%;;+$D(T) = fix.
alors on dit que IT est un
ensemble(TL,!k) s’il existe une mesurede Radon /L dkfinie sur W’” qui a la propriCt6 (I) quand cl = ~1. Le cas !Q = 1: 0 < d < 71, est classique (voiu [7], Chap. I). Maintenant, soit 12un ouvert born6 de W”, X2 E CF. et soit Wi(12) = {P E L2(ft) : 1~j.f E Lz(S2), j = l,...: ~6) l’espace de Sobolev usuel; posons fii(6t) = {f E W:(Q) : tri)cl,f = 0). Soit I’ c 12 un ensemble (d, Q) compact. Si 1 5 ;o _< mY on dksigne par L],(r) l’espace de Banach usuel, Note prkentee 0764-4442/9X/03261
par Gilles PISER. 269 0 AcadCmie
des Sciences/Elsevier,
Paris
1269
D. Edmunds,
H. Triebel
oti la mesure est celle induite par la mesure de Radon ~1. &ant donnC une fonction fr E LP(r) (1 < p < CO), on dkfinit d’une seule man&e tine distribution tempCrCe f par la relation : f(4)
= JI: .f%)(d+
)(--/MdT)>
I$ E s,
oti S est I’espace de Schwartz et c#I/~ = trrd, est la trace de 4. Posons idr : ,fr +--+ f et trr = idrotrr. Alors idr peut se prolonger 21G;(Q).
Soit (-A)-’
l’inverse
B = (-A)-’
du laplacien de Ditichlet,
o trr.
et posons : (2)
THBORLME 1 (Le tambour fractal isotropique). - Soit 0 un ouvert born6 de IR” avec i3R E C”, et soit I? c C? un ensemble (d: 3) compact, uvec n - 2 < d < n (0 < d < 1 si n = 1). Alors B est non nkgutive et compacte dans k:(Q)
: son noyuu est
N(B)={.fEGf~n) :trrf=o}. Ses valeurs propres satisfont :
pi
> 0, chacune rPpe’t& srlon sa multiplicitP
COROLLAIRE 2 (Le tambour rouillk).
et nvec /L~,+~ 2 1~~ (x: E ~1,
I; E N.
/Lk x I;-‘(h-~(~-1))(rL-2)‘;I; De plus, B est associe’e & une forme quadratique
(3)
: si
- Soit 62 satisf~isant
les conditions du the’ot&ne 1 et soit JT c bl
un ensemble (n, 9) compact. Alors B est non-ne’gative et compacte en 6;(O), par (3), et ses valeurs propres pk. > 0 satisfonf :
Ces rksultats
(4)
SOFT
noyau est donnP
gCnCralisent ceux de [7], Chap. 5, oti 9 = 1 et d < 71,.
1. Introduction Let r c UP be compact and let 0 < d < 7~. Then r is called a d-set (see [7]) if there are a Radon measure p on Iw” and numbers cl: c2 > 0 such that for all balls B(r, r) in R” with centre y E r and radius T E (0, l), (1.1) Let -A be the Dirichlet operator
Laplacian on a bounded C” B := (-A)-’
1270
domain f1 in IF!“. The compact non-negative 0 t,rr
(1.2)
Spectral
theory
for isotropic
fractal
drums
in the Sobolev space 6;(Q) was studied in [7], Chapter 5, r c 0 being a d-set with 7%- 2 < n < n, Ol and tr’ being closely related to the trace operator trr of W,(O) on I?. It was shown that the positive eigenvalues pk. of B, ordered by decreasing magnitude and repeated according to multiplicity, satisfy /Lk
x
k-1+(“-*)/“,
that is, qk -I+(TL-2)/d <- ok, 5 ,,k-‘+(“-*)/“!
k E NJ.
(1.3)
where cl, cz > 0 are independent of k. When n = 2 we may think of the XI, := ~~~~~~ as the eigenfrequencies of a vibrating membrane R, fixed at its boundary aR, where the whole mass of the membrane is evenly distributed over the fractal subset I’. Here we find the counterpart of (1.3) for the operator (1.2), where now the compact set l? is a perturbed d-set in the sense that the inequality in (1 .I) is replaced by q&I’(r)
< /~(B(y:,r))
for certain positive, monotone functions Xl!, typically show that if n. - 2 < nl < n, then
5 CUT’% X&(T) = [log,(cr)[“,
ph. x k-‘(kllr(k-1))~n-3~‘d,
(1.4) 0 < c < 1, b E W. We
(1.5)
When 7~ = 2 we again may think of a vibrating drum, the whole mass of which is evenly distributed over I’. Corresponding results are given for d = r~; when IL = 2 this corresponds to a vibrating rusty drum in which the originally evenly distributed mass of the membrane is crumbling. The work rests on estimates for the entropy numbers of compact embeddings between spaces of Besov type on 62 and on l?! which in turn depend on atomic and quarkonial decompositions of elements of these spaces. (For related work see [2], [3], and [4].) 2. Notation
and preliminary
results
Let 4 E S, the usual Schwartz space, with $(x) = 1 if /z/ < 1, 4(x) = 0 if /:I/ > ;3/2; put q& = 4, $1(x) = 4(x/2) - q!(z), &(z) = c,h1(2-“+‘x) for II: E R’“, k E N. DEFINITION 2.1. - A positive monotone function 9 on (0.11 is called admissible if e(2-j) x q(2-“9 (j E N,]). Example. - xl?(t) = (log2(ct)lb,
h E R, 0 < c < 1.
DEFINITION 2.2. -Let s E R, let p: 4 E (0,031, and let 9 be admissible. Then B&“k)(RYt) (see [5], [6] for the case 9 = 1) is the family of all f E S’ such that
(with the usual modification if y = 00) is finite; here A and ” denote the Fourier and inverse Fourier transforms. If fi is a bounded C” domain in R”, Br;‘) (Q) will denote the restriction of B~‘)(iR’“) to R, with the natural quasi-norm. 1271
D. Edmunds,
H. Triebel
DEFINITION 2.3. - Let lI be a compact subset of W’“.
(i) Let 0 < d < U, and let XI! be admissible in the senseof Definition 2.1. Then I? is called a (d; @)-set if there is a Radon measure.11,on W” such that for any ball B(y, 7‘) in W with centre y E r and radius r‘ E (0, l), supp/~ = lY and p(B(~;r)) (ii) Let Q be decreasing and admissible, with l&n+*(“)
x #9(r).
(2.1)
= X. Then I? is called an
(7~!
@)-set if
there is a Radon measure /r on Iw” having property (2.1) when d = II.
3. Spaces on fractals Given a compact (d, *)-set r‘ in R’” and p E [I, CC]?LP(I’) is the usual complex Banach space with respect to the related Radon measure jr. Any fr E Lp(I’) can be interpreted as a (uniquely determined) f E S’ given by: c/JE s.
.f’(y’,) = ,l f’(r)(4lr)(r)ll((lr)~
where (blr is the pointwise trace of (;f,on r. This leads us to identify I,,(r) of B$j”)(IW”) which we now introduce.
(3.1) with certain subspaces
DEFINITION 3.1. - Let I’ be a compact (pi. *)-set r in iw” in the senseof Definition 2.4, let s E Iw
and let p. (I E (0,
:= {f
E By;‘)
: .f($) = 0 if 4 E 5’ and dir- = O}.
THEOREM 3.2. - Let 1 5 p < 3c and let r be a compact (d, *)-set in R” L,,(r)
and E !lF1/~“(2-J)
with cl 2 u. Therl
c Bb~~~.~-‘;;“‘).‘(W,~).
The inclusion in (3.3) can be replaced by equality
(3.2)
(3.3)
(f p > 1 and either (a) d < 11, or (b) d = 11
< 2~:.
J=o
4. Traces Let r be a compact (d. $)-set in W’“. Let tr& be the pointwise trace of (i, E S(W) on !C and supposethat for some space Bgj’@’(W) (in the senseof Definition 2.2), with p, Q < CO,there is a constant c > 0 such that for all 4 E S.
Since S is dense in B&“‘) (!R”), t’rr is extended to B~;“y.g”([wl’)by completion. The statement LP(lY) = trr I+@)(W) P4 must be understood in the sensethat any fI‘ E L,(r) and IlfrlL,,(r) (1 is equivalent to
1272
is the trace on IY of some .(/ E BI”;‘)(IW”)
Spectral
theory
for isotropic
fractal
drums
THEOREM 4.1. -
Let 1 < p < CO. (i) Let !i be a compact (11,q)-ser in R” with d < n. Then
(ii) Let I? be a compact (n, @)-set in R” with CTi,, V’/P(Z-J)
< cx. Then
DEFINJTION 4.2. - Let I? be a compact (cl: *)-set in R” and let 1 < p < IX, 0 < (I 2 0~‘: s > 0 and a E R. Then
equipped with the quasi-norm
(,+yy-yui+“,
where the infimum is taken over all y E B,, (UP”) with trr 9 = J‘. We shall need the classical Sobolev space W: (0) = B:,,(Q) and its subspace W:(e) {f E Wi(12) : t,rijfl f = O}. 5. Entropy
=
numbers
The basic properties of entropy numbers may be found in [l], pp. 7-22. THEOREM 5. I. - Ler IY be a compact (d, *)-set in R” and suppose that pl 9pi E (1, CO), q E (0, CO], a E iw, and s > d($ - &)+. Then the embedding Bkf”)(I’) --+ IlP2(I?) is compact and its entropy numbers satisfy:
r:k(id) x (“i:~(x:-l))-,s’“Il,(n:-l,-“. 6. Fractal
k E N.
drums
Let It be a bounded C” domain in W”, let s E R and p, q E (0, oo], let Q be an admissible function and suppose that r c 52 is a compact (d. @)-set. Then trr% given by:
maps D(12) to D’(0). Of course, cl;(r) = ($jr )(r) = (tr&)(u). Formalizing by idr : fr c--i .f, then tr r = idr o trr. We can extend tjrr by
Let (-A)-’ be the inverse of the Dirichlet Laplacian -A; which spaces (-A)-’ acts. Let B = (-A)-]
o trr.
the interpretation
(3.1)
it will be clear from the context between
(6.1)
1273
D. Edmunds,
H. Triebel
6.1 (Isotropic fractal drum). - Let f2 be a bounded C” domain in W’” (7~1 1) and let fl be a compact (d, *)-set with 71- 2 < d < 71(0 < d < 1 when n. = 1).
THEOREM
r
C
Then B; given by (6. I), is a mm-negative compact operator in &b(n) N(B) = {f E +;(61) : trrf
= 0).
with null-space (6.2)
Its positive eigenvalues pk, repeated uccording to multiplicity and ordered by decreasing magnitude, sati@: 71 -2 j/Q. x k-l(k*(k-l))T, k E N. (6.3) Also, B is generated by a quadratic form in %:($I)
where f,g
:
E G;(Q).
COROLLARY 6.2 (Rusty drum). - Let 62 be as in Theorem 6.1 and let r c 62be a compact (n, !P)-set. Then B, defined by (6. I), is a compuct non-negative operator in &i(O) and its eigenvulues pkr defined us in Theorem 6.1, satisfy. /,,A,x k:-“wJ(k-‘)l-~.
with null-space given by (6.2).
k E N.
References [I ] Edmunds D.E., Triebel H., Function spaces, entropy numbers, differential operators, Cambridge University Press, Cambridge, 1996. [2] Naimark K., Solomyak M., On the eigenvalue behaviour for a class of operators related to self-similar measures on W’, C. R. Acad. Sci. Paris 3 19 Ser. A (1994) 837-842. [3] Naimark K.. Solomyak M., The eigenvalue behaviour for the boundary value problems related to self-similar measures on W”, Math. Res. Lett. 2 (1995) 279-298. [4] Solomyak M., Verbitsky E., On a spectral problem related to self-similar measures, Bull. London Math. Sot. 27 (1995) 242-248. [5] Triebel H., Theory of function spaces, Birkhauser, Basel, 1983. 16) Triebel H.. Theory of function spaces II, Birkhauser, Basel, 1992. 171 Triebel H.. Fractals and spectra, Birkhauser, Basel, 1997.
1274
Spectral David
theory
EDMUNDS
1998
for isotropic
‘, Hans TRIEBEL
fractal
’
” School of Mathematical Sciences, University of Sussex, Brighton BNl E-mail: d.r.edmunds~susspx.a~.uk I’ Fakult&t fiir Mathematik untl Informatik, Friedrich-Schiller-Universit~t E-mail: [email protected] (Rrcu
Ir 15 avril
Abstract.
1998, accept6 If* 27 avril
9QH, IJK jena,
07740 Jrna,
Germany
1998)
We study the behaviour of eigenvalues in problems which correspondto the vibrations of a drum, the whole massof which is concentratedon a fractal subsetof the drum. 0 AcadCmiedes Sciences/Elsevier,Paris
La thkorie RCSUtllC.
drums
spectrale
pour les tambours fiactals
isotropiques
Nous t2udkm.s le comportement des valeurs propres pour drs prohl&mes correspondent aux vibratiorrs d’un tambour dent toute la maLsse est c.oncentr& WI .wwen.semble fractal riu tamhour. 0 Acadhie des ScienceslElsevier, Paris
Version frangaise
qui sur
abr&e
On dit qu’une fonction monotone 9 : (0, l] -+ (0, w) est admissible si X0(2-.‘) x Q(2-“j), j = 0,l: 2,. . _ Par exemple, 9(t) = Ilog,(ct)l” (t’ E Iw, 0 < I: < 1) est admissible. Soit I’ un compact de 03” (r~ _>1) et soit \I/ une fonction admissible. Si 0 < d < u,, alors on dit que I’ est un ensemble (d: $) s’il existe une mesure de Radon 1” dkfinie sur iw” et des nombres cl, c2 > 0 tels que pour toute boule B(y, T-) c Iw” centrke en y E I’ et de rayon ‘r E (U, I).
Si la fonction Q est admissible et dkroissante, avec ,.%;;+$D(T) = fix.
alors on dit que IT est un
ensemble(TL,!k) s’il existe une mesurede Radon /L dkfinie sur W’” qui a la propriCt6 (I) quand cl = ~1. Le cas !Q = 1: 0 < d < 71, est classique (voiu [7], Chap. I). Maintenant, soit 12un ouvert born6 de W”, X2 E CF. et soit Wi(12) = {P E L2(ft) : 1~j.f E Lz(S2), j = l,...: ~6) l’espace de Sobolev usuel; posons fii(6t) = {f E W:(Q) : tri)cl,f = 0). Soit I’ c 12 un ensemble (d, Q) compact. Si 1 5 ;o _< mY on dksigne par L],(r) l’espace de Banach usuel, Note prkentee 0764-4442/9X/03261
par Gilles PISER. 269 0 AcadCmie
des Sciences/Elsevier,
Paris
1269
D. Edmunds,
H. Triebel
oti la mesure est celle induite par la mesure de Radon ~1. &ant donnC une fonction fr E LP(r) (1 < p < CO), on dkfinit d’une seule man&e tine distribution tempCrCe f par la relation : f(4)
= JI: .f%)(d+
)(--/MdT)>
I$ E s,
oti S est I’espace de Schwartz et c#I/~ = trrd, est la trace de 4. Posons idr : ,fr +--+ f et trr = idrotrr. Alors idr peut se prolonger 21G;(Q).
Soit (-A)-’
l’inverse
B = (-A)-’
du laplacien de Ditichlet,
o trr.
et posons : (2)
THBORLME 1 (Le tambour fractal isotropique). - Soit 0 un ouvert born6 de IR” avec i3R E C”, et soit I? c C? un ensemble (d: 3) compact, uvec n - 2 < d < n (0 < d < 1 si n = 1). Alors B est non nkgutive et compacte dans k:(Q)
: son noyuu est
N(B)={.fEGf~n) :trrf=o}. Ses valeurs propres satisfont :
pi
> 0, chacune rPpe’t& srlon sa multiplicitP
COROLLAIRE 2 (Le tambour rouillk).
et nvec /L~,+~ 2 1~~ (x: E ~1,
I; E N.
/Lk x I;-‘(h-~(~-1))(rL-2)‘;I; De plus, B est associe’e & une forme quadratique
(3)
: si
- Soit 62 satisf~isant
les conditions du the’ot&ne 1 et soit JT c bl
un ensemble (n, 9) compact. Alors B est non-ne’gative et compacte en 6;(O), par (3), et ses valeurs propres pk. > 0 satisfonf :
Ces rksultats
(4)
SOFT
noyau est donnP
gCnCralisent ceux de [7], Chap. 5, oti 9 = 1 et d < 71,.
1. Introduction Let r c UP be compact and let 0 < d < 7~. Then r is called a d-set (see [7]) if there are a Radon measure p on Iw” and numbers cl: c2 > 0 such that for all balls B(r, r) in R” with centre y E r and radius T E (0, l), (1.1) Let -A be the Dirichlet operator
Laplacian on a bounded C” B := (-A)-’
1270
domain f1 in IF!“. The compact non-negative 0 t,rr
(1.2)
Spectral
theory
for isotropic
fractal
drums
in the Sobolev space 6;(Q) was studied in [7], Chapter 5, r c 0 being a d-set with 7%- 2 < n < n, Ol and tr’ being closely related to the trace operator trr of W,(O) on I?. It was shown that the positive eigenvalues pk. of B, ordered by decreasing magnitude and repeated according to multiplicity, satisfy /Lk
x
k-1+(“-*)/“,
that is, qk -I+(TL-2)/d <- ok, 5 ,,k-‘+(“-*)/“!
k E NJ.
(1.3)
where cl, cz > 0 are independent of k. When n = 2 we may think of the XI, := ~~~~~~ as the eigenfrequencies of a vibrating membrane R, fixed at its boundary aR, where the whole mass of the membrane is evenly distributed over the fractal subset I’. Here we find the counterpart of (1.3) for the operator (1.2), where now the compact set l? is a perturbed d-set in the sense that the inequality in (1 .I) is replaced by q&I’(r)
< /~(B(y:,r))
for certain positive, monotone functions Xl!, typically show that if n. - 2 < nl < n, then
5 CUT’% X&(T) = [log,(cr)[“,
ph. x k-‘(kllr(k-1))~n-3~‘d,
(1.4) 0 < c < 1, b E W. We
(1.5)
When 7~ = 2 we again may think of a vibrating drum, the whole mass of which is evenly distributed over I’. Corresponding results are given for d = r~; when IL = 2 this corresponds to a vibrating rusty drum in which the originally evenly distributed mass of the membrane is crumbling. The work rests on estimates for the entropy numbers of compact embeddings between spaces of Besov type on 62 and on l?! which in turn depend on atomic and quarkonial decompositions of elements of these spaces. (For related work see [2], [3], and [4].) 2. Notation
and preliminary
results
Let 4 E S, the usual Schwartz space, with $(x) = 1 if /z/ < 1, 4(x) = 0 if /:I/ > ;3/2; put q& = 4, $1(x) = 4(x/2) - q!(z), &(z) = c,h1(2-“+‘x) for II: E R’“, k E N. DEFINITION 2.1. - A positive monotone function 9 on (0.11 is called admissible if e(2-j) x q(2-“9 (j E N,]). Example. - xl?(t) = (log2(ct)lb,
h E R, 0 < c < 1.
DEFINITION 2.2. -Let s E R, let p: 4 E (0,031, and let 9 be admissible. Then B&“k)(RYt) (see [5], [6] for the case 9 = 1) is the family of all f E S’ such that
(with the usual modification if y = 00) is finite; here A and ” denote the Fourier and inverse Fourier transforms. If fi is a bounded C” domain in R”, Br;‘) (Q) will denote the restriction of B~‘)(iR’“) to R, with the natural quasi-norm. 1271
D. Edmunds,
H. Triebel
DEFINITION 2.3. - Let lI be a compact subset of W’“.
(i) Let 0 < d < U, and let XI! be admissible in the senseof Definition 2.1. Then I? is called a (d; @)-set if there is a Radon measure.11,on W” such that for any ball B(y, 7‘) in W with centre y E r and radius r‘ E (0, l), supp/~ = lY and p(B(~;r)) (ii) Let Q be decreasing and admissible, with l&n+*(“)
x #9(r).
(2.1)
= X. Then I? is called an
(7~!
@)-set if
there is a Radon measure /r on Iw” having property (2.1) when d = II.
3. Spaces on fractals Given a compact (d, *)-set r‘ in R’” and p E [I, CC]?LP(I’) is the usual complex Banach space with respect to the related Radon measure jr. Any fr E Lp(I’) can be interpreted as a (uniquely determined) f E S’ given by: c/JE s.
.f’(y’,) = ,l f’(r)(4lr)(r)ll((lr)~
where (blr is the pointwise trace of (;f,on r. This leads us to identify I,,(r) of B$j”)(IW”) which we now introduce.
(3.1) with certain subspaces
DEFINITION 3.1. - Let I’ be a compact (pi. *)-set r in iw” in the senseof Definition 2.4, let s E Iw
and let p. (I E (0,
:= {f
E By;‘)
: .f($) = 0 if 4 E 5’ and dir- = O}.
THEOREM 3.2. - Let 1 5 p < 3c and let r be a compact (d, *)-set in R” L,,(r)
and E !lF1/~“(2-J)
with cl 2 u. Therl
c Bb~~~.~-‘;;“‘).‘(W,~).
The inclusion in (3.3) can be replaced by equality
(3.2)
(3.3)
(f p > 1 and either (a) d < 11, or (b) d = 11
< 2~:.
J=o
4. Traces Let r be a compact (d. $)-set in W’“. Let tr& be the pointwise trace of (i, E S(W) on !C and supposethat for some space Bgj’@’(W) (in the senseof Definition 2.2), with p, Q < CO,there is a constant c > 0 such that for all 4 E S.
Since S is dense in B&“‘) (!R”), t’rr is extended to B~;“y.g”([wl’)by completion. The statement LP(lY) = trr I+@)(W) P4 must be understood in the sensethat any fI‘ E L,(r) and IlfrlL,,(r) (1 is equivalent to
1272
is the trace on IY of some .(/ E BI”;‘)(IW”)
Spectral
theory
for isotropic
fractal
drums
THEOREM 4.1. -
Let 1 < p < CO. (i) Let !i be a compact (11,q)-ser in R” with d < n. Then
(ii) Let I? be a compact (n, @)-set in R” with CTi,, V’/P(Z-J)
< cx. Then
DEFINJTION 4.2. - Let I? be a compact (cl: *)-set in R” and let 1 < p < IX, 0 < (I 2 0~‘: s > 0 and a E R. Then
equipped with the quasi-norm
(,+yy-yui+“,
where the infimum is taken over all y E B,, (UP”) with trr 9 = J‘. We shall need the classical Sobolev space W: (0) = B:,,(Q) and its subspace W:(e) {f E Wi(12) : t,rijfl f = O}. 5. Entropy
=
numbers
The basic properties of entropy numbers may be found in [l], pp. 7-22. THEOREM 5. I. - Ler IY be a compact (d, *)-set in R” and suppose that pl 9pi E (1, CO), q E (0, CO], a E iw, and s > d($ - &)+. Then the embedding Bkf”)(I’) --+ IlP2(I?) is compact and its entropy numbers satisfy:
r:k(id) x (“i:~(x:-l))-,s’“Il,(n:-l,-“. 6. Fractal
k E N.
drums
Let It be a bounded C” domain in W”, let s E R and p, q E (0, oo], let Q be an admissible function and suppose that r c 52 is a compact (d. @)-set. Then trr% given by:
maps D(12) to D’(0). Of course, cl;(r) = ($jr )(r) = (tr&)(u). Formalizing by idr : fr c--i .f, then tr r = idr o trr. We can extend tjrr by
Let (-A)-’ be the inverse of the Dirichlet Laplacian -A; which spaces (-A)-’ acts. Let B = (-A)-]
o trr.
the interpretation
(3.1)
it will be clear from the context between
(6.1)
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D. Edmunds,
H. Triebel
6.1 (Isotropic fractal drum). - Let f2 be a bounded C” domain in W’” (7~1 1) and let fl be a compact (d, *)-set with 71- 2 < d < 71(0 < d < 1 when n. = 1).
THEOREM
r
C
Then B; given by (6. I), is a mm-negative compact operator in &b(n) N(B) = {f E +;(61) : trrf
= 0).
with null-space (6.2)
Its positive eigenvalues pk, repeated uccording to multiplicity and ordered by decreasing magnitude, sati@: 71 -2 j/Q. x k-l(k*(k-l))T, k E N. (6.3) Also, B is generated by a quadratic form in %:($I)
where f,g
:
E G;(Q).
COROLLARY 6.2 (Rusty drum). - Let 62 be as in Theorem 6.1 and let r c 62be a compact (n, !P)-set. Then B, defined by (6. I), is a compuct non-negative operator in &i(O) and its eigenvulues pkr defined us in Theorem 6.1, satisfy. /,,A,x k:-“wJ(k-‘)l-~.
with null-space given by (6.2).
k E N.
References [I ] Edmunds D.E., Triebel H., Function spaces, entropy numbers, differential operators, Cambridge University Press, Cambridge, 1996. [2] Naimark K., Solomyak M., On the eigenvalue behaviour for a class of operators related to self-similar measures on W’, C. R. Acad. Sci. Paris 3 19 Ser. A (1994) 837-842. [3] Naimark K.. Solomyak M., The eigenvalue behaviour for the boundary value problems related to self-similar measures on W”, Math. Res. Lett. 2 (1995) 279-298. [4] Solomyak M., Verbitsky E., On a spectral problem related to self-similar measures, Bull. London Math. Sot. 27 (1995) 242-248. [5] Triebel H., Theory of function spaces, Birkhauser, Basel, 1983. 16) Triebel H.. Theory of function spaces II, Birkhauser, Basel, 1992. 171 Triebel H.. Fractals and spectra, Birkhauser, Basel, 1997.
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