Table Of ContentTeubner-Texte zur Mathematik
Bert-Wolfgang Schulze
Hans Triebel
Symposium; Analysis on
Manifolds with
Singularities"
Breitenbrunn 1990
TEUBNER-TEXTE zur Mathematik· Band 131
Herausgeber/Editors: Beratende Herausgeber/Advisory Editors:
Herbert Kurke, Berlin Ruben Ambartzumian, Jerevan
Joseph Mecke, Jena David E. Edmunds, Brighton
Rudiger Thiele, Leipzig Alois Kufner, Prag
Hans Triebel, Jena Burkhard Monien, Paderborn
Gerd Wechsung, Jena Rolf J. Nessel, Aachen
Claudio Procesi, Rom
Kenji Ueno, Kyoto
Symposium "Analysis on Manifolds
with Singularities",
Breitenbrunn 1990
Edited by Bert-Wolfgang Schulze and Hans Triebel
m
Springer Fachmedien Wiesbaden GmbH 1992
ISBN 978-3-663-11578-6 ISBN 978-3-663-11577-9 (eBook)
DOI 10.1007/978-3-663-11577-9
© Springer Fachmedien Wiesbaden 1992
Ursprünglich erschienen bei B.G. Teubner Verlagsgesellschaft Stuttgart • Leipzig 1992
The present Teubner-Text continues a series of publications on analysis of the Karl-Weierstrass
Institute for Mathematics, Berlin (cf. the volumes 96 (1985/86), 106 (1986/87), 112 (1988), 117
(1990». The volume contains contributions from the International Workshop "Analysis in Domains
and on Manifolds with Singularities" in Breitenbrunn, Germany, April 30 -May 5, 1990.
Der vorliegende Teubner-Text setzt eine Serie von Publikationen zur Analysis des Karl
Weierstra6-Instituts fiir Mathematik, Berlin, fort (vgl. die Bande 96 (1985/86), 106 (1986/87),
112 (1988), 117 (1990». Der Band enthilt Beitrage der Internationalen Konferenz "Analysis in
Domains and on Manifolds with Singularities" in Breitenbrunn, Deutschland, 30. April -5. Mai
1990.
Le Teubner-Text present est Ie prolongement d'une serie des publications sur analyse
mathematique par l'Institut de Mathematiques Karl-Weierstrass, Berlin (cp. les tomes 96
(1985/86), 106 (1986/87), 112 (1988), 117 (1990». Le volume contient des articles du Congres In·
ternational "Analysis in Domains and on Manifolds with Singularities" a Breitenbrunn, Allemagne,
30. avril -5. mai 1990.
lIaHHhIit TeKCT DpO,llOJIlf(8.eT cepHIO DY6n~ HHCTHTYTa MaTeMaTHKH HM. Kapna
BeitepmTpacca r. Bepmma B 06nacTH aBanH3Hca (cp. T. 96 (1985/87rr.), 106
(1986/87rr.), 112 (1988r.), 117 (1990r.». OH CO,llepllGlT ,lIOKn8.,l\bl MelIC.zIYHaPO,II;Hoti
HaY"IHoit KOHcpep~ "Analysis in Domains and on Manifolds with Singularities" B r.
BpeitTeH6pYHH, repMaHHSI, 30 anpenJl DO 5 MaR 1990r.
PREFACE
The present Teubner-Text contains the contributions from the International Workshop "Analysis
in Domains and on Manifolds with Singularities", Breitenbrunn, Germany, 30. April-5. May 1990.
In recent years the analysis on manifolds with singularities became more and more interesting,
not only because of the progress in solving corresponding singular problems in partial differential
equations but also of the new relations to other parts of mathematics such as geometry, topology and
mathematical physics. Other motivations come from concrete models in engineering and applied
sciences which lead to partial differential equations in domains with a piece-wise smooth geometry
(conical points, edges, comers, ... , higher singularities), piece-wise smooth data or boundary and
transmission conditions, degenerate coefficients, and so on. There are natural relations to the
numerical analysis where also the asymptotics of solutions close to the singularities playa role.
As for the smooth cases it is necessary to develop structure principles and unified theories that
cover as much as possible the huge variety of concrete situations, often being treated by individual
papers under very specific assumptions.
Such a framework in the elliptic (and the parabolic) theory is the calculus of pseudo-differential
operators, with the parametrix constructions on symbolic level and the asymptotics of solutions as an
element of tIie regularity. The pseudo-differential calculus is at the same time link to corresponding
extensions of the index theory. Moreover, the micro-local analysis in general is responsible for the
non-elliptic singular problems, similarly to the classical theory of Fourier integral operators in the
case of hyperbolic (or more general classes of) equations.
Most of the basic things is yet to be done. For instance, a general index theory on piece-wise
smooth or non-compact spaces is far from being achieved in a satisfying form. But it is already
obvious by the present knowledge that the structures will be much more complex than in the classical
case.
Also the solvability of elliptic boundary value problems in domains with higher comers, the
functional analysis of spaces with asymptotics in those cases, the problem of branching of exponents
in the asymptotics along edges with complicated model cones, the nature of singular terms for
comers, and many other topics from that area seem to be on a beginning, in spite of the stream of
papers and so much inside and experience in particular cases. In other words the prospects in the
analysis on manifolds with singularities are actually exciting.
The conference in Breitenbrunn was a meeting of outstanding specialists in this field. So this
collection of contributions corresponds to the general intention of this series of publications.
The editors are indebted to the colleagues from the Freie Universitat Berlin,
Dr. B. Schomburg, Ms A. Link and Mr R. Regitz for technical assistance in preparing this volume.
Berlin, Jena October 1991 B.-W. Schulze, H. Triebel
3
CONTENTS
F. ALI MEHMETI
Reflection and refraction of singularities for wave equation. with illt~rlA' ~ ,,,"clitions
given by Fourier integral operators .............. 6
L.A. BAGIROV
On the solutions for a class of elliptic equations degenerate Ilt th~ 1"""101", y 20
M. COSTABEL, M. DAUGE
Edge asymptotics on a skew cylinder . . . . . . . . . . . . . . 28
M. COSTABEL, F. PENZEL, R. SCHNEIDER
A collocation method for a screen problem in 1lI.3 • 43
Yu.V. EGOROV, V.A. KONDRAT'EV
To the study of the negative spectrum of an elliptic operator 51
J. ELSCHNER
On the double layer potential operator over polyhedral dOlIlAill" ... lvl\hility In
weighted Sobolev spaces and spline approximation. 57
A. M. IL'IN
The bisingular problems . . . . . . . . . . . 65
L. JENTSCH
On a bimetal problem for heat conductions. 69
V.A. KOZLOV
Spectral properties of operator pencils, associated with elliptic I"HIIICI",y vl\l1 ... prob-
lems in a cone . . . . . . . . . . . . . . . . . . . . . . . . . . 79
D. LEGUILLON, E. SANCHEZ-PALENCIA
Crack phenomena in heterogeneous media 85
M. LORENZ, B.-W. SCHULZE
Propagation of branching conormal asymptotics for hyperbolic (:/lCl' by pr"hl'·II," 104
M. LORENZ, Y.A. SMAIEL
Reflection of branching asymptotics for hyperbolic problems ll8
R. MAZZEO
Edge operators in geometry .. . . . . . . . . . . . . . . . . . 127
G.A. MENDOZA, F. TREVES
On solvability of systems of vector fields 138
S.A. NAZAROV, B.A. PLAMENEVSKIl
On the Neumann problem in domains with edges ...... . 144
V.S. RABINOVICH
Spectral and scattering theory for acoustic operators in non-holll"j\"""UCI' fluids.
Continuous and discrete models . . . . . . . . . . . . . . . . . 158
JA.A. RolTBERG
The Cauchy problem, boundary value and mixed problems for hype. 1,,,1,, 'y.t,·mB in
the complete scale of Sobolev type spaces . . . . . . . . . . . . . 168
4
JA.A. RoITBERG, Z.G. SHEFTEL
On the density of the solutions of elliptic boundary value problems in the sense of
Petrovsky systems in functional spaces on manifolds . . . . . . . . . . . . . . . . .. 176
A.-M. SANDIG, R. SANDIG
Singularities of non-rotationally symmetric solutions of boundary value problems for
the Lame equations in a 3 dimensional domain with conical points. . . . . . . . .. 181
H. SCHMITZ
A collocation method for potential problems with a mixed Dirichlet-Signorini bound-
ary condition .................................. . . . .. 194
B. SCHMUTZLER
The structure of branching asymptotics for elliptic boundary value problems in do-
mains with edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 201
B.-W. SCHULZE
The Mellin pseudo-differential calculus on manifolds with comers .......... 208
M.A. SHUBIN
A Lefschetz fixed point formula in reduced L2-cohomologies on manifolds with cylinders290
L.R. VOLEVICH
A mixed problem for evolution equations . . . . . . . . . . . . . . . . . . . . . . .. 298
5
REFLECTION AND REFRACTION OF SINGULARITIES
FOR WAVE EQUATIONS WITH INTERFACE CONDITIONS
GIVEN BY FOURIER INTEGRAL OPERATORSI
Felix ALI MEHMETI
Mainz, Fed. Rep. of Germany
o Introduction
Cauchy problems for hyperbolic operators often have the property, that the sin
gularities of the initial data propagate along the bicharacteristic strips of the operator (cf. e.g.
[13]). We consider, in the linear case, the situation where the bicharacteristics hit transversally
a spacelike interface, which is 'active' in the sense that the interface condition is given via cer
tain Fourier integral operators. Taking the identity, we obtain classical transmission conditions.
A suitable functional analytic setting is furnished by the interaction concept [3J, [6], [7J, which
covers very general mutual influences of evolution phenomena on different domains.
We state the laws of reflection and refraction of singularities at the interface in
terms of the canonical relations of the given Fourier integral operators (most proofs are only
sketched). These results are obtained using a modified parametrix construction of J. CJ. Nosmas
[28], the calculus of Fourier integral operators [13J, [14J, [21], arguments involving the Boutet
de Monvel algebra [17], [30], [33] and a nonrecursive formula for the compatibility conditions
for abstract inhomogeneous wave equations [5]. The latter is based on the theory of T. Kato on
time dependent evolution equations [22] and generalizes a regularity result of Lions-Magenes
[24]. The structural similarities on the functional analytic level to interaction in quantum field
theory are underlined by the observation that in both cases new singularities (particles) are
created in the interaction zone.
Our situation is a model for active interfaces; it arises naturally as result of c0-
ordinate transformations of transmission problems; it gives structural insights of the following
kind: classical transmission or boundary opera.tors are an defined via the trace operator, whose
canonical relation governs the laws of reflection and refraction of singulari ties. The latter can
only be changed by introducing further Fourier integral operators with nontrivial canonical
relations. Especially it cannot be changed using standard pseudodifferential operators.
Interaction problems are models for evolutionary or stationary phenomena on
several domains with various kinds of mutual influence (cf. [3], [6], [7]). This concept is
IThis research was supported by the Deutsche Forschungsgemeinschaft under grant Al 289/1-1
6
inspired by methods of B. Gramsch in [15], [16], where he derives a rough asymptotic estimate
for eigenvalues of elliptic transmission or boundary value problems. Important special cases
are transmission problems on ramified spaces (Le. families of domains, where certain parts of
the boundaries are identified, cf. G. Lumer [26], S. Nicaise [27], J.v. Below [9], and many other
contributions of these authors, who considered especially the parabolic case). I have treated
mainly hyperbolic transmission and interaction problems (cf. [1], ... , [5]) from the viewpoint of
evolution equations, obtaining results on local and global existence and regularity. Here arises
also naturally the question of propagation of singularities. This complex has been investigated
for interface problems for wave equations by S. Hansen [18], J.CI. Nosmas [28], V. Petkov [29],
M. Taylor [34].
Consider two half spaces Xl := {x E mn : Xn > O}, X. := {x E mn : Xn < o} and
the interface E := {x E mn : Xn = O}. Let U; : [0,00) x X; ..... m be a distributional solution of
the following Cauchy problem for wave equations with Fourier int°egra l interface conditions:
(W) ~u;(t,x) - a;t.~u;(t,x) + c;u;(t,x) =
"It E [O,oo],x EX;, i = 1,2
(1nit) u;(O,x) = XO,;, f,u;(O,x) = Yo,; "Ix EX;, i = 1,2 (1)
(10) FP(u1(t")IE) = Ff(u.(t,·)IE) "It E [0,00)
(11) F1E(alkul(°t,·) IE) = Ff(a.ku.(t,A) "It E [0,00),
where, for simpl°icity , a; > 0, C; > are constants, xo,; == 0, i = 1,2, YO,l E Hl(Xl) has compact
support, Yo,. == and FP are certain Fourier integral operators associated with local canonical
graphs (cf. assumption 1.2). The (I;) are called interaction conditions.
What will happen to a wa.ve hitting the interface E? Our main results (theorems
5.4,5.7, 5.9) show that the canonical relation of the Fourier integral operator describing the
refraction from Xl into X2 is a composition of canonical relations:
'R:=Bichar(P.)1 otBotc;,EoC;,EoBo(£ll) (2)
RXX2 XI
where £; corresponcls to the pure Cauchy problem for the wave equation with subscript i, B
corresponds to the trace operator on moE, Ci,E can be defined in terms of the canonical
relation of FiE and Bichar(P;) is the bicharacteristic relation of Pi. We have
W F( u.) ~ 'R 0 W F(yo,,) . (3)
A similar inclusion holds for W F( ut) describing the reflection. (2) and (3) mean, that a
singularity in the initial conditions in Xl is propagated along a bicharacteristic strip in the
m
light cone in x X" hits then E, undergoes a modification of its projection to rOE according
Fl'
to the canonical rela.tions of the and is finally sent in a bicharacteristic strip in the light
m m
cone of x X. (further, it is reflected back in x Xl).
In this construction we considered the nondiffractive case, Le. the case that the
bicharacteristics hit E transversally. S. Hansen proved propagation of singularities in the dif
fractive case (where a parametrix construction is no longer possible) for FP = identity. M.
Lorenz [25] investigated the reflection of jumps at boundaries.
7
Propagation of singularities for nonlinear hyperbolic problems without interfaces
has been treated by M. Beals [8], J.M. Bony [10], M. Sable-Tougeron [31] (nonlinear boundary
conditions) and others. A future development might be the extension to interaction problems.
I am grateful to Professor B. Gramsch for valuable support. Further I thank
Professor S. Hansen and Dr. E. Schrohe for many discu88ions.
1 Notations and Assumptions
NOTATION 1.1
(i) Xl := {z E lR:' : Zn > OJ, X2 := {z E lR:' : Zn < O}, E := {z E lR:' : Zn = O} .
(ii) Elements of IR x Xl,2 are denoted by
~:= (Zo,z', x,,),
where Zo E IR is the time variable and Zn E IR is the space variable measuring the distance of
(z', zn) to E. z' = (zt, ... , Zn-l) are the other space variables.
i-:rk,
(iii) Write Di := i = 0, ... , n; D := (DI, ••• , Dn).
(iv) Concerning Fourier integral operators, canonical relations and graphs we use the notations of
[35], vol. II, chapter VIII, for wave front sets cf. [35], vol. I, chapter VI. For pseudo-differential
operators we use also the notations of [35], vol. I.
Further we consider the following operators and forms:
ASSUMPTION 1.2
(i) Consider real numbers do,dl > 0 and realvalued functions al; E C"'(lR:'), I,j = 1, ... ,n,
i = 1,2, such that all derivatives are bounded, with the following properties: strong ellipticity
E al;(z){,{; ~ doEI {; 12 V({lt"',{n) E IRn,z E Xi,i = 1, ... ,n;
IJ=I ;=1
and symmetry
al;(z)=ai'(z) V/,j=l, ... ,n; ZEXi; i=1,2.
ain
Assume == O,j = 1, ... , n - 1; i = 1,2 (this is a technical assumption used only in section 3
which might be released using better variational inequalities). Consider further
Ci E C"'{J1F) with Ci ~ d1 for i = 1,2 .
All derivatives of c; are assumed to be bounded.
Assume that all these functions are constant outside a compact subset of lR:'.
(ii) Consider the differential operator of second order
n
Qi(Z, D) := ~L.J D;aI/'( z)DI + c;(z)
1,j=1
8
