Singularities and Constructive Methods for Their Treatment Proceedin
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IN E L L I P T I C
M. U.E.R.
PROBLEMS
DURAND
de M a t h ~ m a t i q u e s
Universit#
de P r o v e n c e
13331Marseille
Cedex 3
France
We a r e
concerned
space with The f i r s t into to
a crack. one,
obtain
tional
problem.
book,
the
the
are At
to
blems.
Uniform
an i n t e g r a l
are
ones.
able
In
to
framework
Thus
method,
the
of
solved Its
form
methods.
elasticity
is
as e x p l a i n e d
singularities
he o b t a i n s
order
to
all
problem
owing
solution
related
to
to
the
allows the
varia-
pseudo-differential
of
in
the
the
results
the
crack
defined
H s +i (F)
no
usual
the
G.I.
solution about
Eskin's along
the
. Here
the
regularity
= { u ~ Hs(F )
we d e f i n e
6
Hs ( F )
, the
for
a regular
s is
if norm
distribution
integral
curpro-
Then we
pseudo-diffe-
as t h e
some d e f i n i t i o n s
order
and we boundary
integer;
an i n t e r p o l a t e , if
equal
:
s ,
positive
equal is
Hs(r)
limit. of
equations,
by m e t h o d s
of
when
s > O. For e v e r y to to
s
H s (F)
is
posi-
and
boundary. u
in
s < O. IIull
the
a sequence
domain with
, ~ ~ Hs(Rn ) }
as t h e
= (H_s(F))'
we r e c a l l
Hs(F )
go t o
closed
approximated
and o t h e r s .
space of
as u s u a l l y F is
the
computations
clear,
Sobolev
in
of
by r e g u l a r
solve
us t o
limit
occur
Wendland
an i n t e g e r ,
allow
as t h e
numerical W.L.
of
equationsthat
estimates
be q u i t e
is but
integral
operator
of
Hs(F )
In
the
opinion,
infinite
method clear.
perform
methods
the
Hs(?)
our
These o p e r a t o r s
is
~s(F)
system
the
different
the
and S - w a v e s . bilinear
a-priori
Hs(Rn ) tive,
P-waves the
define
obtain
element
three
transforms
we use an a p p r o x i m a t i o n
order
rential
we s u g g e s t
in
wanted.
last, in
in
problem
expansionsof
computes
crack.
it,
The d i f f e r e n t i a l
Wiener-Hopf
C. G o u d j o
edge o f
ves,
In
elasticity
Bamberger,
waves
makes t h e
By u s i n g
that
of
explicit
operators
A.
one.
separation
linear
To s o l v e
due t o
a variational
usual
by t h e
II~lls.
F
u
in ~ i r ) ,
and z e r o
elsewhere.
105 Remark.- I f space
s > 0
is not equal to
k+1/2
Hs(C) i s t h e i r
S
space
Hoo(F).
( w i t h a weaker norm) the L i o n s ' s p a c e
has two s i d e s ,
and
I-
integer,
Hs(F ) i s the
HS(F) d e f i n e d by Lions and Magenes. When s = k=1/2,
integer,
F
, k
~-
on
positive
F , we p u t
For
H-I/2(F).
and n e g a t i v e , [~ =
k positive
s=-1/2, H I/2(F ) contains
if
@ has some t r a c e s
4 + - ~- = T r + ( ~ )
4รท
Tr-(~).
A.BAMBERGER'S METHOD.
To s i m p l i f y t h i s paper, we suppose t h a t the crack is a r e c t i l i n e a r curve ~ in the plane ~2 : F = { ( X l , X 2 ) ~ R2 ; 0 < x I < 1 , x 2 = 0 } We seek f o r the s o l u t i o n of the f o l l o w i n g problem :
nd
u i ~ H~ ( R 2 \ F ) ,
Ij~
(1)g
i=1,2,
such t h a t
~j a i j ( u ) + pw2ui = 0 , u
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