EDGE ASYMPTOTICS ON A SKEW CYLINDER : COMPLEX

EDGE ASYMPTOTICS ON A SKEW CYLINDER :
COMPLEX VARIABLE FORM
Martin Costabel and Monique Dauge
Paris and Nantes
1.
POSITION OF THE PROBLEM
1.a Introduction. It is well known that singularities of the domain give rise to a
loss of regularity for the solutions of any elliptic boundary value problem. The situation
is rather well understood when the singularities are isolated points of the boundary
and are of conical type (see [5], [9]).
When a conical singularity is tensorized with an affine space, one gets an edge.
Regularity results are rather complete in that case [8], [13]. If the operator is translation
invariant along the edge, the asymptotics can be derived in a direct way from the
asymptotics on the corresponding conical domain [3].
But for physical examples in the ordinary three-dimensional space, when a bounded
domain has edges and no corners, then the edges are necessarily curved. The simplest
example is a cylinder with circular basis, cut orthogonally to its generating lines. But
this example is very particular: The opening of the edge is everywhere π/2 and the
curvature of the edge is constant. If one cuts the cylinder by a plane which is skew with
respect to the generating lines, then the edge is elliptic and the opening angle is varying.
This gives rise to difficulties for the precise analysis of the structure of the solution, due
to the fact that the asymptotics in the corresponding two-dimensional domains depend
in a discontinuous way on the opening parameter. In particular, the coefficients of the
singular functions along the edge (stress intensity factors etc.) will blow up at certain
points. Such a behavior causes difficulties also for numerical approximations.
In this note, we consider as model problem the mixed Dirichlet-Neumann problem
for the Laplace operator on a skew cylinder. This problem was posed as a question to
the first author by I. Babuška (Maryland). We already stated results for this problem
in [1]. Here we present improved results. With the introduction of a complex variable
ζ in the normal plane to the edge, we give expressions of the edge asymptotics in a
simplified form. Such a form is based upon the divided differences of the function
λ 7→ ζ λ calculated at the exponents of singularities.
This formulation was inspired by a recent paper by Maz’ya and Rossmann [11]
where a different but related problem was treated, namely the problem of writing the
corner singularities of a two-dimensional Dirichlet problem for the Laplacian in a form
that is stable with respect to variations of the corner angle. It is in fact not hard to
see that our formulation is equivalent to that one given by Maz’ya and Rossmann, if
we consider the edge angle ω(y) as independent unknown instead of the edge variable
y. The possibility of using powers of a complex variable to describe singularities in
piecewise analytic plane domains was shown in the earlier work [7].
The restriction to the case of the Laplace operator is actually not as serious a limitation as it may look : The results we state here can be extended to any strongly elliptic
boundary value problem for a second order operator with real analytic coefficients.
1.b Boundary value problem. Let B be an analytic bounded domain in R2 . Let
Ψ be an affine function (x2 , x3 ) 7→ x1 = Ψ(x2 , x3 ). We assume that
for all (x2 , x3 ) ∈ B,
Ψ(x2 , x3 ) > 0.
We introduce
n
o
Ω = (x1 , x2 , x3 ) ∈ R3 | (x2 , x3 ) ∈ B, 0 < x1 < Ψ(x2 , x3 ) .
This is our skew cylinder. We denote by M the top edge and by ∂1 Ω the side of the
cylinder:
n
o
∂1 Ω = (x1 , x2 , x3 ) ∈ R3 | (x2 , x3 ) ∈ ∂B, 0 < x1 < Ψ(x2 , x3 ) .
The union of the top and the bottom of the cylinder is denoted by ∂2 Ω.
The boundary value problem that we choose to consider here is the following
(PΩ )
(
∆u = f on Ω
u = 0 on ∂1 Ω,
∂u
∂n
= h on ∂2 Ω.
This example is interesting because the singularities appear at a low regularity level :
in a generic way, u 6∈ H 2 (Ω).
1.c Singularities on plane sectors. Let ω ∈ (0, 2π) and let G(ω) denote the
two-dimensional angle with opening ω. We use polar coordinates (r, θ) so that G(ω)
corresponds to r > 0, 0 < θ < ω. The boundary value problem corresponding to (PΩ )
is :
(
∆w = f˜ on G
(PG )
∂w
w = 0 on θ = 0,
= h̃ on θ = ω.
∂n
For any integer k ≥ 1, we set
νk = (2k − 1)
π
.
2ω
It is well known [4] that the singularities of any solution of (PG ) are linear combinations of the functions σk (when νk 6∈ N) and τk (when νk ∈ N) below :
σk (r, θ) = r νk sin νk θ
τk (r, θ) = r νk (log r sin νk θ + θ cos νk θ).
2
Let us set ζ = r eiθ the writing of the plane coordinates in complex form. Then we
have
σk (r, θ) = Im ζ νk
τk (r, θ) = Im ζ νk log ζ.
Proposition 1.1 Let s ∈ R, s > 1/2. Let w be a solution of (PG ) such that f˜ has
1
H s−1 regularity and that h̃ has H s− 2 regularity. Let us assume that w has compact
support. Then w admits the following decomposition,
w = wreg + wsing
and
wsing =
with wreg ∈ H s+1−ε (G)
X
ck σk +
νk <s, νk 6∈N
X
∀ε > 0
ck τk .
νk <s, νk ∈N
1.d Aims. Let us return now to our skew cylinder. We assume that for a fixed real
s > 1/2
f ∈ H s−1(Ω)
1
h ∈ H s− 2 (∂1 Ω).
Then there exists a variational solution u ∈ H 1 (Ω).
For any y ∈ M, let ω(y) be the opening angle of Ω in y. We set ν1 (y) =
s < min ν1 (y), then u ∈ H s+1 (Ω). From now on we assume that
π
.
2ω(y)
If
y∈M
min ν1 (y) < s.
y∈M
Then u can be split into two parts for any ε > 0,
u = ureg + using
where ureg ∈ H s+1−ε (Ω) and using is an asymptotics. Our aim is to describe the structure
of such splittings. In particular, we want to separate as much as possible the roles of
the different variables, the abscissa y along the edge, the distance r from the edge and
the angular variable θ. More precisely, we intend to separate what comes from the
geometrical framework (domain and boundary value problem) and what comes from
the data (f, h).
1. The part that comes from the geometrical framework will be described as special
combinations of real and imaginary parts of powers of ζ and ζ. The introduction
of such combinations allows to get rid of the discontinuity in the expression of
π
the singularities when the function y 7→ νk (y) := (2k − 1) 2ω(y)
crosses an integer.
2. The part that comes from the data will be described by some coefficients c(y)
along the edge. The regularity of c depends on s and on the exponents of the
associated singularity.
3
2.
SIMPLE ASYMPTOTICS
2.a Simple singularities in two-dimensional domains. Let us recall that for
each y in the edge M, ω(y) denotes the opening of Ω at the point y. We can choose
local cylindrical coordinates (y, r, θ) such that r = 0 corresponds to the edge and such
that 0 < θ < ω(y) describes locally the domain Ω.
What can be expected as singularities along the edge for the solution of problem
(PΩ ) is
X
cα (y) Sα(y, r, θ)
(2.1)
α
where the functions Sα do not depend on the data (f, h).
First, we want to link the form of the Sα (y, ·, ·) with the form of singularities of
some two-dimensional boundary value problems on G(ω(y)). Due to the presence of
curvature terms, it is natural to consider instead of ∆ as interior operator on G a more
general second order operator
A(z; ∂z ) =
X
aβ (z)∂zβ
with aβ ∈ C ∞ (G) and aβ (0) = δβ .
(2.2)
|β|≤2
This means that the principal part of A at the vertex 0 is ∆.
Secondly, we want that the functions Sα depend smoothly on y. In a first stage we
search them as powers of ζ and ζ. That is why we will assume that νk 6∈ N to avoid
the discontinuity between σk and τk .
Proposition 2.1 Let A be an elliptic operator satisfying (2.2). Let ω ∈ (0, 2π). We
assume that
kπ
kπ
< 2s,
6∈ N .
(2.3)
∀k ≥ 1 such that
ω
ω
Let w be a solution of problem (PG(ω) ) with ∆ replaced by A and satisfying the same
hypotheses as in Proposition 1.1. Then w can be split in the same way with
wsing =
X
2l
X
ckln Re (e−2inθ ζ νk +l ) + c′kln Im (e−2inθ ζ νk +l ).
k,l : νk +l<s n=0
2.b Simple asymptotics along the edge. The exponents of the singularities
which appear in the asymptotics along the edge depend on the edge parameter y. They
are the same as in two-dimensional problems as considered above in Proposition 2.1 in
an angle with opening ω(y). We can enumerate them using a double index κ = (k, l),
where k ∈ N∗ and l ∈ N :
π
νκ (y) = (2k − 1)
+l.
2ω(y)
We could think that if we assume condition (2.3) for any y, then the functions Sα
will be the real and imaginary parts of
e−2inθ ζ νκ(y) .
4
Indeed, the tangential derivatives ∂y produce logarithmic terms and the Sα are the real
and imaginary parts of
e−2inθ ζ νκ(y) logq ζ .
We have also to note that an asymptotics in tensor product form such as (2.1) is
not convenient in general, since the cα are not regular enough. Therefore we define the
usual (see [6], [3], [10]) regular extension of the coefficients : we introduce a function
Φ(y, r) such that its partial Fourier transform satisfies
Fy→ξ Φ(ξ, r) = φ(r|ξ|)
where φ is a rapidly decreasing function, has a Fourier transform with compact support,
and satisfies for a sufficiently large N
φ(0) = 1,
dn
φ(0) = 0 (n = 1, . . . , N).
dtn
(2.4)
We define the convolution with respect to y,
(c ∗ Φ)(y, r) :=
Z
Φ(y − y ′, r) c(y ′) dy ′.
(2.5)
The following theorem describes our result on the ‘simple’ edge asymptotics in
complex variable form, i. e. when any crossing between exponents νκ themselves and
with integer numbers is excluded.
Theorem 2.2 Let I, I ′ be intervals such that the cylindrical local coordinates (y, r, θ)
are defined in a neighborhood U of I ′ in Ω and I ⊂⊂ I ′ . We suppose there is no
crossing point in I ′ :
∀y ∈ I ′ , ∀k ≥ 1 such that
kπ
< 2s then
ω(y)
kπ
6∈ N
ω(y)
(2.6)
and that the νκ do not cross the value s above I ′ . Then any solution u of problem (PΩ )
1
with f ∈ H s−1 (Ω) h ∈ H s− 2 (∂1 Ω) can be decomposed into
u = ureg + using
and
using =
X
with ureg ∈ H s+1−ε (U)
(2.7)
(2.8)
(cκ,q,n ∗ Φ)(y, r) Re e−2inθ ζ νκ(y) logq ζ
κ,q,n
+
∀ε > 0
(c′κ,q,n ∗ Φ)(y, r) Im e−2inθ ζ νκ(y) logq ζ .
The coefficients cκ,q,n(y) and c′κ,q,n (y) are defined on I and belong to H s−νκ(y)−ε (I) for
all ε > 0. The sum extends over those κ for which Re νκ < s holds on I and to
0 ≤ n ≤ 2l if κ = (k, l).
Simple asymptotics in “real variable form” involve instead of the real and imaginary
parts of e−2inθ ζ νκ(y) logq ζ functions of the type ϕκ,q,n(y, θ) r νκ(y) logq r where the functions ϕκ,q,n (y, θ) are analytic in all their arguments but only implicitly known. Such
asymptotics are given in [6], [12], [10] and [1].
5
2.c The crossing of exponents. Now we have to include all positive integers in
the set of exponents due to the possible interaction between polynomials and singularities. That is the reason for introducing
for κ = (0, l) νκ = l .
The above assumption (2.6) implies that there is no crossing of exponents, i.e., there
are no points y such that for some κ, κ′ with κ 6= κ′ there holds νκ (y) = νκ′ (y). In [6],
[12], [10] such a condition is also required.
For our problem of the skew cylinder, it is impossible to avoid such crossings. For
y0 such that ω(y0) = π/2 (there always exist two such points), we have
ν1,0 (y0 ) = ν0,1 (y0 ) = 1.
The points where crossing of exponents will eventually appear (for large s) are dense
in M, so this phenomenon occurs in a generic way.
In Section 3, we will present the main results of this note for the case when crossing
points are present. Our motivations for their presentation are the following.
1. To give an asymptotics in the neighborhood of crossing points which is as explicit
and as simple as possible
2. To eliminate as many technical hypotheses as possible.
3.
ASYMPTOTICS AT CROSSING POINTS
3.a Ordering the exponents. Let y0 be a crossing point, i. e., a point where there
exist distinct κ and κ′ such that
νκ (y0 ) = νκ′ (y0 ) < s.
(3.1)
Since we assume that our cylinder is actually skew, crossing points are isolated, so
there exist open intervals I and I ′ with y0 ∈ I, I ⊂ I ′ , and there is no other crossing
point in I ′ .
Let Ky0 be the set of indices,
n
o
Ky0 := κ = (k, l) ∈ N2 | νκ (y0 ) < s .
We denote by µ01 , . . . , µ0j0 the distinct elements of the set
o
n
νκ (y0 ) | κ ∈ Ky0 .
Since y0 is a crossing point, the cardinality of Ky0 is strictly larger than j0 . For each
j, let Ky0 ,j be the subset of Ky0 ,
n
o
Ky0 ,j := κ ∈ Ky0 | νκ (y0 ) = µ0j .
6
The µ0j are either crossing exponents (if #Ky0 ,j > 1) or simple exponents (if
#Ky0 ,j = 1).
For each κ, we call multiplicity of νκ the maximal power of log ζ which appears in
the asymptotics (2.8) along with the term ζ νκ(y) for y ∈ I \ {y0}. Then we denote by
(κqj )1≤q≤qj an enumeration of Ky0 ,j , repeating each term according to its multiplicity.
Finally, we set for y ∈ I ′ :
ν j (y) := max νκ (y).
(3.2)
κ∈Ky0 ,j
3.b Divided differences. What essentially changes from the simple asymptotics
(2.8) is the behavior of the functions of ζ. Instead of having separately the terms
ζ νκ(y) logq ζ, we have now special combinations of these terms which cannot be separated, namely the divided differences of the function λ 7→ ζ λ at some of the points
νκ1 (y), . . . , νκq (y) for each y ∈ I. Let us recall that, when µ1 , . . . , µK are all distinct,
the divided difference of the function w at the K-tuple µ1 , . . . , µK is defined by the
classical recursion formula :
w[µ1] = w(µ1 )
and for ℓ = 2, . . . , K
w[µ1 , . . . , µℓ ] =
1
(w[µ1 , . . . , µℓ−1] − w[µ2 , . . . , µℓ ]) .
µ1 − µℓ
Moreover for analytic functions w one has for any µ1 , . . . , µK not necessarily distinct
w[µ1, . . . , µK ] =
1 Z
w(λ)
dλ
K
2iπ γ Y
(λ − µℓ )
(3.3)
ℓ=1
where γ is a simple curve surrounding all µℓ . Thus, for any ζ we consider
1
ζλ
S[µ1 , . . . , µK ; ζ] =
dλ .
K
2iπ γ Y
(λ − µℓ )
Z
(3.4)
ℓ=1
We see that S[µ1 (y), . . . , µK (y); ζ] with analytic µ1 (y), . . . , µK (y) is a linear combination of terms of the form ζ µℓ (y) logq ζ with coefficients that are meromorphic in y. If all
µℓ (y) are equal to the same µ(y) then
S[µ, . . . , µ; r] =
|
{z
}
q+1 times
1 µ q
r log r .
q!
(3.5)
When all the µℓ (y) are distinct, we obtain
S[µ1 , . . . , µK ; ζ] =
K
X
ℓ=1
ζ µℓ
q
Y
k=1
k6=ℓ
7
(µℓ − µk )
.
(3.6)
Theorem 3.1 Let I, I ′ be intervals satisfying the same assumption as in Theorem 2.2
except that hypothesis (2.6) is replaced by
y0 ∈ I is the only crossing point in I’.
(3.7)
1
Then any solution u of problem (PΩ ) with f ∈ H s−1 (Ω) h ∈ H s− 2 (∂1 Ω) can be decomposed into
u = ureg + using with ureg ∈ H s+1−ε (U) ∀ε > δ(I)
(3.8)
where δ(I) is a continuous function of I which tends to 0 when the length of I tends
to 0 and
using =
X
j,q,n
+
(dj,q,n ∗ Φ)(y, r) Re e−2inθ S[νκ1j (y), . . . , νκqj (y); ζ]
((d′j,q,n
−2inθ
∗ Φ)(y, r) Im e
(3.9)
S[νκ1j (y), . . . , ν (y); ζ] .
κqj
The coefficients dj,q,n(y) and d′j,q,n(y) are defined on I and belong to H s−ν j −ε (I) for all
ε > 0. The index n spans {0, 1, . . . , 2ℓj } where
ℓj := max{l ∈ N | ∃k ∈ N : νkl (y0 ) = µ0j } .
Remark 3.2 If there is no crossing in I ′ , then this statement yields the same result as
Theorem 2.2. Indeed, the sets Ky0 ,j are all reduced to one element and the functions
in ζ are all of the form S[ν, . . . , ν; ζ], i. e. ζ ν logq ζ according to (3.5).
Remark 3.3 If y and the multiplicities in (3.9) are such that all νκ1 , . . . , νκq are
different, then one can write the singular function in (3.9) as
−2inθ
e
n
S[νκ1 (y), . . . , νκq (y); ζ] = ζ ζ
−n
q
X
aℓ (y) ζ νκℓ (y)
ℓ=1
with the coefficients
aℓ (y) =
q
Y
r=1
r6=ℓ
1
.
νκℓ (y) − νκr (y)
3.c The first singularity. Let us illustrate our statements by considerations about
the first singularity of our problem (PΩ ). We take s > 1 and consider those points y0
and y0′ such that ω(y0), ω(y0′ ) = π2 . Since there is one zero Dirichlet condition, ν(0,0)
does not appear. With that choice of s, the first exponents in the neighborhood of
y0 are ν(1,0) and ν(0,1) and they cross each other in y0 and the same holds for y0′ . For
simplicity let us suppose that s < 2 so that only ν(1,0) and ν(0,1) are relevant and let us
denote
π
2ω(y)
ν2 (y) := ν(0,1) (y) = 1.
ν1 (y) := ν(1,0) (y) =
8
For y 6= y0 and y in a neighborhood of y0 , the simple asymptotics (2.8) holds.
Indeed the terms involved in using are only the imaginary parts of ζ νj and the only
contribution of n is n = 0 : The “simple asymptotics” of u is
c1 ∗ Φ Im ζ ν1 + c2 ∗ Φ Im ζ ν2 .
(3.10)
The function Im ζ ν1 corresponds to the first corner singularity and the function Im ζ ν2 is
polynomial. Here it is possible to compute c2 (y) since it depends only on the pointwise
value of the Neumann boundary datum h on the edge :
c2 (y) =
h(y, 0)
.
cos ω(y)
s−1
Then c2 ∈ Hloc
(M \ {y0 , y0′ }) and c2 generally blows up in y0 and y0′ .
In order to get the representation of using according to Theorem 3.1 at the crossing
points, we only need again the imaginary parts of two basis functions, for instance:
S[ν1 (y); ζ] = ζ ν1(y)
ζ − ζ ν1(y)
S[ν1 (y), ν2(y); ζ] =
.
1 − ν1 (y)
Then the asymptotics of u can be written
d1 ∗ Φ Im S[ν1 ; ζ] + d2 ∗ Φ Im S[ν1 , ν2 ; ζ].
(3.11)
We also could have chosen S[ν1 (y), ν2 (y); ζ] and S[ν2 (y); ζ] as basis functions.
At the limit when ω → π2 , Im S[ν1 (y), ν2(y); ζ] tends to the logarithmic singularity
Im ζ log ζ.
Now we can compare the two above representations (3.10) and (3.11) of a singular
part. There hold the following relations between the coefficients.
d 1 = c1 + c2
, c1 = d 1 −
d2 = c2 (1 − ν1 ) , c2 =
d2
(1 − ν1 )
d2
.
(1 − ν1 )
3.d Aims for the future. It would be interesting to know whether asymptotics
for general elliptic equations and systems can be described by similar formulas, i. e. by
divided differences of some analytic generating functions.
For general second order elliptic equations we have obtained asymptotics with “geometric terms” Sα in the form
ψ(y, θ) S[νκ1 (y), . . . , νκq (y); r]
with analytic ψ(y, θ). Such formulas rely upon the fact that there exist analytic choices
for the exponents. But such an analytic choice is generically impossible for fourth order
9
operators such as the bilaplacian. The basic problem is the expression of the roots of
a polynomial whose coefficients depend analytically on a parameter. The roots are
algebraic but, in general, non-analytic functions of the parameter. Such situations of
bifurcations are studied in [14]. In the general case there appear combinations of both
crossings and bifurcations. We think that even then it will be possible to reach the aims
we described at the end of the first section, i. e., to separate all that can be separated.
Note. The detailed proofs of the results presented in this paper can be found in
the Paris Preprint [2].
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[10] V. G. Maz’ya, J. Rossmann. Über die Asymptotik der Lösungen elliptischer
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[11] V. G. Maz’ya, J. Rossmann. On a problem of Babuška (Stable asymptotics
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[12] A. Nikishkin. Singularities of the solution of the Dirichlet problem for a second
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Teubner Verlagsgesellschaft, Leipzig 1991. To appear.
Martin Costabel
Laboratoire d’Analyse Numérique
Université Pierre et Marie Curie
4, place Jussieu
75252 PARIS Cedex 05 (FRANCE)
Monique Dauge
Département de Mathématiques
Université de Nantes
2 rue de la Houssinière
44072 NANTES Cedex 03 (FRANCE)
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