INCREASED STABILITY OF SOLUTIONS TO THE HELMHOLTZ EQUATION
A Thesis by
Deepak Aralumallige
B.E., Bangalore University, 2001
Submitted to the College of Liberal Arts and Sciences
and the faculty of Graduate School of
Wichita State University in partial fulfillment of
the requirements for the degree of
Master of Science
December 2005
INCREASED STABILITY OF SOLUTIONS TO THE HELMHOLTZ EQUATION
I have examined the final copy of this Thesis for form and content and recommend that it be
accepted in partial fulfillment of the requirements for the degree of Master of Science with a
major in Mathematics.
-------------------------------------------------------------Dr. Victor Isakov, Committee Chair
We have read this thesis
and recommend its acceptance:
------------------------------------------------------------Dr. Thomas DeLillo, Committee Member
-------------------------------------------------------------Dr. Hamid M. Lankarani, Committee Member
ii
DEDICATION
I would like to dedicate this work to my advisor Dr. Isakov and to my parents.
iii
ACKNOWLEDGEMENTS
I like to thank my advisor Dr. Victor Isakov for his patient guidance and support. Without his
valuable suggestions and motivation, this research would not have been successful.
I would like to take this opportunity to thank Dr. Kenneth Miller, Graduate Coordinator, for his
support and guidance over the past two years.
I wish to extend my gratitude to the members of my committee, Dr. Thomas DeLillo and Dr.
Hamid M. Lankarani for their helpful comments and suggestions for my thesis.
I would also like to thank Chetan V. Gubbi and Arun D. Satish. Together their friendship and
selfless role modeling have contributed to my professional development.
Finally, I would like to thank all my friends at Wichita State University for their kind
cooperation during my academic years.
iv
ABSTRACT
Study of the Cauchy problem for Helmholtz equation is stimulated by the inverse scattering
theory and more generally by remote sensing. This thesis explains the increased stability of the
Cauchy problem for Helmholtz equation when the frequency increases. The stability estimate is
obtained inside the whole domain.
v
TABLE OF CONTENTS
CHAPTER
PAGE
1. Introduction
1
1.1 Cauchy Problem. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Problem Description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2. Auxiliary Results
9
2.1 Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9
3. Stability Estimate
11
3.1 Main Result . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Interpolation . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13
Bibliography
17
vi
CHAPTER 1
INTRODUCTION
1.1
Cauchy Problem
As explained in [1] finding the solution u of a partial differential equation from the given data gj is the well known Cauchy Problem.
Lu = f on Ω,
∂νj u = gj , j ≤ m − 1 on Γ,
where Lu =
X
aα ∂ α u, Ω is a domain in Rn , Γ ∈ C m−1 , is a part of
|α|≤m
∂Ω, the boundary of the domain Ω and ν is the outward normal to the
boundary.
A Cauchy problem is said to be well-posed in the sense of
Hadamard if the following conditions hold:
1. u ∈ U exists for any gj ∈ G, f ∈ F ;
2. u ∈ U is determined uniquely by gj ∈ G, f ∈ F ;
3. u ∈ U depends continuously on gj ∈ G, f ∈ F ,
where U is the space of all solutions u and G × F is the space of all
data gj , f prescribed on the boundary Γ and on the domain Ω. In other
1
words a Cauchy problem is well posed, if the operator A : U → G × F
defined as Au = {g, f } has a continuous inverse from G × F onto U .
U , G, and F are open subsets of classical spaces C k (Ω), C k (Γ), Hkp (Ω),
Hkp (Γ) or their closed subspaces of finite codimension.
If one of the above 3 conditions is not satisfied, then such problems
are called ill posed problems in the sense of Hadamard. For ill posed
problems u may not exist. If u exists continuous dependence of u on gj ,
f may not be guaranteed. For example, consider the classical example
of Hadamard [1] : the Cauchy problem for the Laplace equation
∂x2 u + ∂y2 u = 0 in R2+ = {(x, y)|y > 0},
u = 0, ∂y u = g1 ,when y = 0.
If g1 (x) = n−2 cos(nx), then u exists (since g1 is entire analytic) and
u = n−3 cos(nx)sinh(ny). But one can note that there is no continuous
dependence of u on g1 . In applications (inverse problems) this continuous dependence (stability estimates for u) is of most importance.
Only this conditions guarantees the convergence of the solutions u while
using computational algorithms.
A stability estimate is defined [3] as a function ω such that
ku − u∗ kU ≤ ω(kAu − Au∗ kG×F )
2
An important condition for stability estimate is that limτ →0 ω(τ ) = 0
and also ω is increasing monotonically. Depending on this function ω
we have three kinds of stability estimates:
1. if ω() = C, then the solution u depends Lipschitz continuously
on gj , f ;
2. if ω() = Cκ , 0 < κ < 1, then the solution u depends Holder
continuously on gj , f ;
3. if ω() =
C
, then the solution u depends logarithmic continu|log|
ously on gj , f which is much weaker kind of continuity.
For applications mere continuity is not good, to develop efficient
numerics we expect that u depends Holder continuously on data (the
best case would be Lipschitz continuity). Cauchy problems where u
depends on data Holder continuously are said to be well behaved [1].
This can be achieved by assuming that the solution u and first few
derivatives of u are bounded. Hence we consider a restricted solution
space UM where M is the apriori bound.
Now let us consider an example of a problem which is not well be-
3
haved [1] :
∂x2 u + ∂y2 u = ∂t2 u in Ω × (−T, T ),
u = g0 and ∂ν u = g1 on ∂Ω × (−T, T ).
Let the solution u and first few derivatives be bounded and the domain
Ω be given by
Ω = {(x, y) : 0 ≤ x2 + y 2 < 1}.
It is shown in [1] that at any point inside the cylinder Ω × (−T, T )
the solution u depends Holder continuously on g0 , g1 (or even Lipschitz
continuously [3] if appropriate norms are selected), but outside the
cylinder Ω × (−T, T ) the dependence of the solution u on g0 , g1 is
logarithmic continuous.
The Lipschitz continuous dependence of solution u on the given data
can be explained as follows. From known theory of hyperbolic problem
there is a u1 ∈ H21 (Ω × (−T, T )) solving
∂tt u1 = ∆u1 in Ω × (−T, T ),
u1 = ∂t u1 = 0 on Ω × {0},
u1 = g0 on ∂Ω × (−T, T )
4
and satisfying the bound
ku1 k(2) (Ω × (−T, T )) ≤ Ckg0 k( 23 ) (Γ),
where Γ = ∂Ω × (−T, T ). Let u2 = u − u1 , then
∂tt u2 = ∆u2 in Ω × (−T, T ),
u2 = 0 on ∂Ω × (−T, T ).
So from Theorem 3.4.5 of [3] we can conclude that
ku2 k(1) (Ω × (−T, T )) ≤ Ckg1 k(0) (Γ).
Provided T > L, where L = sup|l| and l(x) = x − x0 , x0 is some point
in Rn . Finally,
kuk(1) (Ω × (−T, T )) = ku1 + u2 k(1) (Ω × (−T, T ))
≤ (ku1 k(1) + ku2 k(1) )(Ω × (−T, T ))
≤ (ku1 k(2) + ku2 k(1) )(Ω × (−T, T ))
3
≤ C kg0 k( 2 ) (Γ) + kg1 k(0) (Γ)
5
1.2
Problem Description
The goal of this thesis is the study of the Cauchy problem for the
Helmholtz equation
∆ + k 2 u = f in Ω
(1.1)
u = g0 and ∂ν u = g1 on Γ,
(1.2)
with Cauchy data
where Ω is a subset of a cylinder Ωh in Rn , Ωh = {x : 0 < xn <
h, |x0 | < r}, x0 = (x1 , x2 , ..., xn−1 ), and Γ is the Lipschitz open part of
the boundary ∂Ω contained in the layer {0 < xn < h}.
We assume that Ω = {x : 0 < xn < ω(x0 ), x0 ∈ Ω0 } , ω > 0, ω ∈ C 1 (Ω0 ).
T
Let Ω(d) = Ω {x : xn > d}, M1 = kuk(1) (Ω), F = kf k(Ω) + kuk(Γ) +
k5uk(Γ) and F (k, d) = kf k(Ω)+d−0.5 (k +d−1 )kuk(Γ)+k5uk(Γ), here
kuk(l) (Ω) is the norm in the Sobolev Space H(l) (Ω) and kuk = kuk(0) .
The constant C depends on Ω and Γ, any additional dependence is
indicated.
In [2] a stability estimate for the solution u of (1.1) and (1.2) is obtained
inside the domain Ω(d) which is as follows:
M 1−λ F (k, d)λ
kuk(Ω(d)) ≤ C F + 1 2−2λ
d
k
6
(1.3)
xn
h
Γ
d
r
x'
Ω (d )
Figure 1.1: Domain Ω(d)
where
2r2 d + 83 d3
λ= 2
4r h + h2 d + 45 d2 h + 83 d3 + 3r2 d
(1.4)
and C = C(Ω, Γ). One important observation is that the stability and
hence the resolution in the Cauchy problem increases as the frequency
k increases. The main aim of this thesis is to obtain a stability estimate
for the solution u of (1.1) and (1.2) inside the whole domain Ω. First,
we obtain simpler upper bound for (1.3) which is given by
M 1−λ1 F λ1
√
kuk(Ω(d)) ≤ C F +
d2 k
7
(1.5)
using the fact that d is small (d < 1) and assuming that F < 1, 1 ≤ M1 ,
λ<
1
2
and
5
(h2 + 3r2 )d + d2 h < 4r2 h,
4
(1.6)
where
λ1 =
d
< λ.
4h
Indeed from (1.4) we can conclude that
2r2 d
λ> 2
4r h + (h2 + 3r2 )d + 45 d2 h
From the assumptions we have λ >
8
d
= λ1
4h
(1.7)
CHAPTER 2
AUXILIARY RESULTS
2.1
Trace
Lemma 2.1.1 (Bound on Trace) There exists a constant C = C(Ω, Γ)
such that
kuk(0) (S(d)) ≤ Ckuk(1) (Ω),
(2.1)
where S(d) = {x : |x0 | < r, xn = d} ∩ Ω.
Proof: From Trace Theorem [3] we can conclude that
kuk(0) (S(d)) ≤ C(d) kuk(1) (Ω(d)).
We need to show that the constant C(d) does not depend on d.
Consider the extended domain
Ω∗ = {(x0 , xn )| p x0 p< r, 0 < xn < 2h}.
Form Calderon’s extension theorem [5] we have the existence of the continuous operator E : H(1) (Ω) → H(1) (Ω∗ ) such that Eu(x) = u(x) for all x ∈
Ω(d). Let v = Eu.
Now, consider a sub-domain
Ω∗ (d) = {(x0 , xn ) : kx0 k < r, d < xn < d + h}
9
and translate this sub-domain Ω∗ (d) downwards by d. The translation function is defined as td (x) = (x1 , x2 , ..., xn − d). Hence the new
translated domain is
Ω∗ (0) = {(x0 , xn ) : kx0 k < r, 0 < xn < h}.
Let us define the function
vd∗ = (v ◦ td ).
Applying trace theorem to vd∗ in Ω∗ (0) we obtain
kvd∗ k(0) (S(0)) ≤ Ckvd∗ k(1) (Ω∗ (0).
Since vd∗ = (v ∗ ◦ td ) and also the norms are translation invariant
kvd∗ k(0) (S(0)) = kvk(0) (S(d)),
kvd∗ k(1) (Ω∗ (0)) = kvk(1) (Ω∗ (d)).
Since v(x) = u(x) for all x ∈ Ω(d), we have
kuk(0) (S(d)) ≤ C kuk(1) (Ω).
10
CHAPTER 3
STABILITY ESTIMATE
3.1
Main Result
Theorem 3.1.1 There exists a constant C = C(Ω, Γ) such that for
any solution u to (1.1) and (1.2)
1
2δ 4
k
√
h
√
kuk2 (Ω(0)) ≤ CM12 δ 2 + δk2 e−
− 21
!
+ δk
(3.1)
F
M1
where δk = (−lnδ)(lnk) and δ =
Proof: Let F (d) = kuk2 (Ω(d)). We can write F (d) as
Z
Z
ω(x0 )
F (d) =
(x0 ,d)∈Ω
|u|2 dxn dx0
d
We have
F 0 (d) = −
Z
|u(x0 , d)|2 dx0 = −kuk2 (S(d))
(3.2)
(x0 ,d)∈Ω
From the mean value theorem we have
F 0 (d∗ ) =
F (d) − F (0)
d
where 0 < d∗ < d. Hence,
|F (0)| ≤ |F (d)| + |F 0 (d∗ )|d
11
(3.3)
From (1.5) we have
1
kuk2 (Ω(d)) ≤ C F 2 + M12 δ 2λ1 4
kd
where λ1 =
d
4h .
Using the above equation, (3.3), and (2, 1) we conclude
that
2
2
kuk (Ω(0)) ≤ C F +
M12 δ 2λ1
1
2
+ M1 d .
kd4
(3.4)
Let
g(d) = δ 2λ1
1
+ d.
kd4
Our aim is to minimize this function with respect to d > 0. Motivated
by minimization of g(d) we let
1
d=
1
2
(lnk) (−lnδ)
1
2
.
(3.5)
Hence
1
(−lnδ) 2
1
2h(lnk) 2
−
g(d) = (−lnδ)2 (lnk)2 e
!
+lnk
1
1
+ (−lnδ)− 2 (lnk)− 2 .
√
Since 2 AB ≤ A + B, we have
1
2
2(−lnδ) (lnk)
h
1
2
! 12
1
≤
(−lnδ) 2
1
2h(lnk) 2
+ lnk
and hence
−
g(d) ≤ (−lnδ)2 (lnk)2 e
1
1
2(−lnδ) 2 (lnk) 2
h
12
!1
2
1
1
+ (−lnδ)− 2 (lnk)− 2 .
Setting δk = (−lnδ)(lnk) we have the final result
1
2δ 4
√k
h
√
kuk2 (Ω(0)) ≤ CM12 δ 2 + δk2 e−
− 21
!
+ δk
3.2
Interpolation
Here the author would like to mention the well know interpolation
inequalities for intermediate derivatives. The main idea is that if u is
bounded in H(s1 ) and H(s2 ) then u is bounded in all the intermediate
Sobolev spaces H(s) where s1 < s < s2 . The bound as explained in [3]
is given by:
θ
kuk(s) (Ω) ≤ Ckuk1−θ
(s1 ) (Ω)kuk(s2 ) (Ω)
(3.6)
where s = (1 − θ)s1 + θs2 , 0 < θ < 1 and C = C(Ω, s1 , s2 , θ).
If k > n2 , then by known Sobolev embedding theorems [4]
kuk∞ (Ω) ≤ Ckuk(k) (Ω).
(3.7)
For n = 3 we can take k = 2.
The above result is important, since we have bound on the supremum
of u. Once we have bounds of supremum norm on the domains Ωα ,
13
Figure 3.1: Union of domains Ω1 ,Ω2 ,...,Ωα .
α ∈ A, where A is some index set, then we have the bound of u in the
S
domain Ω = Ωα .
Let us assume that
kuk( n+1 ) (Ω) ≤ Mn
2
Then from (3.6) we have
θ
kuk(s) (Ω) ≤ Ckuk1−θ
n+1 (Ω)kuk(0) (Ω).
( 2 )
where s = (1 − θ)
n+1
2
and 0 < θ < 1. If s =
n
2
from (3.7) we can conclude that
kuk∞ (Ω) ≤ Ckuk(s) (Ω)
14
+ 41 and θ =
(3.8)
1
2(n+1) then
Using (3.1) and (3.8) in the above equation one can conclude that
1
2δ 4
√k
h
√
δ 2 + δk2 e−
kuk∞ (Ω) ≤ CMn1−θ M1θ
− 12
!θ
(3.9)
+ δk
where C depends only on the domain Ω and ∂Ω.
Let us assume that kuk(4) ≤ M4 and from the embedding theorems for
Sobolev spaces we can conclude that
k∂ν uk∞ ≤ Ckuk(3)
Using the interpolation inequality (3.6) we can write,
k∂ν uk∞ ≤ Ckuk(3)
θ
≤ Ckuk1−θ
(4) kuk(0) .
Here s1 = 4, s2 = 0, s = 3, and hence θ = 14 . Therefore
3
1
4
4
k∂ν uk∞ ≤ Ckuk(4)
kuk(0)
1
2δ 4
k
√
h
√
3
1
≤ CM44 M12
δ 2 + δk2 e−
15
−1
+ δk 2
! 14
.
Bibliography
16
BIBLIOGRAPHY
[1] John, Fritz , ‘’Continuous Dependence on data for solutions of
partial differential equations with a prescribed bound,” Communications on Pure and Applied Mathematics, 13,(1960): 551–585.
[2] Hrycak T. and Isakov V. , “Increased stability in the continuation of solutions to the Helmholtz equation,” Inverse Problems,
20,(2004): 697–712.
[3] Isakov, V , Inverse Problems for Partial Differential Equations,
Springer, New York, 1998.
[4] Lions, J.L., Magenes, E. , Non-Homogeneous Boundary Value
Problems and Applications, Springer, 1972.
[5] Morrey, C.B., Multiple Integrals in the Calculus of Variations,
Springer, 1966
17