On the crack inverse problem for pressure waves in half-space Darko Volkov

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On the crack inverse problem for pressure waves in

half-space

Darko Volkov ∗

March 13, 2023

Abstract

After formulating the pressure wave equation in half-space minus a crack with a zero

Neumann condition on the top plane, we introduce a related inverse problem. That

inverse problem consists of identifying the crack and the unknown forcing term on that

crack from overdetermined boundary data on a relatively open set of the top plane. This

inverse problem is not uniquely solvable unless some additional assumption is made.

However, we show that we can diﬀerentiate two cracks Γ1and Γ2under the assumption

that R3\Γ1∪Γ2is connected. If that is not the case we provide counterexamples that

demonstrate non-uniqueness, even if Γ1and Γ2are smooth and “almost” ﬂat. Finally,

we show in the case where R3\Γ1∪Γ2is not necessarily connected that after excluding

a discrete set of frequencies, Γ1and Γ2can again be diﬀerentiated from overdetermined

boundary data.

MSC 2010 Mathematics Subject Classiﬁcation: 35R30, 35B60, 35J67.

Keywords: Nonlinear inverse problems, Overdetermined elliptic problems and unique con-

tinuation, Domains with cusps.

1 Introduction

In this paper, we study an inverse problem consisting of identifying a crack in half-space and

the unknown forcing term on that crack from overdetermined boundary data on a relatively

open set of the top plane. For the forward problem, the governing equations involve the

Helmholtz operator, a zero Neumann condition on the top plane, and continuity of the

normal derivative across the crack. A jump across the crack constitutes the forcing term.

Some decay at inﬁnity is enforced by requiring that the solution lie in an adequately weighted

Sobolev space.

Closely related inverse problems have been extensively studied in the steady state case. In

fact, the steady state case has been investigated for the Laplace operator [9, 19] and for

∗Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609. Corre-

sponding author email: darko@wpi.edu.

arXiv:2210.02510v2 [math.AP] 9 Mar 2023

the linear elasticity operator [3, 2, 21, 22, 18, 14, 20, 16, 17, 15]. [9, 15] cover a related

eigenvalue problem derived from stability analysis. In [3], the direct crack problem for half

space elasticity was analyzed under weaker regularity conditions (weaker than H1regular).

In [2], this direct problem was proved to be uniquely solvable in case of piecewise Lipschitz

elasticity coeﬃcients and general elasticity tensors. Both [2] and [3] include a proof of

uniqueness for the related crack (or fault) inverse problem under appropriate assumptions.

[14] and [19] focus on the Lipschitz stability of the reconstruction of cracks based on the

assumption that only planar cracks are admissible. The seismic model introduced in [21]

was used in [22] to address a real life problem in geophysics consisting of identifying a fault

using GPS measurements of surface displacements. [18, 20] feature statistical numerical

methods for the reconstruction of cracks based on a Bayesian approach. In [16], a related

parallel accept/reject sampling algorithm was derived. The numerical method in [17] is

entirely diﬀerent, it is based on deep learning.

Here, we analyze a direct and an inverse problem that generalize the Laplace based case to

the wave case. The waves considered here are time harmonic pressure waves and can be

modeled by an inhomogeneous Helmholtz equation. We can actually model heterogeneous

media by assuming that the wavenumber k2is an L∞function, as long as it is non-negative,

bounded away from zero, and equal to a constant k0outside a bounded set. In section 2, we

prove that the direct pressure wave problem in half space minus a crack is uniquely solvable

and well posed in the space of functions

{v∈H1

loc(R3−\Γ) : v

√1 + r2,∇v

√1 + r2,∂v

∂r −ik0v∈L2(R3−\Γ)},

where Γ is the crack and ris the distance to the origin. The proof of uniqueness for the inverse

problem relies on unique continuation for elliptic equations from Cauchy data. The earliest

such continuation results relied on properties of analytic functions. Later, Nirenberg proved

that for second order PDEs whose leading term is the Laplacian, it suﬃces to assume that

solutions are C1with piecewise continuous second derivatives for the unique continuation

property to hold [13].This result was further improved by Aronszajn et al. where in [1] it was

extended to such PDEs with only Lipschitz coeﬃcients. Unfortunately, demanding Lipschitz

continuity is impractical in applications since it does not even cover the piecewise constant

case. More recently, Barcelo et al. [4] proved a stronger result. In particular, their unique

continuation result implies that a solution to the pressure wave equation (∆ + k2)u= 0 in

an open set of R3satisﬁes the unique continuation property if k2is in L∞

loc(R3). This unique

continuation property will help us show in section 2.2 a uniqueness result for the pressure

wave inverse problem in the half-space {x:x3<0}minus a crack: if Γi,i= 1,2 are two

cracks where the forcing terms gi,i= 1,2 deﬁning pressure discontinuity across Γihave full

support in Γileading to the solution uiof the forward problem, if R3\Γ1∪Γ2is connected,

and the Cauchy data for u1and u2are the same on a relatively open set of the top boundary

{x:x3= 0}then Γ1= Γ2and g1=g2.

In section 3, we show counterexamples where uniqueness for the crack inverse problem fails

if R3\Γ1∪Γ2is not connected. In a ﬁrst class of counterexamples, Γ1∪Γ2is a sphere

and we use the ﬁrst Neumann eigenvalue for the Laplace operator inside an open ball that

is odd about the equator. Such a function is necessarily zero on the equator and thus the

values on the top half sphere can be extended by zero to the lower half sphere without losing

its H1

2character. One might argue that this ﬁrst counterexample is unsatisfactory in the

sense that it involves a geometry that is quite “round”: if nis the normal vector on Γ1

pointing up the range of n·e3is (0,1]. For that reason we provide a second counterexample

where that range can be made arbitrarily narrow, and such that Γ1can be continued into

a plane in a C1regular fashion. Constructing this family of counterexamples requires using

arguments borrowed from the analysis of elliptic PDEs on domains with cusps. To make our

argument easier to follow, this example is ﬁrst constructed in dimension 2, then generalized

to the three dimensional case using cylindrical coordinates. Let Γabe the open curve x2=

a(x1−1)2(x1+ 1)2−2, x2∈(−1,1), where a > 0 is a ﬂattening parameter. In ﬁgure 1

we sketched Γafor a= 1, .25, .05. We also sketched in the same ﬁgure Γ0

a, obtained from

Γaby symmetry about the line x2=−2. We show that for some values of kand some

choice of forcing terms gon Γaand g0and Γ0

a, the half space crack PDE leads to the same

Cauchy data everywhere on the top boundary x2= 0. Next, this geometry is rotated in

three dimensional space, and we show how to construct a counterexample to uniqueness

using rotationally invariant forcing terms g1, g2. The values of g1, g2on a cross-section are

not those from the two dimensional case: slight adjustments have to be made as the volume

element in cylindrical coordinates is rdrdθdx3and the Laplace operator for a function which

is independent of θis 1

∂

∂r r∂

∂r +∂2

x3.

In section 4, our last result states that the pressure wave crack inverse problem is uniquely

solvable within the class Pof Lipschitz open surfaces that are ﬁnite unions of polygons,

except possibly for a discrete set of frequencies. A change of frequency amounts to changing

the wavenumber k2to t2k2, for some t > 0. If Γ1and Γ2are in P, and if tis not in some

discrete set, we show that if the values of the corresponding solutions u1, u2to the pressure

wave crack inverse problem in half space for the wavenumber t2k2, are equal on a relatively

open set of the top boundary then Γ1= Γ2, as long as the jump of uihas full support across

Γi.

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-3

-2.5

-2

-1.5

-1

-0.5

0.5

Figure 1: The shapes Γa(solid curves) and Γ0

a(dashed curves) for three values of a. We

prove that it is impossible to distinguish Γafrom Γ0

aat some wavenumbers and some forcing

terms on Γaand Γ0

afor the half space crack PDE from Cauchy data, even if the Cauchy data

is given everywhere on the top boundary.

2 The direct and inverse crack problems for pressure

waves in half space

2.1 Problem formulation and physical interpretation

Let R3−be the open half space {x= (x1, x2, x3) : x3<0}. Let Γ be a Lipschitz open

surface in R3−. Assume that Γ is strictly included in R3−so that the distance from Γ to

the plane {x3= 0}is positive and let Dbe a domain in R3−with Lipschitz boundary such

that Γ ⊂∂D. The trace theorem (which is also valid in Lispchitz domains, [7, 8]), allows us

to deﬁne an inner and outer trace in H1

2(∂D) of functions deﬁned in R3−\∂D with local

H1regularity. Let ˜

2(Γ) be the space of functions in H1

2(∂D) supported in Γ. Let kbe in

L∞(R3−) such that,

(H1) kis real-valued,

(H2) there is a positive constant kmin such that k≥kmin almost everywhere in R3−,

(H3) there is an R0>0 and a k0>0 such that if |x| ≥ R0, and x∈R3−\Γ, k(x) = k0.

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Onthecrackinverseproblemforpressurewavesinhalf-spaceDarkoVolkov*March13,2023AbstractAfterformulatingthepressurewaveequationinhalf-spaceminusacrackwithazeroNeumannconditiononthetopplane,weintroducearelatedinverseproblem.Thatinverseproblemconsistsofidentifyingthecrackandtheunknownforcingtermonthatcrac...

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On the crack inverse problem for pressure waves in half-space Darko Volkov

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