Singular solutions for complex second order elliptic equations and their application to time-harmonic diffuse optical tomography

We construct singular solutions of a complex elliptic equation of second order, having an isolated singularity of any order. In particular, we extend results obtained for the real partial differential equation in divergence form by Alessandrini in 1990. Our solutions can be applied to the determination of the optical properties of an anisotropic medium in time-harmonic Diffuse Optical Tomography (DOT).


Introduction
In this paper we construct singular solutions to the complex elliptic equation Lu = −div (K∇u) + qu = 0, ( with an isolated singularity of an arbitrary high order.The study of singular solutions of elliptic equations having isolated singularities goes back to the 1950's and 60's with the works of John [10], Bers [3], Gilbarg and Serrin [6] and Marcus [12].In 1990 Alessandrini [1] constructed singular solutions with an isolated singularity of an arbitrary high order for a real equation of type (1.1), with q = 0 and K ∈ W 1,p (Ω) in a ball B ⊂ R n , p > n.Such solutions were applied to the so-called Calderòn's inverse conductivity problem for the stable determination of the conductivity K from the so-called Dirichlet-to-Neumann (D-N) boundary map Λ K in some domain Ω ⊂ R n , with n ≥ 2 (see the review papers [4], [13] on this inverse problem).We also recall the paper of Isakov [11], who employed singular solutions to determine (the discontinuities of) the conductivity K in (1.1) from Λ K .Here we extend the construction of singular solutions in [1] to the complex equation (1.1).A partial extension to the complex case, where K was assumed to be a multiple of the identity matrix I near the boundary ∂Ω, was obtained in [5].In the current work we completely remove this assumption for K near the boundary.The application we have in mind is the time-harmonic Diffuse Optical Tomography (DOT) in the presence of anisotropy up to the surface of the medium Ω under consideration.DOT is a minimally invasive imaging modality with significant potentials in medical applications.An object of unknown internal optical properties is illuminated with a near-infrared light through its boundary and the scattered and transmitted light is measured ideally everywhere on the boundary.The optical properties of the material (the absorption and scattering coefficients) are then estimated from this boundary information ( [2], [7], [8], [9]), typically encoded in a boundary map, e.g. the D-N map (see definition 3.1).In [5], stability estimates of the derivatives (of any order) of the absorption coefficient at the boundary were established in terms of the D-N map corresponding to (1.1), under the assumption that the medium Ω under inspection was isotropic (K = aI, with a scalar function) near its boundary.This limitation in the anisotropic structure of K was needed for the construction of singular solutions to the complex (1.1) with isolated singularities y 0 , such that K(y 0 ) = a(y 0 )I.Here we remove this assumption and extend the construction of singular solutions to the complex (1.1) with no restriction on K near its singularity y 0 .This allows for the extension of the stability of the derivatives of the absorption coefficient in [5] to the full anisotropic DOT case in the low frequencies regime, which suits the DOT experiment we have in mind.
The paper is organised as follows.In section 2, we rigorously formulate the problem, state and prove our main result (Theorem 2.2) on the construction of singular solutions to (1.1) with complex coefficients.A crucial step for the machinery of the proof of Theorem 2.2 is Claim 2.1.Section 3 is devoted to the application of Theorem 2.2 to the DOT problem.We show that by feeding the singular solutions constructed in section 2 into the so-called Alessandrini Identity (3.7), Hölder type stability estimates of the derivatives (of any order) of the absorption coefficient of the medium in terms of the D-N map (see definition 3.1) can be established at the boundary, therefore extending results in [5] to the full anisotropic DOT problem.

Formulation of the problem and main result
For n ≥ 3, a point x ∈ R n will be denoted by x = (x ′ , x n ), where x ′ ∈ R n−1 and x n ∈ R.Moreover, given a point x ∈ R n , we will denote with B r (x), B ′ r (x ′ ) the open balls in R n , R n−1 , centred at x and x ′ respectively with radius r and by Q r (x) the cylinder . We consider the operator L introduced in (1.1) on B R (y), y ∈ R n .K and q are a complexsymmetric, matrix-valued and complex-valued function respectively.Setting K = K R + iK I , we assume that K R K I = K I K R , as this fits the application in section 3, and that there are positive constants λ, Λ, E and p > n such that Setting q = q R + iq I , (1.1) is equivalent to the system for u = (u 1 , u 2 ) where 2) imply that system (2.5) satisfies the strong ellipticity condition Below we introduce some notation adopted through this paper: 1. C y,r = {x ∈ R n | r < |x − y| < 2r}, for y ∈ R n and r > 0.
2. K −1 (n) (y), K −1 nn (y) denote the last row, last entry in the last row of matrix K −1 (y), respectively, for y ∈ R n .

zx,y :=
5. C will always denote a positive constant, which may vary throughout the paper.
6.For z ∈ C, we denote by ℜz and ℑz the real and imaginary part of z, respectively.
Next, we state and prove our main result.
Proof of Theorem 2.2.
is the fundamental solution, with a pole at y 0 , of it was shown in [5], via an inductive argument, that m (z x,y 0 ). (2.11) Note that L 0 u m = 0, in R n \ {y 0 }.To find w solution to satisfying (2.8), (2.9), we estimate the L p -norm of the right hand side of (2.12) as (see [5] for details), where C depends n, p, R, λ, Λ, and E.
We start by showing that there exists To prove (2.14), we need the following result, which is crucial in the machinery of our complex singular solutions' construction, as it extends results in [ [1]).Therefore, by the Bernstein-Walsh lemma [14, Section 4.6], for any R 0 > 1, we have (2.17) 3), there exists R 0 > 1, with R 0 depending on λ, Λ, such that for any x, y ∈ R n \ {0} as the term on the left hand side of inequality (2.18) is bounded, and where C > 0 is a constant constant depending on λ, Λ.Hence, for |y| < C/R 0 |x|, the fundamental solution of L 0 defined in (2.10) with y 0 = y, has the expansion where P j (x, y) y) is a fundamental solution for L 0 as well.To show this, it is enough to prove that L 0 P j = 0, for any j.With the linear change of variables J : C n → C n , defined by x −→ x := Jx, where J is the complex-symmetric and invertible matrix such that JJ = K −1 (y), we have Defining the open and connected set and denoting by U y its left-hand projection to C n , for any y ∈ C n , with ℜ(ỹ • ỹ) > 0, we have R n \ {0} ⊂ U y and J(R n \ {0}) ⊂ U y , where the latter inclusion is due to the ellipticity assumption (2.2).For x, y ∈ R n \{0}, the right-hand side of (2.21) is zero (see [1,Lemma 2.3]).We also have that for any y ∈ C n , with ℜ(ỹ • ỹ) > 0, the expression on the right-hand side of (2.21) is analytic in x on U y , therefore by analytic continuation, the right hand side of (2.21) is zero also on U y , hence L 0 P j = 0, for any j.
We define where C depends only on n, λ, Λ, and s, concluding the proof of the claim.

Application to anisotropic time-harmonic Diffuse Optical Tomography
We consider the inverse problem in Diffuse Optical Tomography of determining the absorption coefficient in an anisotropic medium Ω, represented by a domain in R n , for n ≥ 3. We assume that the boundary of Ω, ∂Ω, is of Lipschitz class with constants r 0 , L > 0, i.e. that for any P ∈ ∂Ω there exists a rigid transformation of coordinates under which we have P = 0 and Ω ∩ , where ϕ is a Lipschitz function on B ′ r 0 satisfying ϕ(0) = 0 and ϕ C 0,1 (B ′ r 0 ) ≤ Lr 0 (see for example [5]).
Under the so-called diffusion approximation ([2], [8]), Ω is interrogated with an input field that is modulated with a fixed harmonic frequency ω = k c , where c is the speed of light and k is the wave number.In this setting, in the anisotropic case, for a fixed k, the photon density u in Ω solves (1.1) with K , q defined by for any x ∈ Ω, (3.1) respectively.Here I denotes the n × n identity matrix, B (encompassing the anisotropy of Ω) is known and I − B is positive definite ( [2], [8]).µ a and µ s (the optical properties of Ω) are the absorption and scattering coefficients respectively.Assuming that there are positive constants λ, E, E and p > n such that λ −1 µ s (x), µ a (x) ≤ λ, for a.e x ∈ Ω, ( K, q in (3.1) satisfy (2.1)-(2.4) on Ω, hence the resulting system in (2.5) satisfies the strong ellipticity condition (2.6) on Ω.Here we assume that µ s is known and we address the inverse problem of determining µ a from the Dirichlet-to-Neumann boundary map defined as follows.We denote K , q with K µa , q µq respectively, to emphasise their dependence on the unknown µ a .for any f , g ∈ H 1 2 (∂Ω), where u ∈ H 1 (Ω) is the weak solution to −div(K µa (x)∇u(x)) + (µ a − ik)(x)u(x) = 0 in Ω, u| ∂Ω = f in the trace sense and ϕ ∈ H 1 (Ω) is any function such that ϕ| ∂Ω = g in the trace sense.