Exercise 2.1.2 A Nonhyperbolic Cauchy Problem for the Wave Equation)
5.8 Maxwell’s and elasticity systems
Recently, uniqueness results for the Schr¨odinger equation were generalized onto two classical systems of mathematical physics, which was not a simple task because the available proofs for the scalar equations made a substantial use of substitution (5.2.1). This substitution relies on commutativity properties of scalar differential operator that are not valid for most matrix differential operators, in particular for the classical elasticity system. Maxwell’s system is not elliptic, so there are additional difficulties in this case.
The stationary electromagnetic field (E,H) of frequencyωin the mediumof permittivity, conductivityσ, and magnetic permeabilityàsatisfies Maxwell’s equations
curlE=iωàH, curlH= −iω(+iσ/ω)Ein. (5.8.1)
We assume that, à, andσ are inC3()∩C(), all of them are nonnegative, and, àare strictly positive on. It is known that with the exception of some discrete set of values of ω with accumulation point at infinity, for any func- tion g0∈T H(1/2)(∂) (the space of tangential vector fields on∂ with com- ponents inH(1/2)(∂)) there is a unique (weak) solution (E,H)∈L2()×L2() to Maxwell’s system with prescribed tangential componentγτE=g0on∂, so we have the well-defined map:g0→γτHfromT H(1/2)(∂) intoT H(−1/2)(∂).
We refer for proofs and definitions to the paper of Ola, P¨aiv¨arinta, and Somersalo [OPS].
Theorem 5.8.1. Assume that a frequencyωis not in the exceptional set. Assume that, àare constants0, à0on∂, andσis zero on∂.
Then the operatoruniquely determines the coefficients, σ, andà.
This result in an important particular case has been obtained by Colton and P¨aiv¨arinta [CoP] and in the general case by Ola, P¨aiv¨arinta, and Somersalo [OPS].
Their proofs are quite ingenious. They are based on appropriate orthogonality relations that claim thatuniquely determines the integrals
I =
((à−à0)HãH0−(+iσ/ω−0)EãE0)
for all solutionsE,Hto Maxwell’s system and all solutionsE0,H0 to this sys- tem withà=à0, =0, andσ =0 with given boundary data. The next step is constructing the special almost exponential solutions
E=eiζã(ζ(E)+W(E)), H=eiζã(ζ(H)+W(H))
similar to functions (5.3.3) containing a large parameter (“frequency”)|ζ|, which is a delicate part since our system is not elliptic. We assume that
ζ ∧ζ(E)=ωà0ζ(H), ζ∧ζ(H)= −ω0ζ(E), ζãζ =k2 withk=ωà00.
5.8. Maxwell’s and elasticity systems 167 ThenEandHwith W(E)=W(H)=0 solve Maxwell’s system with constant parametersà=à0, . . ..
LetG be a fundamental solution for the operator−−k2,α= ∇lnγ, β=
∇lnà, γ =+iσ/ω. Then one can show that a solution (E,H) to the (vector) equation
E H
=eiζã
ζ(E) ζ(H)
+G
k2γ•E+ ∇αãE+iωà0∇ ∧(à•H) k2à•H+ ∇βãH−iω0∇ ∧(γ•E)
withγ•=(γ−0)/0, à•=(à−à0)/à0is a solution to Maxwell’s system. This equation can be written as
E H
=eiζã ζ(E)
ζ(H)
+G(V +S) E
H
, (5.8.2)
where V =
ω2(àγ−à00)+t(∇α) iω/γ∇(àγ)∧
−iω/à∇(àγ)∧ ω2(àγ−à00)+t(∇β)
and
S=
αã ∇ 0 0 βã ∇
.
to solve equation (5.8.2) by contraction arguments as in Section 5.3 it is crucial to use the following matrix identity discovered by Colton and P¨aiv¨arinta:
[,M]=M(S+Q) with
M =
γ1/2 0 0 à1/2
, Q=
(γ1/2)/γ1/2 0 0 (à1/2)/à1/2
. By using this identity and applying the operators−(+k2) andMto (5.8.2), one obtains the following analogue to equation (5.3.11):
W(E) W(H)
=(M−1M0−I) ζ(E)
ζ(H)
+M−1GζM(V −Q)
W(E) W(H)
, (5.8.3)
whereGζ is the regular fundamental solution for the operator−(+2iζ ã +k2), which can be solved by a contraction argument like that in Section 5.3.
Choosingζ =(τ,i(τ2+R2)1/2,(R2+k2)1/2) and
ζ(E)=(1,1,−(ζ1+ζ2)/ζ3), ζ(H)=(ωà0)−1ζ ∧ζ(E),
we satisfy all the conditions above. We define the free space solutionsE0,H0, replacingζ, ζ(E), ζ(H) in the formulae forE,Hby
ζ0 =τ∗−ζ, ζ0(E)=(1,−1,(ζ1+ζ2)/ζ3), ζ0(H)=(ωà0)−1ζ0ζ0(E) and lettingW =0.
Finally, studying the asymptotic behavior of I as the parameter R goes to
∞ by means of equation (5.8.3), one obtains the partial differential equation u+F(u,v)=pu. Then interchanging ζ(E) and ζ(H) one gets the equa- tion v+F(v,u)=qv. Here u =(à/à0)1/2 and v=(+iσ/k)1/2, where F(x,y)=C(1/2−x3y−x2/2+x/y+x) and p,q are functions determined by the data of the inverse problem. Moreover,u,v are given outside , so by applying uniqueness of continuation for elliptic equations we conclude that they are uniquely determined inside.
This outline can only illuminate the proof but not explain it in sufficient detail given in the original paper [OPS] or in the paper [OS].
For second order systems the situation is more complicated. Eskin [Es1] obtained necessary and sufficient conditions for elliptic matrix equations with the diagonal principal part−:
−u+B∇u+Cu=0 in (5.8.4)
whereu=(u1, . . . ,um),B=(B1, . . . ,Bn),Bj arem×m-matrices with entries inC∞(),Cis am×mmatrix with entries inL∞(). The pair (B,C) is gauge equivalent to the pair (B•,C•) if there is am×minvertible matrixG∈C∞() such that
A•=G−1AG+1/2G−1∇G, (1/4B2j,•−1/2∂jBj,•)+C•)=G−1(
(1/4B2j−1/2∂jBj)+C)Gon where the sums are over j =1, . . . ,n. We will assume that the Dirichlet problem for the system (5.8.4) with the datau=g0 on∂ has a unique solution. It is known that this is true for almost all coefficients and it is not hard to give sufficient conditions by assuming some positivity of matricesB,Clike in [LU]. Then we have a well-defined Dirichlet-to Neumann map:g0→∂νuon∂∈Li pwhich is a continuos linear operator fromH(1/2)(∂) intoH(−1/2)(∂).
Theorem 5.8.2. Let n≥3. Let be a ball in Rn. Let matrix-functionsB,C, B•,C• have supports in . Let , • be the Dirichlet-to-Neumann operators corrsponding to these sets of coefficients.
If=•, then the the matrix coefficients(B,C)are gauge equivalent to the marix coefficients(B•,C•).
Now we consider the stationary elasticity system Aeiu=
∂j(ci j klεkl)=0 in (the sum is over j,k,l =1, . . . ,n), (5.8.5)
whereεkl=1/2(∂luk+∂kul) is the linear strain andci j klis the elastic tensor with C∞()-components, they obtained uniqueness at∂of this tensor in the case of the classical elasticity (ci j kl =λδi kδjl+à(δi kδjl+δilδj k),δi jis the Kronecker delta). The natural analogue of the Dirichlet-to-Neumann map λ,à maps the
5.8. Maxwell’s and elasticity systems 169 displacement vectoru=(u1, . . . ,un)=gat the boundary to the stress
i(g0)=
νjci j kl2−1εkl (the sum over j,k,l=1, . . . ,n)
at the boundary. The boundary reconstruction is considered in [NaU1], [NaU2].
Partial uniqueness results for determination of elastic parameters insidefrom all boundary observations were obtained by Eskin and Ralston [ER4] and Nakamura and Uhlmann [NaU3]. In the multidimensional case currently the following result is available.
Letn ≥3. Let the Lame parametersλ, à∈C∞() satisfy the following strong convexity assumption:
à >0,nλ+2à >0 on. Let|∇à| ≤ε0for some small positiveε0.
Then the elastic Dirichlet-to-Neumann mapλ,àuniquely determinesλandà in.
Proofs are based on the following orthogonality relations for two possible solu- tionsλ1, à1, λ2, à2of the inverse problem:
((λ1−λ2) divu1ãdivu2+2(à1−à2)(ε(u1)ãε(u2)))=0 (5.8.6)
for all solutionsujto the elasticity systems (5.8.5) withλ=λj, à=àjand further use of almost exponential solutions as in Section 5.3. However, one cannot reduce the classical elasticity system to a diagonal operator with constant coefficients plus zero-order operator and to construct almost complex exponential solutions some smallness assumtions are needed. The first step is to form a fourth-order system by multiplying A3from the right by the special matrix Ae,co (a second-order matrix operator) of “cofactors” ofAeto obtain
A=(à(λ+2à))−1AeAe,co=2+A1+A2,
whereAjis a (matrix) differential operator of jth order. Then one looks for solu- tionseiζãV, so one is using the operatorsAζ =e−iζãAeiζã. By using the pseudodif- ferential operatorPζ with the symbol (|ξ|2+ |ζ|2), we reduceAζto a second-order pseudodifferential operator
A•ζ =•2ζ +Bζ•ζ +Cζ
with•ζ =Pζ−1ζ and zero-order pseudodifferential operatorsBζ,Cζ.
A crucial step is a diagonalization of this operator by introducing a new extended vector-functionV∗=t (V, •ζV) satisfying the first-order systemA∗ζV∗=0 with A∗ζ =•ζ +A∗0, A∗ a zero-order 6×6-matrix differential operator. It turns out that A∗ζ is completely diagonalizable:
A∗ζA0ζ =Bζ0•ζ (modulo smoothing operators)
for some zero-order pseudodifferential operators A0ζ,Bζ0. Moreover, all these op- erators are computable when|ζ| → ∞. Finally, our approximate solutions will
be of the form Ae,co(eζãV∞(v)), wherev solves a diagonal system with constant coefficients and the operatorV∞is some standard fundamental solution.
Calculating the limit of the integrals (5.8.6) as|ζ| → ∞and using the inverse Fourier transform, one gets second-order hyperbolic equations forλandàwith respect to space variables. Since the Cauchy data of the Lame parameters on∂ can be found by boundary reconstruction, these equations uniquely determineλ andà.
A version of the orthogonality relations (5.8.6) can be used to estimate size of an inclusion with different elastic properties as for inverse conductivity problem in section 4.6. We refer to the review paper of Alessandrini, Morassi, and Rosset in [I3].