Exercise 2.1.2 A Nonhyperbolic Cauchy Problem for the Wave Equation)
9.4 Lateral overdetermination: many measurements
9.4 Lateral overdetermination: many measurements
This section is devoted to identification problems when one is given the Neu- mann data (9.0.6) for all (regular) Dirichlet data (9.0.3). In other words, we know the so-called lateral Dirichlet-to-Neumann mapl :g0→g1. We will assume that unknown coefficients are regular; in particular, the principal coefficientsa must be at leastC1. Discontinuous principal coefficients are to be considered in section 9.5.
Theorem 9.4.1. Let a0=1,b=0, and let a be a scalar matrix. Let f =0and u0 =0. Let=∂.
Then the lateral Dirichlet-to-Neumann map l uniquely determines a∈ H2,∞()and c∈ L∞().
PROOF. We will make use of stabilization of solutions of parabolic problems when t → ∞, reducing our inverse parabolic problem to inverse elliptic problems with parameter, where one can use results from Chapter 5.
The substitution
(9.4.1) u=veλt
transforms equation (9.0.1) into the equation
(9.4.2) ∂tv−div(a∇v)+(c+λ)v=0 onQ.
Choosingλlarge, we can achieve that 0≤c+λ. It is clear that the lateral Dirichlet- to-Neumann map forvis given as well.
Letg0 =g0φ, whereg0 is any function inC2() andφ(t)∈C∞(R) satisfies the conditionsφ(t)=0 on (−∞,T/4) andφ(t)=eλt on (T/2,+∞). Since the coefficients of equation (9.4.2) and the lateral boundary condition do not depend on t>T/2, the solutionu(t,x) is analytic with respect tot >T/2 as well as
∂νu(t,x),x∈∂.
Since our equation (9.4.2) satisfies all the conditions of the maximum principle, and its coefficients and the lateral boundary condition do not depend ont >T/2, Theorem 9.6 guarantees that
(9.4.3) v(,t)−v0∞()→0 ast → ∞, wherev0is a solution to the (stationary) Dirichlet problem (9.4.4) −div(a∇v0)+(c+λ)v0 =0 in,v0=g0on∂.
The maximum principle for parabolic equations yields v∞(×R+)≤C, so the local Lp-estimates for parabolic boundary value problems (Theorem 9.1) givev2,1;p(×(s−1,s+1))≤C. Differentiating the equation with re- spect to t and again using local estimates, we derive from the previous bound that ∂tv2,1;p(×(s,s+1))≤C. So trace theorems givev(,t)(2)()≤C. From this estimate and from the estimate (9.4.3), by interpolation theorems it
follows that
(9.4.5) v(,t)−v0(1)()→0 ast → ∞.
Again by trace theorems, we obtain that∂νv(,t)→∂νv0inH(−1/2)(∂) ast →
∞. Since a∂νv(,t) on ∂is given, we conclude thata∂νv0 on∂is given as well. Thus, we are given the Dirichlet-to-Neumann map for the elliptic equation (9.4.4) (for all λ≥ −c). As in Theorem 5.1.1, a and∇a on∂ are uniquely determined.
The known Riccati substitutionv0=a−1/2was in Section 5.2 transforms equa- tion (9.4.4) into the Schr¨odinger equation
(9.4.6) −w+a−1/2(a1/2+a−1/2(c+λ))w =0 in.
By Theorem 5.3.1 (3≤n) and Corollary 5.5.2 (n=2), the coefficientc∗ofw in equation (9.4.6) (for anyλ) is uniquely determined by its Dirichlet-to-Neumann map. Hencea−1/2is uniquely determined as the coefficient ofλinc∗, and therefore cis uniquely determined as well.
The proof is complete.
With the appropriate stabilization theory for equations withL∞() coefficienta the method of the proof of Theorem 9.4.1 and Theorem 5.4.1 will imply uniqueness ofaprovidedb=0,c=0.
The method of proof of this theorem and Theorem 5.5.1 can be used to solve the following exercise.
Exercise 9.4.2. Leta=1 and 3≤n. Show thatluniquely determines the coeffi- cientsa0(x)∈L∞(),curlb(b∈H2,∞()) and 4c+bãb−2 divb,c∈L∞(), in equation (9.0.1) with f =0, provided thatb1,∞() is small.
The stabilization technique also enables one to obtain stability estimates. Let us consider two lateral Dirichlet-to-Neumann mapsl1andl2corresponding to equation (9.0.1) with the coefficientsa0=a01 anda02,a =1,b=0,c=c1and c2that depend only onxandf =0. Letεbe the operator norm ofl1−l2(from L2((0,T);H(1/2)(∂)) intoL2((0,T);H(−1/2)(∂)).
Theorem 9.4.3. Assume that n=3and
(9.4.7) a0j∞,1()+ cj∞,1()<M. Then there are constants C, δ∈(0,1)such that
(9.4.8) a01−a02∞()+ c1−c2∞()<C|lnε|−1/6. PROOF. Using the substitution (9.4.1), we can achieve that 0≤cj+λa0j.
Letg0∈C2(∂),g0(1/2)(∂)<1. We considerg0=g0φ, whereφis as in- troduced in the proof of Theorem 9.3.1. Let u1,u2 be solutions to the initial boundary value problems (9.0.1)–(9.0.3) with the coefficientsa01,a02, . . ., with zero initial condition (9.0.2), and with lateral Dirichlet datag0. By Theorem 9.5,
9.4. Lateral overdetermination: many measurements 277 uj(x,t) has a complex-analytic extension onto the sectorS= {t =t1+i t2:|t2| ≤ (t1−T/2)}of the complext-plane. Moreover, the extension operator is bounded, souj(,t)(1)()≤CeCt1 whent ∈S, withC possibly depending on, φ, and M. By trace theorems,
(9.4.9) c−Ct∂ν(u1−u2)(,t)(−1/2)(∂)≤Cwhent ∈S.
Since the norm of a complex-analytic function with values in a Banach space is a subharmonic function in the plane, we conclude that the function s(t)= lne−Ct∂ν(u1−u2)(−1/2)(∂) is subharmonic on S. Letà(t) be the harmonic measure of the interval IT =(7T/8,T) in S. We recall that à is a harmonic function on S\IT that is continuous in S\IT and is equal to zero on∂S, to 1 on IT, and tends to zero whent goes to infinity. By using conformal mappings onto a standard annulus, it is not difficult to show that à exists. Observe that using the conformal mappingτ1=(t−T/2)2ofSonto the half-plane{0<Rτ1} and then the conformal mapping τ =2/(1+τ1), one concludes thatà (inτ- variables) is harmonic in B1\γ, where B1is the disk{|τ −1|<1}andγ is the subinterval of (0,1) that is the image of (7T/8,T) under the map t→τ. In addition,à=0 on∂B1andà=1 onγ. Sinceàachieves its minimum atτ =0, from the maximum principles for elliptic equations it follows that ∂1à(0)>0, so à(τ)> τ/C when 0< τ for some constantC. Returning tot-variables, we conclude thatt−2/C < à(t) whenT <t.
We claim that
s(t)≤Cà(t) lnε+(1−à(t)) lnCwhent ∈S.
Indeed, whent∈∂S, this follows from inequality (9.4.9). Whent ∈IT, it fol- lows from our definition ofε. Since the right side is harmonic and the left side is subharmonic inS\IT, the maximum principles imply the inequality inS. Taking exponents of both parts of this inequality, we obtain
(9.4.10) ∂ν(u1−u2)(−1/2)(∂≤CeCtεà(t)C1−à(t) ≤C2eCtεt−2/C,T <t, when we use one of the properties ofàand drop theàin the exponent. On the other hand, by Theorem 9.6 we have exponential stabilization of the solutionuj(,t) of our parabolic problem to the solutionuj(,t) of our parabolic problem to the solutionu0j to the steady-state elliptic problem
(9.4.11) −u0j+(cj+λa0j)u0j=0 in,u0j =g0on∂,
which givesuj(,t)−u0j∞()≤Ce−θt, where positiveθdepends only on and the upper bounds on the coefficients of the operator. From this bound and from interior Schauder-type estimates (Theorem 9.1) for the differencesuj−u0j
in the domains ×(, +1) we obtain the bounds of the H(1)()-norms of these differences. By using trace theorems inwe get
(9.4.12) ∂ν(uj(,t)−u0j)(−1/2)(∂)≤Ce−θt.
The triangle inequality for norms yields
∂ν(u01−u02)(−1/2)(∂)
≤ ∂ν(u1(,t)−u2(,t))(−1/2)(∂)+ ∂ν(u1(,t)−u01)(−1/2)(∂) + ∂ν(u2(,t)−u02)(−1/2)(∂)≤C(eCtεt−2/C+e−θt),
where we also have made use of (9.4.10) and (9.4.12). To make the last two terms equal we lett =(−lnε)1/3/(2C). Denoting byε1the norm of the difference of the Dirichlet-to-Neumann operators of equations (9.4.11), we conclude that
ε1≤Ce−(−lnε)1/3/C.
By Theorem 5.2.3 we have
(c2−c1)+λ(a02−a01)∞≤C(−lnε1)−1/2.
By using the bound onε1, we can replaceε1byε, provided that12is changed to 16 andCdenotes possibly a larger constant. From the last estimate with two different λ(say,λ=1 andλ=2) the bound (9.4.8) follows.
The proof is complete.
At present, use of stabilization does not produce uniqueness results when mea- surements are implemented at a part of the boundary (=∂), because such results are not available for elliptic equations. This “local” case can be handled by again using the transform (9.2.1) and the recent method of Belishev for hyperbolic problems. We observe that the transform (9.2.1) applied to the parabolic equation (9.0.1) with time-independent coefficients and Theorem 9.2.1 imply that given the local (corresponding toγ ⊂∂) parabolic Dirichlet-to-Neumann map uniquely determine the local Dirichlet-to-Neumann map for the corresponding hyperbolic equation.
Indeed, the set of functions g0(x)φ∗(t) with g0∈C20(γ) and bounded φ∗ ∈C2(R+), φ∗(0)=φ∗(0)=0, is complete in the subspace of functions in C([0,T∗];H(1)(∂)) that are zero at {t=0}and outsideγ ×(0,T∗). A func- tionφdetermined byφ∗ via (9.2.1) is analytic with respect tot, so a solution to the parabolic problem will be analytic with respect tot as well. Then as above, the Dirichlet data for the hyperbolic problem generate the Dirichlet data for the parabolic problem whose solution is analytic with respect to time, so its Neumann data given on γ×(0,T) determine the Neumann data onγ×R+ and due to uniqueness of the back transform∂νu→∂νu∗uniquely determine the hyperbolic Neumann data for anyT∗. By applying Theorem 8.4.1, we obtain the following corollary.
Corollary 9.4.4. Letγ be any nonempty open part of∂and let the parabolic equation(9.0.1)have coefficients a=1,b=0,c=0, f =0.
Then the diffusion coefficient a0 ∈C∞()is uniquely determined by the local parabolic Dirichlet-to Neumann mapl :g0→g1onγ×(0,T),suppg⊂γ× (0,T).