Exercise 2.1.2 A Nonhyperbolic Cauchy Problem for the Wave Equation)
4.2 Reconstruction of lower-order terms
f2()≤C(g0(3/2)(∂)+ g1(1/2)(∂)).
{Hint: By using extension theorems find a function v bounded by the given norms of the boundary value data foru that has the same boundary data asu, subtractvfromuand repeat the uniqueness proof with (Au,Av)2() in the right side of (4.1.9) instead of zero, and make use of the Schwarz inequality.}
Existence and stability of the solution (u, f) to the inverse source problem in case (4.1.5), (4.1.8) follows from the relations
A2u =0 in, u=g0, ∂νu =g1on∂,
which constitute the first boundary value problem for the fourth-order elliptic equation. This problem is known to be Fredholm, so uniqueness of its solution implies its existence and Lipschitz stability estimate in both the H¨older and Sobolev classes. We refer to the book of Morrey [Mor], Theorems 6.5.3, 6.4.8.
The elliptic equation A(U−ε)u =0 (with small positiveε) can be used for finding an approximation fεof a solution f when some of theεj jare zero.
4.2 Reconstruction of lower-order terms
Now we give an example of an inverse problem that is well-posed. We consider equation (4.0.1) with coefficients that do not depend onxnin the domain=G× (−H,0), whereGis a bounded domain inRn−1withC2+λ-boundary (0< λ <1) andHis a positive number. We assume thatAu= Au−∂2u/∂xn2, whereAdoes not involve derivatives with respect toxn.
We let=G× {0}, and we are looking for the coefficientc≥0 of equation (4.0.1) whose solutionusatisfies the Dirichlet boundary data (4.0.2) and the addi- tional Neumann data (4.0.6). Letujbe a solution to the Dirichlet problem (4.0.1), (4.0.2) withc=cj andg1,j the corresponding Neumann data (4.0.6), j =1,2.
We assume thatg0∈C2+λ(∂), 0≤g0, on∂;g0=0 onG× {−H};∂ng0≥0 and∂2g0/∂xn2=0 on∂G× {−H}; 0≤ Ag0 on; and Ag0 =0 on∂. About the coefficients ofAwe assume thata,b∈C1+λ();c∈Cλ().
Observe that the elliptic theory briefly described by Theorem 4.1 guarantees uniqueness and existence of a solutionu ∈C2+λ(). Indeed, we have a unique solutionu∈ H(1)() that is inC2+λ(\V), whereV is some neighborhood of the corner points∂× {0,−H}. To show thatuisC2+λ-smooth innear× {−H} one can use the even extension with respect to (xn+H), obtaining anH(1)-solution in the extended domain, and then apply the local regularity claim of Theorem 4.1. To show smoothness near× {0}one can solve the Dirichlet problem for (smoothly extended)Ain aC2+λ-domain containing, withC2+λ-extended data g0; subtract thisC2+λ()-solution fromu; and apply to the difference the odd extension with respect toxntogether with the previous argument.
Theorem 4.2.1. Under the given assumptions on g0 the following stability esti- mate holds:
|c2−c1|λ(G)≤C|g1,2−g1,1|1+λ(), (4.2.1)
where C depends only on|cj|λ(G),|a|1+λ(G),|b|λ(G),|g0|2+λ(∂), and on the ellipticity constant of A.
In particular, we have uniqueness of recovery ofcfrom the additional Neumann data.
This result follows from uniqueness and stability in the inverse source problem (A+c2)u=αf, ∂nf =0 in,
u =0 on∂,
∂νu =g1on, (4.2.2)
whereα=u1and f =c1−c2. The relations (4.2.2) are the results of subtraction of the equations foru1from the equations foru2. It is interesting to observe that nowαdepends on the boundary datag0for the direct problem, so unlike the inverse source problem with fixedα, by changingg0we obtain more information about f. This observation will be essential in Chapter 5, which is devoted to many boundary measurements.
The idea of the proof of (4.2.1) given in the paper of Khaidarov [Kh1] is first to establish Schauder-type estimates for the inverse source problem (4.2.2). This can be done by “freezing coefficients,” differentiating the equation with respect toxn, and applying standard elliptic estimates. We explain this idea referring for details to the cited paper.
Letx0 ∈andεsmall and positive. We write equation (4.2.2) in∩B(x0; 2ε) asα−1A0u =α−1(A0−(A+c2))u+ f, whereA0is the operatorAwith the co- efficients at the pointx0. Differentiating this equation gives the following boundary value problem:
∂n(α−1A0u)=∂n(α−1(A0−A−c2))uin∩B(x0; 2ε), ∂nu
=g1on∩B(x0; 2ε).
Schauder-type estimates of Theorem 4.1 for equations in variational form give
|∂nu|1+λ(∩B(x0;ε))≤C(|α−1(A0−A)u|λ(∩B(x0; 2ε))+ |g1|1+λ() + |u|1+λ(∩B(x0; 2ε))
≤C(ελ|u|2+λ()+ |g1|1+λ()+C(ε)|u|0()).
Using this estimate and the Dirichlet boundary condition onfrom relation (4.2.2) on, we then obtain that|f|λ(∩B(x0;ε)) is bounded by the right side of this inequality. Since this is true for any x0∈and does not depend on xn on the whole ofwe bound the norm of f on. Hence the Schauder-type estimates for the problem (4.2.2) bound by the same quantity the norm|u|2+λ(). Choosing ε small we eliminate the term with|u|2+λ() on the right side and obtain the
4.2. Reconstruction of lower-order terms 99 Schauder-type estimate|u|2+λ()≤C(|g1|1+λ()+ |u|0()). The term|u|0() can be eliminated by using compactness-uniqueness arguments as in the proof of Theorem 3.4.11, provided that we have uniqueness of a solution.
Under the assumptions that the coefficients of A are xn-independent and the weight functionαis monotone with respect toxn, one proves uniqueness in the inverse source problem by a modification of Novikov’s orthogonality method, which we describe briefly below. We will setH=× {−H},0=∂\(∪ H).
Lemma 4.2.2. If the coefficients of A do not depend on xn, and∂nα >0on, then f =0on.
PROOF. We multiply equation (4.2.2) byv∈ H(2)() and apply Green’s formula.
We obtain
∪0∪H
(∂ν(A)vu−v∂ν(A)u) d+
u A∗v=
αf v.
If A∗v=0 onandv=0 on0, then using the boundary conditions (4.2.2) whereh=0, we yield
αf v = −
H
v∂νu.
By using the odd extension with respect toxnwe conclude that finite differences (which satisfy the same equation asvbecause the coefficients arexn-independent) of v with respect toxn can be used instead of v. Passing to the limit in finite differences, we conclude that we can replacevby∂nv. When in addition,∂nv=0 onH, we conclude that the integral ofαf∂nvoveris zero. Integrating by parts with respect toxn, we obtain
vαf −
v f∂nα=0.
(4.2.3)
Intending to show as in the proof of Theorems 4.1.1 and 4.1.6 that the left side is positive for somev, we introduce the set+= {f >0}and the set−= {f <0}.
Since f does not depend on xn, these sets are cylinders +×(−H,0), −× (−H,0). We can assume that both+and−have nonvoid interiors; otherwise, αf is nonnegative (or nonpositive), and we will obtain a contradiction by using Giraud’s maximum principle on.
Letvbe a solution to the mixed problem
A∗v=0 in, v=0 on0, ∂νv=0 onH, v=g0on.
Let us assume thatg0∈C02(),0≤g0 ≤1, on, andg0=0 on an open subset 0−of−that does not depend ong0. By using comparison with a fixed solution to the same boundary value problem one can show thatv(x)≤1−ε(x), whereε is a positive continuous function onthat does not depend ong0. Indeed, letv0be the solution to the above mixed problem for the operatorA∗inwith boundary
datag00∈C2() that are 1 outside0−and 0 on an open nonvoid subset of0−. As shown above, this solution exists and is inC1( ¯). By the maximum principles, 0<v0(x)<1 on. Again by maximum principles (in the form of comparison), v≤v0, so we can letε(x)=1−v0(x). By maximum principles we also have 0≤von. Since∂nα >0 on, the left side of (4.2.3) is not less than
gαf −
+
(1−ε)f∂nα.
We can approximate inL1() by suchg0the function that is 1 on+and zero on
\+, and conclude that 0≥
+
αf −
+
f∂nα+
+
εf∂nα=
+
εf ∂nα >0.
In the last equality we used that the integral over + combined with the first integral over+is zero becauseα=u1=0 onHdue to the Dirichlet boundary condition. The contradiction shows that the initial assumption was wrong, and so
f =0.
The proof is complete.
Theorem 4.2.1 will follow from Schauder estimates and from Lemma 4.2.2 if we show that∂nα=∂nu1>0 on. Differentiating the equation foru1with respect toxn and using that the coefficients ofAand the coefficientc1do not depend on xn, we obtain the equation (A+c1)∂nu1=0 in. Differentiating the boundary condition on∂G×(−H,0) gives∂nu1 =∂ng0 ≥0 there. OnG× {−H}we have from the differential equation,
∂n(∂nu1)=Au1+c1u1=Ag0+c1g0=0,
and similarly, on G× {0},∂n(∂nu1)≥0, due to the conditions on g andc1. By applying maximum principles, we conclude that ∂nu1 >0 in . Indeed, ∂nu1
attains its minimum on ¯. If this minimum is negative, then it cannot be achieved either insideor onG× {−H},∂G×(−H,0). So it must be achieved at some pointG× {0}. At this point, by Giraud’s maximum principle (Theorem 4.2), we have∂n(∂nu1)<0, which contradicts the properties of this function.
A similar problem for parabolic equations (with final overdetermination) is discussed in more detail in Section 9.1, where there are some existence results and monotone iterative algorithms suitable also for numerical solution of the inverse problem. Similar results can be obtained for the elliptic inverse problem under consideration.
In this problem the unknown term does not depend on one of the space variables, which is not quite natural in many applications.
If one has additional boundary data onG× {−H}, it is possible to recover in a stable way two terms (e.g.,aand the right side f, which do not depend onxn).
For proofs and further information we refer to the paper of Khaidarov [Kh1].
Whenc(and other coefficients of A) does not depend onx1, and∂G∩V ⊂ {x1=0}, whereVis a ball centered at a point of∂G, then uniqueness ofcon the set
4.2. Reconstruction of lower-order terms 101 {x∈: (0,x2, . . . ,xn−1)∈V}can be obtained by the method of Carleman-type estimates originated by Bukhgeim and Klibanov [BuK]. The essential condition is thatg0≥0 on∂and>0 on∩V. This inverse problem is not well-posed, in contrast to the problem withxn-independentc. One can obtain only conditional H¨older-type stability estimates as in the Cauchy problem for elliptic equations. Due to space limitations we will not discuss this here in detail, referring to the review paper of Klibanov [K1] and to the recent book [KlT]. For hyperbolic equations the method of Carleman estimates is demonstrated in Section 8.2.
Identification of the coefficient c is of interest for theory of semiconductor devices. We refer to more detail to the paper of Burger, Engl, Markovich, and Pietra [BuEMP]. One of accepted mathematical models for semiconductors is represented by a system of three quasilinear elliptic partial differential equations of second order. In particular, it is of interest [BuEMP], p.1777, to identify the so-called doping profilec∗∈ L∞() from the elliptic equation di v(a∇v)=0 in with c∗= −loga+a−a−1. The well known substitution v=a−1/2u transforms the equation forvinto−u+cu =0 withc=a−1/2a1/2. The additional data for determiningccan be few sets of the Cauchy foruor all possible Cauchy data on a part of∂.
The coefficientcwhich is a function of all space variables can not be uniquely determined from few sets of the boundary data. The assumiption thatcdoes not depend on one of space variables leads to a nice theory described above, but it is artificial in many applications. Another option ( piecewise constant doping profiles of importance for semiconductors theory) is to consider the equation
−u+kχ(D)u =0 in , (4.2.4)
where a domain D is to be found from few sets of the boundary Cauchy data for u. Global uniqueness results for this inverse problem are still not available.
Linearizations aroundk=0 lead to the inverse source problem with the unknown source term u0χ(D) whereu0 is the harmonic function,u0=u on∂. For an appropriate choice of the Dirichlet data foru one can derive global uniqueness results forDwithin star-shaped orxn-convex domains. WhenDis close to a fixed domainD0, one can obtain uniqueness results by using a device from the theory of variational inequalities like in [Is4], Corollary 5.1.4.
Turning to inverse problems for nonlinear elliptic equations
−u+c(u)=0 in, (4.2.5)
we observe that at present there are only local uniqueness results for (small)c when in addition to the Dirichlet datag0we prescribe the Neumann data on∂ as well. We refer to the paper of Pilant and Rundell [PiR2], where an essential assumption is thatg0is in a certain sense monotone on∂.
However, there is an important inverse problem for equation (4.2.5) originating in magnetohydrodynamics whereuis constant on∂. Equation (4.2.5) with these boundary data describes the plasma in equilibrium. This inverse problem was studied by Beretta and Vogelius [BerVo]. For natural configurations (is a ball or a torus) there are examples of nonuniqueness. Indeed, whenis a ball known results
for nonlinear elliptic equations imply thatu depends only on the distance to the center of the ball. In resulting ordinary differential equation (in polar coordinates) one can not find a functionc(u) from one additional number ( the derivative of solution at the endpoint) even assuming thatcis analytic. Uniqueness is shown in [BerVo] for analyticcwhenhas an analytic corner whose interior angle is not π/2 orπ. Their proof consists in demonstrating that compatibility conditions and additional Neumann data determine all derivatives ofcat a corner point. For recent results about this problem and for another approach based on complex variables we refer to Demidov and Moussaoui [DM].