On an inverse boundary value problem

On an inverse boundary value problem^*

Alberto P. Calderón

Department of Mathematics, The University of Chicago

ABSTRACT

This paper is a reprint of the original work by A. P. Calderón published by the Brazilian Mathematical Society (SBM) in ATAS of SBM (Rio de Janeiro), pp. 65-73, 1980. The original paper had no abstract, so this reprint to be truthful to the original work is published with no abstract.

Mathematical subject classification: 35J25, 35Q60, 47F05, 86A20, 86A22.

Key words: inverse problems, boundary value problems, identification problem, elliptic equations.

In this note we shall discuss the following problem. Let D be a boundedDomain in ⁿ, n > 2, with Lipschitzian boundary dD, and g be a real bounded measurable function in D with a positive lower bound. Consider the differential operator

L_g (w) = Ñ . (g Ñ w)

acting on function of H¹(D) and the quadratic form Q_g(f) , where the functions in H¹(ⁿ) , defined by

The problem is then to decide whether g is uniquely determined by Q_g and to calculate g in terms Q_g, if g is indeedDetermined by Q_g.

This problem originates in the following problem of electrical prospection. If D represents an inhomogeneous conducting body with electrical conductivity g, determine g by means of direct current steady state electrical measurements carried out on the surface of D, that is without penetrating D. In this physical situation Q_g(f) represents the power necessary to maintain an electrical potential g in ¶D.

In principle Q_g can be determined through measurements effected on dD and contains all information about g which can be thus obtained.

But let us return to our mathematical problem. Let us introduce the following norms in the space of functions g on dD and in the space of quadratic forms Q(f)

Then the mapping

F : g ® Q_g

is bounded and analytic in the subset of L^¥(D) consisting of functions which are real and have a positive lower bound. Our goal is then to determine whether F is injective, and invert F if this is the case. This we are yet unable to do and is, as far as we know, an open problem. However we shall show that dF|_{g = const} is indeed injective, that is, the linearized problem has an affirmative answer. If dF|_{g = const}, which is a linear operator, had a closer range, one could conclude that F itself is injective in a sufficiently small neighborhood of g = const. But the range of dF is not closed and the desired conclusion cannot be obtained in this fashion. Nevertheless, as we shall see below, if g is sufficiently close to a constant, it is nearly determined by Q_g and in some cases it can be calculated with an error much smaller than ||g - const||L^¥.

To show this let us first obtain an expression for the solution of the equation

L_g (W) = Ñ . (g Ñ W), W|dD = f. g = 1+d

Let W = u + v, where Du = L₁u = 0, u|_dD = f. Then

L_g(W) = L_1+g (u+v) = L₁v + L_gv + L_gu = 0

Since u|_dD = W|dD we have v|_dD = 0 and v Î (D), the closure in H¹(ⁿ) of functions of C^¥ with support in D. But L₁, as an operator from (D) into H^-1(D), has a bounded inverse G, and from the last expression we obtain

v + GL_dv = - GL_du

and

Since for W Î (D), if A denotes the norm of G, the series above will converge for A < 1 and

From (1) it follows that f is analytic at g = 1. The same argument would show that f is analytic at any other point g.

Next let us calculate d f|_{g = 1}. We have

The contribution of the second term in the integrand of the last integral vanishes on account of the fact that Du = 0. Furthermore, from (1) one sees readily that the parts linear in d of the last two terms in the integrand vanish. Thus setting d g = d we obtain

To show that d Q_g(f) |_{g = 1} is injective, we merely have to show that if the last integral vanishes for all u with Du = 0 then d = 0. But if the integral vanishes for all such u, then we also have

whenever Du₁ = Du₂ = 0 in D. Now let Z be any vector in ⁿ and a another vector such that |a| = |Z|,( a.Z) = 0 . Then the functions

are harmonic, and substituting in (3) we obtain

whence it follows that d = 0.

Now let us return to Q_g (W) . We set again g = 1+d and introduce the bilinear form

and setting W_j = u_j + v_j, j = 1,2, Du_j = 0, u_j|_dD = f_j we obtain

Now, substitution of the exponentials in (5) for u₁ and u₂ in the preceding expression (taking a to be a function of Z such that |a| = |Z|, (a.Z) = 0 ) yields

where (Z) is a Fourier transform of g extended to be zero outside D, the function

is known and, as follows readily from (2),

provided that A

< 1-e, where C depends only on D and e, and r is the radius of the smallest sphere containing D. Now R(Z) is too large to permit estimating g(x). However, under favorable circumstances it is still possible to obtain satisfactory information about g. Choose a, 1 < a < 2, then for

we have |R(Z)| < C. Let h be a function such that Î C^¥, supp Ì {|Z| < 1} (0) = 1, and let h_s = sⁿh(sZ). Then we have

and

where * denotes convolution and

where C₁ depends only on D, a e e.

Thus if is sufficiently small, (9) gives an approximation for g*h_s with an error which is much smaller than . Clearly, if is small then s is large and g*h_s is itself in some sense, a good approximation to g.

Approximations to the function g itself may be obtained if one assumes that g, extended to be equal to 1 outside D, is in C^m, m > n. In this case one obtains

(Z) = ₁ (Z) + (Z)

where F₁ is known and R(Z) is the same as in (6). One then calculates d(x) by integrating over |Z| < s with s as in (8) and estimates the error by using the decay of at ¥. Thus one obtains

g(x) = F₂(x) + s(x)

where F₂(x) is known and

where M is a bound for the derivatives of order m of g.

REFERENCES

We have been unable to find treatments of the problem discussed above in the literature, at least not in the general setting in which we are interested. Similar problems have been studied in the papers listed below.

Received: 14/XI/06

#693/06.

• [1] J.R. Cannon, Determination of certain parameters in heat conduction problems. J. Math. Anal. Appl., 8 (1964), 188-201.
• [2] J.R. Cannon, Determination of an unknown coefficient in a parabolic differential equation. Duke Math. J., 30 (1963), 313-323.
• [3] J.R. Cannon, Determination of the unknown coefficient k(u) in the equation Vˇk(u) Ńu = 0 from overspecified boundary data. J. Math. Anal. Appl., 18 (1967), 112-114.
• [4] J.R.Cannon and Paul Du Chateau, Determining unknown coefficients in a non-linear heat conduction problem. SIAM J. Appl. Math., 24(3) (1973), 298-314.
• [5] Jim Douglas, Jr and G.F. Jones, Determination of a coefficient in a parabolic differential equation, Part. II: numerical approximation. J. Math. Mech., 11 (1962), 919-926.
• [6] B.F. Jones, Various methods for finding unknown coefficients in parabolic differential equations. Comm. Pure Appl. Math., 16 (1963), 33-44.

*

We thank the Brazilian Mathematical Society (SBM) for the permission to reproduce this article originally published at the "Seminar on Numerical Analysis and its Application to Continuum Physics'' in the ATAS of SBM (Rio de Janeiro), 1980, pp. 65-73.

Publication Dates

• Publication in this collection
  19 Mar 2007
• Date of issue
  2006

History

• Received
  14 Nov 2006