偏微分方程正则化：反问题理论详解

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偏微分方程是数学分析中的一个核心分支，它研究函数对多个变量的偏导数之间的关系。在给定的文献列表中，涵盖了广泛的主题，从基础理论到实际应用，以满足不同层次的研究者和工程师的需求。以下是部分书籍概述及其涉及的知识点： 1. "John: Partial Differential Equations, 4th ed." - 这本书是偏微分方程的经典教材，重点在于解析解的求解方法、分类（如椭圆、双曲、抛物型方程）以及边界值问题和特征值问题等内容。 2. "Sirovich: Techniques of Asymptotic Analysis." - 这本书探讨了近似分析技术，对于理解偏微分方程的渐近行为和极限解有重要意义，尤其是在处理复杂物理系统时。 3. "Hale: Theory of Functional Differential Equations, 2nd ed." - 功能微分方程涉及依赖于过去状态的动态系统，此书深入讨论了这些系统的稳定性、周期性和非线性特性。 4. "Percus: Combinatorial Methods" - 介绍应用于偏微分方程中的组合数学工具，可能用于离散化、近似或数值解的构建。 5. "von Mises/Friedrichs: Fluid Dynamics" - 关注流体力学中的偏微分方程，如纳维-斯托克斯方程，涉及连续介质动力学和边界层理论。 6. "Freiberger/Grenander: A Short Course in Computational Probability and Statistics" - 虽然不是专门针对偏微分方程，但可能涉及概率统计在解决随机偏微分方程中的应用。 7. "Pipkin: Lectures on Viscoelasticity Theory" - 讨论了粘弹性材料中偏微分方程的应用，特别是在材料力学和工程中的波动问题。 8. "Giacaglia: Perturbation Methods in Non-linear Systems" - 提供了非线性系统中的变分法和扰动理论，有助于理解和求解复杂系统的边界条件和初始值问题。 9. "Friedrichs: Spectral Theory of Operators in Hilbert Space" - 关键的数学工具，阐述了谱理论在偏微分方程中的作用，特别是关于算子和其特征函数的研究。 10. "Stroud: Numerical Quadrature and Solution of Ordinary Differential Equations" - 尽管是针对常微分方程，但书中提到的方法可以扩展到偏微分方程的数值求解，如有限元和差分方法。 11. "Wolovich: Linear Multivariable Systems" - 对于控制理论中的线性多变量系统建模，这可能与偏微分方程在系统动力学中的应用有关。 12. "Berkovitz: Optimal Control Theory" - 描述了最优控制问题中的偏微分方程，包括如何设计控制策略以达到特定性能指标。 13. "Bluman/Cole: Similarity Methods for Differential Equations" - 提供了解决非线性偏微分方程的一种特殊技巧，即相似变换，常用于简化和解析求解过程。 14. "Yoshizawa: Stability Theory and the Existence of Periodic Solution and Almost Periodic Solutions" - 稳定性理论对于确定偏微分方程长期行为至关重要，这本书探讨了周期解和准周期解的存在性。 15. "Braun: Differential Equations and Their Applications, 3rd ed." - 全面介绍偏微分方程的各个方面，包括理论背景和实际应用案例。 16. "Lefschetz" - 可能是指乔治·莱夫谢茨，他在微分几何和偏微分方程领域做出了重要贡献，提及他的名字通常与经典问题和理论联系在一起。这些书籍构成了一个丰富的学习资源库，涵盖了偏微分方程的理论基础、数值方法、实际应用、解析技巧和控制理论等多个方面，为深入研究该领域的学生和专业人士提供了全面的学习路径。

1.1. The inverse problem of gravimetry 3

Prescribing the Cauchy data u, u



at t = 1 is equivalent to the prescription of two

integrals



(1 − y) f (y)dy and



f (y)dy.

We cannot determine more given the Cauchy data at t =−1, 1, no matter what is

the original Cauchy data. The same information about f is obtained if we prescribe

any u on ∂ and if in addition we know u



on ∂. In particular, nonuniqueness

is substantial: one cannot ﬁnd a function from two numbers. If we add to f any

function f

such that



v(y) f

(y)dy = 0

for any linear function v (i.e., for any solution of the adjoint equation −v



= 0),

then according to the above formulae we will not change the Cauchy data on ∂.

The situation with partial differential equations is quite similar, although more

complicated.

If ∇u is given on , then u can be found uniquely outside  by uniqueness in

the Cauchy problem for harmonic functions using the assumptions on the behavior

at inﬁnity. Observe that given u on ∂ ⊂ R

one can solve the exterior Dirichlet

problem for u outside  and ﬁnd ∂

u on ∂ ∈ Li p, so in fact we are given the

Cauchy data there.

Exercise 1.1.1. Assume that  is the unit disk {|x| < 1} in R

Show that a solution f ∈ L

∞

() of the inverse gravimetry problem that satisﬁes

one of the following three conditions is unique. (1) It does not depend on r =|x|.

(2) It satisﬁes the second-order equation ∂

f = 0. (3) It satisﬁes the Laplace

equation  f = 0in.

In fact, in the cases (2) and (3),  can be any bounded domain with ∂ ∈ C

with

connected R

. {Hint: to handle case (1) consider v = r∂

u − 2u and observe

that v is harmonic in . Determine v in  by solving the Dirichlet problem and

then ﬁnd f . In cases (2) and (3) introduce new unknown (harmonic in ) functions

v = ∂

u and v = u.}

Exercise 1.1.2. In the situation of Exercise 1.1.1 prove that a density f (r ) creates

zero exterior potential if and only if



rf(r)dr = 0.

{Hint: make use of polar coordinates x = r cos θ , y = r sin θ and of the expres-

sion for the Laplacian in polar coordinates,

 = r

−1

(∂

(r∂

) + ∂

−1

∂

)).

Observe that for such f the potential u does not depend on θ , and perform an

analysis similar to that given above for the simplest differential equation of second

order.}

4 1. Inverse Problems

What we discussed brieﬂy above can be called the density problem. It is linear

with respect to f . The domain problem when one is looking for the unknown D is

apparently nonlinear and seems (and indeed is) more difﬁcult. In this introduction

we simply illustrate it by recalling that the Newtonian potential U of the ball

D = B(a; R) ⊂ R

of constant density ρ is given by the formulae

U (x; ρχ(B(a; R))) =



ρ/3|x − a|

−1

when |x − a|≥R;

ρ/2 − ρ/6|x − a|

when |x − a| < R.



(1.1.3)

These formulae imply that a ball and its constant density cannot be simultane-

ously determined by their exterior potential (|x − a| > R). One can only ﬁnd

ρ. Moreover, according to (1.1.2) and (1.1.3), the exterior Newtonian poten-

tial of the annulus A(a; R

, R

) = B(a; R

B(a ; R

)is(R

− R

)ρ/3|x − a|

so only ρ(R

− R

) can be found. In fact, in this example the cavity of an annulus

further deteriorates uniqueness. The formulae (1.1.3) can be obtained by observing

the rotational (around a) invariancy of the equation (1.1.1) when f = ρχ(B(a; R))

and using this equation in polar coordinates together with the continuity of the po-

tential and ﬁrst order derivatives of the potential at ∂ D.

We will give more detail on interesting and not completely resolved inverse

problem of gravimetry in Section 4.1, observing that starting from the pioneering

work of P. Novikov [No], uniqueness and stability results have been obtained by

Prilepko [Pr], [PrOV], Sretensky, Tikhonov, and the author [Is4].

There is another interesting problem of potential theory in geophysics, that of

ﬁnding the shape of the geoid D given the gravitational potential at its surface.

Mathematically, like the domain problem in gravimetry, it is a free boundary

problem that consists in ﬁnding a bounded domain D and a function u satisfying

the conditions

u = ρ in D ⊂ R

,u = 0 outside

u, ∇u ∈ C(R

), lim u(x) = 0as|x|→∞,

u = g

on ∂ D,

where g

is a given function. To specify the boundary condition, we assume that

D is star-shaped, so it is given in polar coordinates (r,σ) by the equation r <

d(σ ), |σ |=1. Then the boundary condition should be understood as u(d(σ )σ ) =

(σ ), where g

is a given function on the unit sphere. This problem is called

the Molodensky problem, and it was the subject of recent intensive study by

both mathematicians and geophysicists. Again, despite certain progress, there are

many challenging questions, in particular, the global uniqueness of a solution is

not known.

To describe electrical and magnetic phenomena one makes use of single- and

double-layer potentials

(1)

(x; gd ) =





K (x, y)g(y)d(y)

1.2. The inverse conductivity problem 5

and

(2)

(x; gd ) =





∂

ν(y)

K (x, y)g(y)d(y)

distributed with (measurable and bounded) density g over a piecewise-Lipschitz

bounded surface  in R

. As in inverse gravimetry, one is looking for g and 

(or for one of them) given one of these potentials outside a reference domain .

The inverse problem for the single-layer potential can be used, for example, in

gravitational navigation: it is probably more efﬁcient to look for a single layer dis-

tribution g instead of the volume distribution f . As a good example of a practically

important problem about double layer potentials we mention that of exploring the

human brain to ﬁnd active parts of its surface 

(cortical surface). The area of

active parts occupy not more than 0.1 of area of 

. They produce a magnetic ﬁeld

that can be described as the double-layer potential distributed over 

with density

g(y), and one can (quite precisely) measure this ﬁeld outside the head  of the

patient. We have the integral equation of the ﬁrst kind

G(x) =





∂

ν(y)

K (x, y)g(y)d(y), x ∈ ∂,

where 

isagivenC

-surface,



⊂ , and g ∈ L

∞

(

) is an unknown function.

In addition to its obvious ill-posedness, an intrinsic feature of this problem is the

complicated shape of 

. There have been only preliminary attempts to solve it

numerically. No doubt a rigorous mathematical analysis of the problem (asymptotic

formulae for the double-layer potential when 

is replaced by a closed smooth

surface or, say, use of homogenization) could help a lot.

In fact, it is not very difﬁcult to prove uniqueness of g (up to a constant) with

the given exterior potential of the double layer.

We observe that in inverse source problems one is looking for a function f

entering the partial differential equation −u = f when its solution u is known

outside . If one allows f to be a measure or a distribution of ﬁrst order, then the

inverse problems about the density g of a single or double layer can be considered

as an inverse source problem with f = d or f = g∂

(d).

1.2 The inverse conductivity problem

The conductivity equation for electric (voltage) potential u is

div(a∇u) = 0in.(1.2.1)

For a unique determination of u one can prescribe at the boundary the Dirichlet

data

u = g

on ∂.(1.2.2)

Here we assume that a is a scalar function, 0 <ε

≤ a, that is measurable and

bounded. In this case one can show that there is a unique solution u ∈ H

(1)

()tothe

6 1. Inverse Problems

direct problem (1.2.1)–(1.2.2), provided that g

∈ H

(1/2)

(∂) and ∂ is Lipschitz.

Moreover, there is stability of u with respect to g

in the norms of these spaces. In

other words, we have the well-posed direct problem.

Often we can assume that a is constant near ∂. Then, if g

∈ C

(∂), the

solution u ∈ C

near ∂, so the following classical Neumann data are well-deﬁned:

a∂

u = g

on ,(1.2.3)

where  is a part of ∂ ∈ C

. In general case, ∂

∈ H

(−1/2)

(∂), so the data (1.2.3)

are still well-deﬁned.

The inverse conductivity problem is to ﬁnd a given g

for one g

(one boundary

measurement) or for all g

(many boundary measurements).

In many applied situations it is g

that is prescribed on ∂and g

that is measured

on . This makes some difference (not signiﬁcant theoretically and computation-

ally) in the case of single boundary measurements but makes almost no difference

in the case of many boundary measurements when  = ∂, since actually it is the

set of Cauchy data {g

, g

} that is given. The study of this problem was initiated

by Langer [La] as early as in the 1930s.

The inverse conductivity problem looks more difﬁcult than the inverse gravi-

metric one: it is “more nonlinear.” On the other hand, since u is the factor of a in

the equation (1.2.1), one can anticipate that many boundary measurements pro-

vide much more information about a than one boundary measurement. We will

show later that this is true when the dimension n ≥ 2. When n = 1, the amount

of information about a from one or many boundary measurements is almost the

same.

This problem serves as a mathematical foundation to electrical impedance to-

mography, which is a new and promising method of prospecting the interior of

the human body by surface electromagnetic measurements. On the surface one

prescribes current sources (like electrodes) and measures voltage (or vice versa)

for some or all positions of those sources. The same mathematical model works

in a variety of applications, such as magnetometric methods in geophysics, mine

and rock detection, and the search for underground water.

In the following exercise it is advisable to use polar coordinates (r,θ)inthe

plane and separation of variables.

Exercise 1.2.1. Consider the inverse conductivity problem for  ={r < 1} in R

with many boundary measurements when a(x) = a(r). Show that this problem is

equivalent to the determination of a from the sequence of the Neumann data w



(1)

of the solutions to the ordinary differential equations −r(arw



)



− k

aw = 0on

(0, 1) bounded at r = 0 and satisfying the boundary condition w(1) = 1.

We will conclude this section with a discussion of the origins of equation (1.2.1),

which we hope will illuminate possible applications of the inverse conductivity

problem.

1.3. Inverse scattering 7

The ﬁrst source is in Maxwell’s system for electromagnetic waves of frequency

ω:

curl E =−iωµH,

curl H = σE + iωE,(1.2.4)

where E, H are electric and magnetic vectors and σ, , and µ are respectively

conductivity, electric permittivity, and magnetic permeability of the medium. In

the human body µ is small, so we neglect it and conclude that curl E = 0in.

Assuming that this domain is simply connected, we can claim that E is a potential

ﬁeld; i.e., E =∇u. Since it is always true that div curl H = 0, from second equation

(1.2.4) we obtain for u equation (1.2.1) with

a = σ + i ω.(1.2.5)

Observe that in medical applications σ and  are positive functions of x and ω.

In certain important situations one can assume that  is small and therefore obtain

equation (1.2.1) with the real-valued coefﬁcient a = σ , which is to be found from

exterior boundary measurements. This explains what the problem has to do with

inverse conductivity. An important feature of the human body is that conductivites

of various regions occupied by basic components are known constants, and actually

one is looking for the shapes of these regions. For example, conductivities of

muscles, lungs, bones, and blood are respectively 8.0, 1.0, 0.06, and 6.7.

In geophysics the same equation is used to describe prospecting by use of

magnetic ﬁelds. Moreover, it is a steady-state equation for the temperature u. In-

deed, if at the boundary of a domain  we maintain time-independent temperature

g(x), x ∈ ∂, then (Section 9.0) a solution of the heat equation ∂

U = div(a∇U )

in ,0< t, is (exponentially) rapidly convergent to a steady-state solution u to

the equation (1.2.1) with the Dirichlet boundary condition (1.2.2). The function a

then is called the thermal conductivity of the medium and is to be found in several

engineering applications.

So, the inverse conductivity problem applies to a variety of situations when

important interior characteristics of a physical body are to be found from boundary

experiments and observations of fundamental ﬁelds.

1.3 Inverse scattering

In inverse scattering one is looking for an object (an obstacle D or a medium

parameter) from results of observations of so-called ﬁeld generated by (plane)

incident waves of frequency k. The ﬁeld itself (acoustic, electromagnetic, or elastic)

in the simplest situation of scattering by an obstacle D is a solution u to the

Helmholtz equation

−u − k

u = 0inR

D(1.3.1)

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