Exercise 2.1.2 A Nonhyperbolic Cauchy Problem for the Wave Equation)
7.2 The energy integral methods
|K| ≤C|x−y|−2, |∇xK| ≤C|x−y|−3,
so it is anN(1,1)-kernel by [Mi], p. 23. As shown in the book of Miranda ([Mi], p.
27), the integral operator with kernelKis continuous fromC() intoCλ() and for anyλ, 0< λ <1. Since the embedding ofCλintoCis compact, the equation is Fredholm. The estimate of Corollary 7.1.7 also follows from the above-mentioned
continuity of this operator.
We observe that the weighted Radon transform is defined for a more general family of hypersurfaces than hyperplanes. In the papers of Bukhgeim and Lavren- tiev [LaB] and of Beylkin [Bey] there are results similar to Corollary 7.1.7 in a more general situation and also for the plane case, where on has to use (−)1/2, which is a first-order pseudo-differential operator.
When volis small, the same argument implies that the integral operator is a contradiction, and therefore we have uniqueness and stability of inversion of the attenuated Radon transform.
As a useful fact we give (without proof, which one can find in the paper of Isakov and Sun [IsSu1]) the following uniqueness and stability results on the weighted X-ray transform
Pρ•f(x, σ)=
l(x,σ)
ρ(l,)f dl,
wherel(x, σ) is the straight line passing throughxwith directionσ. To formulate it, we introduce a ballBinR3and the manifold (h)= {(x, σ) : the intersection of the plane through x with the normalσ with B is at positive distance from {x3≥ −h}}. LetB−(h) beB∩ {x3<−h}.
Theorem 7.1.8. Let the weight functionρ∈C2(R6)andρ > δ >0. Then there is a constant C(δ,|ρ|2)such that
|f|0(B−(h+ε)≤Cε−4|Pρ•f|3/2( (d)) for all functions f ∈C02(B).
7.2 The energy integral methods
Let γ(x,y) be a C2-smooth regular curve with endpoints x,y∈⊂R2. We assume that the family of curves{γ(x,y)}has the following properties: (a) For any x,y∈there is a uniqueγ(x,y). (b) This family is locally diffeomorphic to the family of straight lines between the same points. We refer for more detail concerning condition (b) to the papers of Muhometov [Mu1], [Mu2] and the book of Romanov [Rom]. Lety(x3) be a parametrization of∂,x3∈I =[0,S]. Let us
introduce the function
(7.2.1) F(x,y)=
γ(x,y)
ρ(v,y)f(v)dγ(v).
Letρ∗(v,x3)=ρ(v,y(x3)) and F∗(x,x3)=F(x,y(x3)). Letτ =(cosφ,sinφ) be the tangent direction toγ(x,y(x3)) at a pointx.
The following result is due to Muhometov [Mu1].
Theorem 7.2.1. Assume that the weight functionρsatisfies the following condi- tions:ρ∗∈C2(×I), ρ∗ > ε >0, and
(7.2.2) |∂3lnρ∗|< ∂3φon×I. Then
(7.2.3) f2()≤C∇F∗2(∂×I),
where C depends on the family{γ}and onρ. Moreover, C=(2π)−1/2whenρ=1.
PROOF. Let us consider the function (7.2.4) u(x1,x2,x3)=
γ((x1,x2),y(x3))
ρ(y,x3)f(y)dγ(y).
This function satisfies the first-order partial differential equation (7.2.5) τ1∂1u+τ2∂2u=ρf in×I.
The function f does not depend onx3, so dividing both parts byρand differenti- ating with respect tox3we get the homogeneous second-order equation
(7.2.6) ∂3(ρ−1τ1∂1u+ρ−1τ2∂2u)=0 in×I.
To make use of the energy integrals method we multiply this equation by the nonstandard factor (−ρτ2∂1u+ρτ1∂2u) and integrate over×I, observing that ρ∂3ρ−1= −∂3lnρ. We obtain
0=
×I
((∂3lnρ)(τ2∂1u−τ1∂2u)(τ1∂1u+τ2∂2u) +(−τ2∂1u+τ1∂2u)(∂3τ1∂1u+∂3τ2∂2u) +(−τ2∂1u+τ1∂2u)(τ1∂3∂1u+τ2∂3∂2u)).
Using the notationτ⊥=(τ2,−τ1), multiplying the terms in parentheses and inte- grating by parts the term
−τ2τ1∂1u∂3∂1u = −τ1τ2∂3(∂1u)2, τ1τ2∂2u∂3∂2u=τ1τ2∂3(∂2u)2/2 of the last parentheses, and using that due to the S-periodicity with respect to x3 there will be no boundary terms resulting from this integration by parts, we
7.2. The energy integral methods 203 obtain
×I
(∂3lnρ)τ⊥ã ∇uτã ∇u
=
×I
(−τ2∂3τ1(∂1u)2+(τ1∂3τ1−τ2∂3τ2)∂1u∂2u+τ1∂3τ2(∂2u)2 +1
2∂3(τ1τ2)(∂1u)2+τ12∂2u∂1∂3u−τ22∂1u∂2∂3u−1
2∂3(τ1τ2)(∂3u)2) (7.2.7)
Integrating by parts with respect to x3 in the term τ12∂2u∂1∂3u, we obtain the term −2τ1∂3τ1∂2u∂1u−τ12∂2∂3u∂1u. Collecting all terms involving ∂1u∂2u, we obtain the term−(τ1∂3τ1+τ2∂3τ2)∂1u∂2u, which is zero because the factor in the parentheses is 12∂3(τ12+τ22)=12∂31=0. So the last integral is reduced to
×I
(1
2(τ1∂3τ2−τ2∂3τ1)((∂1u)2+(∂2u)2)−(τ12+τ22)∂1u∂2∂3u).
Now we will make use of the identityτ12+τ22=1, the identities∂3τ1 = −τ2∂3φ,
∂3τ2 =τ1∂3φ, and the relations
−
×I
∂1u∂2∂3u= −1 2
×I
∂1u∂2∂3u−1 2
∂×I
∂1u∂3uν2
+1 2
∂×I
∂2u∂3uν1−1 2
×I
∂2u∂1∂1∂3u.
The sum of the first and the last integrals on the right side is zero because
∂1u∂2∂3u+∂2u∂1∂3u =∂3(∂1u∂2u), so the integral of this function over×I is zero due to theS-periodicity with respect tox3.
Summing up and using that (τ1∂3τ −τ2∂3τ1)=(τ12+τ22)∂3φ, from (7.2.7) we conclude that
×I
∂3φ((∂1u)2+(∂2u)2)+2
×I
(∂3lnρ)τ⊥ã ∇uτã ∇u
= −
∂×I
(−ν2∂1u+ν1∂2u)∂3u.
Due to the assumption (7.2.2),∂3φ >|∂3lnρ| +εon×I. By using this in- equality as well as the known inequality
−2|τ⊥ã ∇uτ ã ∇u| ≥ −|τ⊥ã ∇u|2− |τ ã ∇u|2
= −((∂1u)2+(∂2u)2), we conclude that
ε
×I
((∂1u)2+(∂2u)2)≤
∂×I
|ν⊥ã ∇u∂3u|.
From equation (7.2.5) we have|f|2≤ |ρ|−2((∂1u)2+(∂2u)2). In addition,ν⊥ã ∇u is the tangential component of∇u, so the last integral is bounded byCu(1)(∂× I).
Thus we obtain the bound (7.2.3) in the general case.
When∂3ρ=0 we can repeat the previous argument and use the relations
×I
∂3φ((∂1u)2+(∂2u)2)≥
×I
∂3φρ2f2
=
ρ2f2
I
∂3φd x3
d x=2π
ρ2f2, which follow from equation (7.2.5) and the independence of f onx3, to complete
the proof
This proof essentially belongs to Muhometov. In fact, Theorem 7.2.1 is only formulated in his paper [Mu1], where the basic idea of the given proof is actually applied to a more difficult problem of determining a Riemannian metric from the lengths of its geodesics, as described below. More detail are given in [Mu2]. One of the crucial steps is to find an appropriate factor by which to multiply equation (7.2.5). It seems that this multiplying factor has origin in Friedrich’s theory of symmetric positive systems [Fri] and that it is unique. Multidimensional versions are given by Beylkin, Gerver, Markushevich, Muhometov, and Romanov [Rom].
They are quite similar to Theorem 7.2.1, and their main condition is thatγ(x,y) are geodesics of a RiemannianC2-metric that form a regular family inandis convex with respect to these geodesics. Regularity of geodesics loosely speaking means that locally their family is diffeomorphic to the family of straight lines.
For other possibilities of the two-dimensional case we refer to the paper of Gelfand, Gindikin, and Shapiro [GeGS].
Vector (and tensor) versions of this theory are obtained and presented in the book of Sharafutdinov [Sh] where one can find a necessary preliminary on Riemannian geometry and multidimensional versions of Mukhometov’s theory, as well as many applications to problems of geophysics and vector tomography. We will briefly describe some of recent findings of Pestov and Uhlmann [PU] based on this theory.
A compact Riemannian manifold,gwith the boundary is simple ifis simply connected, any geodesic has no conjugate points, and∂is strictly convex (in the metricg). Then the distanced(x,y) between two pointsx,y∈is well defined.
Letdjcorresponds to the metricgj, j=1,2.
Theorem 7.2.2([PU]). Let(,gj),j=1,2be two two-dimensional simple Rie- mannian manifolds with the boundary. Let
d1(x,y)=d2(x,y), when(x,y)∈∂×∂
Then there is a diffeomorphismofonto itself identical on∂and such that g2=∗g1.
We remind thatg2=∗g1ifg2(x)ξãη=g1((x))((x)ξ)ã((x)ζ) for all vectorsξ, ηof tangent space toat allx∈.
We mention ideas of the proof in [PU]. Sharafutdinov [Sh] showed that the conclusion of Theorem 7.2.2 follows from Theorem 7.2.2 with the additional