A generalization on the solvability of integral geometry problems along plane curves

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R E S E A R C H

Open Access

A generalization on the solvability of integral

geometry problems along plane curves

Zekeriya Ustaoglu

*

*_{Correspondence:}

[email protected] Department of Mathematics, Bulent Ecevit University, Zonguldak, 67100, Turkey

Abstract

This paper is concerned with a general condition for the solvability of integral geometry problems along the plane curves of given curvatures. As two important results, the solvabilities of integral geometry problems along the family of circles with ﬁxed radius and along the family of circles of varying radius centered on a ﬁxed circle are given. By using some extension of the class of unknown functions, the proofs are based on the solvabilities of equivalent inverse problems for transport-like equation. MSC: 35R30; 53C65; 65N30

Keywords: integral geometry problem; inverse problem; Galerkin method; transport-like equation

1 Introduction

The problems of integral geometry are to determine a function, given (weighted) integrals of this function over a family of manifolds, and there has been signiﬁcant progress in the classical Radon problem when manifolds are hyperplanes and the weight function is unity, there are interesting results in the plane case when a family of curves is regular or in the case of a family of straight lines with arbitrary regular attenuation [, Chapter ]. It is assumed that the basis of the integral geometry problems is the Radon transform []. The Radon transformRintegrates a functionf onRn_{over hyperplanes. Let}_H₍_s_,_{) =}_{_x_∈_Rn_:

x·=s}be the hyperplane perpendicular to∈Sn–(unit sphere) with signed distance s∈R _{from the origin, and the Radon transform (}_Rf₎₍_s_,_{) is deﬁned as the integral of}_f

overH(s,),i.e.,

(Rf)(s,) =

H(s,)

f(x)dx

(see [, Chapter ]).

The problems of integral geometry have important applications in imaging and provide the mathematical background of tomography, where the main goal is to recover the inte-rior structure of a nontransparent object using external measurements. The object under investigation is exposed to radiation at diﬀerent angles, and the radiation parameters are measured at the points of observation. The basic problem in computerized tomography is the reconstruction of a function from its line or plane integrals, and there are many appli-cations related with computerized tomography: medical imaging, geophysics, diagnostic radiology, astronomy, seismology, radar and many other ﬁelds (see,e.g., []).

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From the applied point of view, the importance of integral geometry problem over a family of straight lines in the plane is indicated in [], where the problem models X-rays, and applicable to the problems of radiology and radiotherapy. Because of their many prac-tical applications, a considerable attention has been devoted to other family of curves in the plane as well as straight lines. Invertibility of the Radon transforms on some families of curves in the plane is given with explicit inversion formulas via circular harmonic decom-position in [] and for the explicit inversion formulas of the attenuated Radon transform, see,e.g., [, ]. Note that the circle is the simplest non-trivial curve in the plane next to the straight line, and the representation of a function by its circular Radon transform also arises in applications. In [], invertibility of the Radon transforms over all translations of a circle of fixed radius and circles of varying radius centered on a fixed circle is considered, where the proofs require microlocal analysis of the Radon transforms and a microlocal Holmgren theorem. In [], some existence and uniqueness results on recovering a func-tion from its circular Radon transform with partial data are presented and the relafunc-tions to applications in medical imaging are described. There are several other ways related to the selection of a family of curves, such as circles of varying radius centered on a straight line or a fixed curve, circles passing through a fixed point, along paths that are not on the zero sets of harmonic polynomials, circular arcs having a chord of fixed length rotat-ing around its middle pointetc., which are meaningful in applications on thermo-acoustic and photoacoustic tomography, synthetic aperture radar, Compton scattering tomogra-phy, ultrasound tomographyetc.(see,e.g., [–] and the references therein).

In fact, since the seminal work of Radon [], the various integral geometry problems with numerous applications have been considered in several important aspects which are not mentioned here, but for a comprehensive list, see,e.g., [–] and the references therein. Furthermore, the problems of integral geometry and inverse problems for transport equa-tions are interrelated and the latter are also of great importance in theory and applicaequa-tions; see,e.g., [–] and for the derivation and applications of transport equations, see,e.g., [–].

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solvability covers those of [–], in the manner indicated by Remark  in Section .. Moreover, these previous solvability conditions do not hold for IGP along the above given two families of circles, and this is the main importance and motivation of this study.

In Section , IGP and its reduction to the equivalent inverse problem for a transport-like equation, the method to overcome the difficulty on investigating the solvability arising from overdeterminacy of these problems are presented and some definitions and nota-tions which will be used throughout the paper are introduced. Section  is devoted to the statements of main results, and finally in Section , proofs of the results are given.

The investigation of approximate solutions of the concerned integral geometry prob-lems is beyond the scope of this paper, but similar procedures as in [] can be carried out by using the Galerkin method or the ﬁnite diﬀerence method.

2 Statement of the problem and overdeterminacy 2.1 Statement of the problem

LetDbe a bounded domain inR_{. It is assumed that in}_D_{, a family of regular curves is}

given by curvatureK(x,ϕ) which is the curvature of the curve passing from the pointx= (x,x)∈Din the directionν= (cosϕ,sinϕ), and there exists a unique suﬃciently smooth

curve of this family which is passing from any pointx∈Din the arbitrary directionν, with the endpoints on the boundary ofD. Suppose that the lengths of these curves inD are bounded above by the same constant. Let us denote the family of these curves by{}. IGP is stated below.

IGP Determine a functionλ(x)in the domain D from the integrals ofλalong the curves of a given family of curves{}.

Suppose thatλ(x)∈C(R_{) vanishes outside}_D_{, and let us introduce an auxiliary function}

u(x,ϕ) =

γ(x,ϕ)

λdσ, ()

whereγ(x,ϕ) is a part of the curve that belongs to{}, with one end of it being the pointx and the other one on∂D, anddσis the arc length element alongγ(x,ϕ).

Investigating the uniqueness of a solution of a problem of integral geometry by reducing it to the equivalent inverse problem for a diﬀerential equation was ﬁrst carried out in []. Similar reduction is demonstrated for IGP formulated below.

Diﬀerentiating () in the directionνatx, we obtain the following transport-like equation:

Lu≡uxcosϕ+uxsinϕ+K(x,ϕ)uϕ=λ(x). ()

From (),uis π-periodic with respect toϕ, and since the integrals ofλalong the curves of{}are known,uis known on∂D×(, π),i.e.,

u|∂D×(,π)=u(x,ϕ), u(x,ϕ) =u(x,ϕ+ π) ()

(see [] and [, p.]). So, we have the following inverse problem.

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2.2 Overdeterminacy

Generally, in the theory of integral geometry, reconstruction of a function ofnvariables from a function ofm>nvariables is said to be an overdetermined problem of integral geometry (see,e.g., [, Chapter ]), and in the theory of inverse problems, overdetermi-nation usually means that the numbermof independent variables in the data exceeds the number nof independent variables in the unknown target function (see, e.g., [, Sec-tion .]). However, since the dataugiven on the two-dimensional surface∂D×(, π)

andλ(x) is a function of two variables, these are not the cases for IGP or Problem . Here, since the operator given in () (forγ(x,ϕ), where (x,ϕ)∈∂D×(, π)) is compact, its in-verse is unbounded, and therefore it is not possible to prove a general existence result. So, IGP and Problem  are called overdetermined in this sense. Hence, because of the overdeterminacy, the initial data for these problems should not be arbitrary and satisfy some ‘solvability conditions’ (see [, p.] and [, p., Theorem .]) which are diﬃ-cult to establish. It should be noted that the set of functions ufor which IGP is

solv-able is not everywhere dense in any of the spacesL(∂D×(, π)),Cm(∂D×(, π)) and

Hm_(∂_D_×_{(, π}_{)). Moreover, the data in problems of integral geometry are of}

quasiana-lytic character,i.e., their values speciﬁed in a domain of the Lebesgue measure can be as small as desired, determine their values in an essentially larger domain (see [, Chapter , Section ] and [, Chapter , Section ]). In particular, this implies that it is impossible to avoid overdeterminacy of the problem by specifying the data on a part of the boundary rather than on the whole boundary. Even if it were possible to ﬁnd the solvability condi-tions for the mentioned overdetermined problems, since the real data usually have some errors in practice, and thus fall out of the data class for which the existence of a solution is established, it appears that these conditions would not always be satisfactory in appli-cations. Therefore, to prove the existence results, such special conditions on the datau

have to be posed.

Let us propose the procedure for establishing the solvability of IGP. Assume that the unknown functionλin IGP depends not only upon the space variablesx, but also upon the directionϕin some special manner,i.e., considerλ(x,ϕ), where this dependence uponϕis impossible to be arbitrary, for in the opposite case the problem would be underdetermined and the examples on the nonuniqueness of a solution can be easily constructed. Herein the special dependence ofλ(x,ϕ) upon the direction means thatλ(x,ϕ) satisfies a certain differential equation (Lˆλ= , where the expression ofLîs given in Section .) with the following properties:

() The IGP or Problem  with the functionλ(x,ϕ)becomes a determined one. () The suﬃciently smooth functionsλdepending only onxsatisfy this equation.

Suppose that a diﬀerential equation forλ(x,ϕ) satisfying properties () and () has been found and thata priorithe functionue_, which represents the exact data of IGP related to a functionλdepending only onx, is known. Then, utilizingue_, a solutionλ˜to IGP can be constructed. By uniqueness of a solution,λ˜andλ(x) coincide. At the same time, knowing the approximate dataua

withue–uaH(∂D×(,π))≤ε, an approximate solutionλa(x,ϕ)

can be constructed such thatλ–λaL_()≤εC. Recall that if λdepends only onxand

ua

does not satisfy the ‘solvability conditions’, the solutionλadepending onlyxdoes not

exist. Here the data are speciﬁed on∂D×(, π) andC>  is independent ofue

andua.

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In general, the equation with the properties () and () for the same problem is not uniquely deﬁned. Hence, the class of unknown functionsλextends so that IGP for this

class becomes a determined problem and all suﬃciently smooth functions inx belong

to it. On using some extension of the class of functionsλ, the overdetermined Problem  is replaced by a determined one (Problem  in Section .).

The above method of solvability of the IGP or Problem  leads to the Dirichlet-type problem with conditions () for the third-order equation of the formAu≡ ˆLLu=F. In investigating and proving the solvability of IGP over any regular family{}of curves with curvatureK, since the quadratic formJ(∇u) in () is required to be positive deﬁnite, the construction ofLˆis important and to be able to this, in Theorem  condition () is given.

2.3 Deﬁnitions and notations

In this section, some notations are given based on []. Let D⊂R _{with the boundary}

∂D∈C_,₌_{₍_x_,_{ϕ) :}_x_∈_D_,_ϕ_∈_{(, π}₎_}_and¯ _{be the closure of}_{. By (}_u_,_v₎

L()we denote

a scalar product of functionsuandvinL() and byC∞ () the set of all functions deﬁned

inwhich have continuous partial derivatives of order up to allk<∞, whose supports are compact subsets of. For a diﬀerential expressionA, byA∗we denote the conjugate ofAin the sense of Lagrange. Forx= (x,x),

∂xiu= ∂u ∂xi

=uxi (i= , ), ∂ϕu= ∂u ∂ϕ=uϕ,

|∇xu|=u_x_+u_x_ and |∇x,ϕu|=|∇xu|+uϕ.

LetCπ() denote the set of real-valued functionsu∈C() that are π-periodic with respect toϕin the domain,i.e.,∂α

x∂ α

x∂ α

ϕ u(x, ) =∂ α

x∂ α

x∂ α

ϕ u(x, π), whereαiare

non-negative integers such that ≤α+α+α≤.

The proof of Theorem  involves energy-like estimates and the Galerkin method (see, e.g., [, Chapter , Section .], [, Chapter ]), and therefore some class of functions are introduced below. InC

π(), let us introduce the scalar product

(u,v),=

uv+



i=

(uxivxi+uxiϕvxiϕ) +uϕvϕ+uϕϕvϕϕ

d,

whered=dxdxdϕand setu,= [(u,u),]/. LetH,π () andHmπ() be the

com-pletions ofC

π() with respect to the norms · ,and · Hm₍₎(m= , , ), respectively (for the spaceHm_{, see,}_e.g._{, [, ]).}

LetCπ={w:w|∂D×(,π)= ,w∈Cπ()}, andH˚,π () andH˚mπ() be the completions

of Cπ with respect to the norms · , and · Hm₍₎ (m= , , ). Let us take a set

{w,w, . . .} ⊂Cπwhich is complete and orthonormal inL(), then we may assume that

the linear span of this set is everywhere dense inH˚π

,(). SinceH˚,π ()∩H˚() is

separa-ble, there exists a countable set{ϕi}∞_i_=⊂C

πwhich is everywhere dense in this space and

this set up can be extended to a set which is everywhere dense inL().

Orthonormaliz-ing the latter inL(), we obtain{w,w, . . .}. We denote byPnthe orthogonal projector

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Let

ˆ

Lu= ∂

∂l

∂ ∂ϕu

,

∂

∂l= (sinϕ) ∂ ∂x

– (cosϕ) ∂ ∂x

+g ∂ ∂ϕ +gϕ,

whereg(x,ϕ)∈C

π() and it can be easily veriﬁed that

∂ ∂l

∗

= –(sinϕ) ∂ ∂x

+ (cosϕ) ∂ ∂x

–g ∂ ∂ϕ.

The existence of the functiongin the expression of ∂

∂l leads to a generalization on con-ditions for the solvability of integral geometry problems. In Theorem , it is shown that if there exists a functiongsatisfying condition () which depends on the curvatureKand the domainD, the solvability holds.

Let Au≡ ˆLLu and(A) be the set of all functionsu∈L() such that for anyu∈

(A) there exists y∈L() such that Au=yin the generalized functions sense,i.e.,

(u,A∗η)L()= (y,η)L()holds for everyη∈C∞(). Take a subset(A)⊂(A) such that

for anyu∈(A) there exists a sequence{uk} ⊂Cπsuch thatuk→uweakly inL() and

(Auk,uk)L()→(Au,u)L()ask→ ∞. If we denote the closure ofCπwith respect to the

normu(A)=uL()+AuL()by(A), then we have(A)⊂(A)⊂(A) and it

can be shown that the inclusionsH˚π()⊂(A)∩H˚,π ()⊂(A)⊂L() hold.

3 Statements of results

3.1 Solvability of IGP along plane curves

Since Problem  is overdetermined, as indicated in Section ., we consider the following determined problem.

Problem  Determine a pair of functions (u,λ) deﬁned inthat satisﬁes

Lu=λ(x,ϕ) ()

provided thatLˆλ= ,uis π-periodic with respect toϕ,u|∂D×(,π)=uandKare known.

In (), it is assumed that the unknown functionλdepends also onϕand the condition

ˆ

Lλ=  holds in the generalized functions sense.

Ifu∈C(∂D×(, π)) and∂D∈C, then there exists a functionG∈Cπ() (see [,¯ p., Theorem ]) such that G|∂D×(,π) =u, and we can consider the new unknown

functionu¯=u–G. Hence from () we obtain the equationLu¯=λ+F, whereF= –LGand

¯

u|∂D×(,π)= . Let us denoteu¯again byufor simplicity, then Problem  can be reduced to

Problem  given below (see [, p.]) and the solvability of the former follows from that of the latter and does not depend on the choice ofG.

Problem  Determine a pair of functions (u,λ) deﬁned inthat satisﬁes

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provided that Lˆλ= ,u is π-periodic with respect to ϕ, u|∂D×(,π)= ,K andF are

known.

The existence, uniqueness and the stability of the solution of Problem  are given by the following theorem.

Theorem  If F∈Hπ

()and a function g(x,ϕ)∈Cπ()exists such that gxcosϕ+gxsinϕ+gϕK–Kxsinϕ+Kxcosϕ–Kϕg>K

₊_g_, _()

for all(x,ϕ)∈ ¯D×(, π),then Problemhas a unique solution(u,λ)such that u∈(A)∩

˚

Hπ

(),λ∈L()and

u_H˚π

()+λL()≤C

FL_()+FϕL() ()

holds,where C> depends on K and the Lebesgue measure of D.

Remark  In fact, without being aware of (), the functiong, with the appropriate choices

of it, was used previously in [, , ]. The convenience of () with those of previous solvability results is indicated below.

(i) In [], when the curvatureKis suﬃciently smooth andπ-periodic, the condition for the solvability is–(sinϕ)∂K

∂x+ (cosϕ) ∂K

∂x >K

_{for all}₍_x_,_ϕ)_{∈ ¯}_D_×_{(, π}₎_{, where}

this condition holds forg= in ().

(ii) In [], whenK(x,ϕ) =f(x)cosϕ–f(x)sinϕ, wheref(x)andf(x)are suﬃciently

smooth functions, the condition for the solvability is ∂f ∂x+

∂f

∂x > for allx∈ ¯D,

which holds forg(x,ϕ) =f(x)cosϕ+f(x)sinϕin ().

(iii) In [], whenK(x,ϕ) =f(x,ϕ)cosϕ+f(x,ϕ)sinϕ, wheref(x,ϕ)andf(x,ϕ)are

suﬃciently smoothπ-periodic functions, the condition for the solvability is

∂f ∂x –

∂f ∂x+f

∂f

∂ϕ –f ∂f

∂ϕ > for all(x,ϕ)∈ ¯D×(, π), and this condition holds for g(x,ϕ) =f(x,ϕ)sinϕ–f(x,ϕ)cosϕin ().

Note that in none of the above cases, the solvability conditions hold for the IGP along the family of circles of varying radius centered on a ﬁxed circle and the family of curves of constant curvatures,i.e., the family of circles with ﬁxed radius where the curvature is a nonzero constant or the family of straight lines where the curvature is zero. The former cases are investigated in Section . below and the latter case is considered in [], where the termKuϕ in the expression ofLuwill not be present sinceK =  and the proof of solvability is given forg= . In fact, the strict inequality in () can be written as a non-strict inequality, with the equality only forK=g= .

3.2 Solvability of IGP along the family of circles

Since Theorem  was given for IGP along a regular family of plane curves for the general case, in this section the given results on the solvabilities of IGP along the family of circles depend on ﬁnding an appropriate functiongsatisfying () for the given curvatureKand

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.. Solvability of IGP along the family of circles with ﬁxed radius LetD⊂R_and_M₌_sup

(x,x)∈D(x



+x). Let us take the family of curves in IGP as the

fam-ily of circular arcs (the segments of circles insideDwith the endpoints on∂D) with ﬁxed radiusr, passing from the point (x,x)∈Din the directionν= (cosϕ,sinϕ) and denote

this family of circles by{r}. Since the curvature of a circle is deﬁned to be the reciprocal of the radius of this circle, the curvature of the elements of the family{r}becomesK=

r.

The solvability of IGP along the curves of{r}follows from the following lemma.

Lemma  Let us deﬁne g on D×(, π)as

g(x,x,ϕ) =



M(xcosϕ+xsinϕ). ()

If r> √M,then()holds for K=_r.

Remark  The above choice ofgis not unique, and since the curvatureK=_ris constant, and henceKx=Kx=Kϕ= , condition () reduces to

gxcosϕ+gxsinϕ+gϕ  r>

 r +g

_, _()

for all (x,ϕ)∈ ¯D×(, π), which depends onrand the domainD.

.. Solvability of IGP along the family of circles of varying radius centered on a ﬁxed circle

LetD⊂R_and_M₌_sup

(x,x)∈D(x



+x). LetDRbe a ﬁxed disk of radiusR>  and, without

any loss of generality, centered at the origin such thatR>√M, and take the family of curves in IGP as the family of circular arcs (the segments of circles insideDwith the endpoints on∂D) of varying radiusr(x,x,ϕ) centered on∂DR, passing from the point (x,x)∈D

in the directionν= (cosϕ,sinϕ) and denote this family of circular arcs by{R}. It can be shown that the radius of the circles of the family{R}is deﬁned by the function

r(x,x,ϕ) =xsinϕ–xcosϕ+

R– (xcosϕ+xsinϕ) /

, ()

and hence the curvature of the elements of the family{R}is deﬁned by

K(x,x,ϕ) =

xsinϕ–xcosϕ+

R– (xcosϕ+xsinϕ) / –

()

onD×(, π). Note that sinceR>√M, we haver(x,x,ϕ) >  andK(x,x,ϕ) >  on

D×(, π).

The solvability of IGP along the curves of{R}follows from Lemma  given below.

Lemma  If the function g is deﬁned on D×(, π)as

g(x,x,ϕ) =



M(xcosϕ+xsinϕ), ()

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4 Proof of results

Proof of Theorem First, we will prove the uniqueness of the solution (u,λ) of Problem  under the assumptions of the theorem. To this end, it is suﬃcient to show that the corre-sponding homogeneous problem has only a trivial solution. Then, by taking into account thatLˆλ=  andF= , from () we obtainLLuˆ =Au= .

Sinceu∈(A), there exists a sequence{uk} ⊂Cπsuch thatuk→uweakly inL()

and (Auk,uk)L()→(Au,u)L()=  ask→ ∞. Now we want to decompose the product (Auk)ukinto the sum of a positive deﬁnite quadratic form and a divergence form. For this

purpose, we have the following identities:

(Auk)uk= (LLuˆ k)uk

= ∂ ∂l ∂ ∂ϕLuk

uk

= ∂

∂ϕLuk

∂ ∂l

∗ uk+

∂ ∂x

uk

∂ ∂ϕLuk

sinϕ

– ∂

∂x

uk

∂ ∂ϕLuk

cosϕ + ∂ ∂ϕ uk ∂ ∂ϕLuk

g () and  ∂ ∂ϕLuk

∂ ∂l

∗

uk= (ukxϕcosϕ–ukxsinϕ+ukxϕsinϕ+ukxcosϕ

+Kϕukϕ+Kukϕϕ)(–ukxsinϕ+ukxcosϕ–gukϕ)

=u_kx_+u_kx_+ Kukϕ(ukxcosϕ+ukxsinϕ)

+ gukϕ(ukxsinϕ–ukxcosϕ) + (gxcosϕ+gxsinϕ

+gϕK–Kxsinϕ+Kxcosϕ–gKϕ)u



kϕ

+ ∂

∂x

ukxukϕ+Ku



kϕsinϕ–gu



kϕcosϕ

– ∂

∂x

ukxukϕ+Ku



kϕcosϕ+gukϕsinϕ

+ ∂

∂ϕ

–u_kx_sinϕcosϕ+u_kx_sinϕcosϕ+ukxukxcosϕ – Kukxukϕsinϕ+ Kukxukϕcosϕ–gKukϕ .

If () holds, then the quadratic formJ(∇uk) inukx,ukx,ukϕis positive deﬁnite, where

J(∇uk) =ukx+u



kx+ Kukϕ(ukxcosϕ+ukxsinϕ)

+ gukϕ(ukxsinϕ–ukxcosϕ) + (gxcosϕ+gxsinϕ



kϕ

=u_kx_+u_kx_+ ukϕukx(Kcosϕ+gsinϕ)

+ ukϕukx(Ksinϕ–gcosϕ) + (gxcosϕ+gxsinϕ



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Indeed, we can estimate the terms ukϕukx(Kcosϕ+gsinϕ) and ukϕukx(Ksinϕ–gcosϕ) as follows:

ukϕukx(Kcosϕ+gsinϕ)≥–εu



kx–ε

–₍_K_cos_ϕ₊_g_sin_ϕ)_u

kϕ,

ukϕukx(Ksinϕ–gcosϕ)≥–εu



kx–ε

–₍_K_sin_ϕ_–_g_cos_ϕ)_u

kϕ,

where  <ε< , and we obtain

J(∇uk)≥ukx+u



kx–εu



kx–εu



kx–ε

–₍_K_cos_ϕ₊_g_sin_ϕ)_u

kϕ

–ε–(Ksinϕ–gcosϕ)u_kϕ+ (gxcosϕ+gxsinϕ +gϕK–Kxsinϕ+Kxcosϕ–gKϕ)u



kϕ

= ( –ε)u_kx_+u_kx_ +gxcosϕ+gxsinϕ+gϕK –Kxsinϕ+Kxcosϕ–gKϕ–ε

–_K₊_g _u

kϕ.

Moreover, whenever () holds, for suﬃciently close value ofεto , there exists anα∈R such thatgxcosϕ+gxsinϕ+gϕK–Kxsinϕ+Kxcosϕ–Kϕg–ε

–₍_K₊_g₎_≥_α_{>  in}_,

and we obtain

J(∇uk)≥( –ε)|∇xuk|+αukϕ≥β|∇x,ϕuk|,

whereβ=min{( –ε),α}.

Since the domainDis bounded,uk=  on∂D×(, π) andJ(∇uk) is positive deﬁnite,

we have

uk_L_₍)≤C

|∇xuk|d≤C

J(∇uk)d,

whereC=Cβ–andC>  depends on the Lebesgue measure ofDand does not depend

onk.

Thus, sinceuk∈Cπ,Kandgare π-periodic with respect toϕ, after integrating ()

over, the divergent terms disappear and we obtain

(Auk,uk)L()=

J(∇uk)d. ()

Sinceu∈(A), from () we have

u_L_₍₎≤ lim

k→∞

uk_L_₍₎

≤C lim

k→∞

J(∇uk)d= C lim

k→∞(Auk,uk)L()= , ()

which implies thatu= , and sinceF= , from () we getλ= . So, the uniqueness part of the proof is completed.

Now we will prove that there exists a solution (u,λ) of Problem  in ((A)∩H˚π())×

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Determine u deﬁned inthat satisﬁes

Au=F, ()

u|∂D×(,π)= , u(x, ) =u(x, π), ()

whereF=LFˆ .

The solutionuof problem ()-() will be approximated by

uN = N

i=

αNiwi(x,ϕ); αN= (αN,αN, . . . ,αNN)∈R

N_,

construction of which is based on ﬁnding the vectorαNfrom the system of linear algebraic

equations

(AuN,wj)L()= (F,wj)L(), j= , , . . . ,N, ()

where the system of functions{wj} is taken as indicated in Section . anduN =  on

∂D×(, π).

We must show that the solution of system () exists and is unique for anyF∈Hπ

().

To demonstrate this, let us assume that the homogeneous version of (),i.e., the system

(AuN,wj)L()= , j= , , . . . ,N,

has a nonzero solutionα¯N= (α¯N,α¯N, . . . ,α¯NN). Substitutingα¯NforαN, multiplying thejth equation of the above homogenous system by α¯Njand summing with respect tojfrom  toN, we obtain

(Au¯N,u¯N)L()= , ()

whereu¯N = N

i=α¯Niwi. So, from () and () we obtain (Au¯N,u¯N)L()=

J(∇ ¯uN)d= ,

and since the quadratic formJ(∇ ¯uN) deﬁned in () is positive deﬁnite and u¯N =  on

∂D×(, π), we haveu¯N=  in. But{wi}is linearly independent and this implies that ¯

αNi= ,i= , , . . . ,N, which contradicts with the assumptionα¯N= . So, it is shown that system () has a unique solutionαN for anyF∈Hπ().

Now we estimate the solution uN of system () in terms ofF. For this purpose, we

multiply both sides of the jth equation of () by αNj and sum the obtained equations with respect tojfrom  toNto obtain

(AuN,uN)L()= (F,uN)L()= (LFˆ ,uN)L(). () SinceuN ∈Cπ, applying integration by parts, the right-hand side of () can be estimated

as

(LFˆ ,uN)L()≤γ

F_ϕd+γ–

∂ ∂l

∗ uN



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forγ> , and from () and () we have

J(∇uN)d≤γ

Fϕd+γ–

∂ ∂l

∗ uN



d. ()

It can be veriﬁed that for suﬃciently largeγ> , from () we obtain

uN_H˚π

()≤CFϕL(), ()

where the constantCis independent ofN. This implies that{uN}∞N=is bounded inL()

andH˚π

(), and sinceL() andH˚π() are Hilbert spaces, it is weakly compact inL()

andH˚π

(). Therefore, there exists a subsequence, which we again denote by{uN}, such

thatuN →uweakly inL() andH˚π() asN→ ∞and

u_H˚π

()≤CFϕL()

holds. Since uN|∂D×(,π) =  and uN →u weakly in H˚π(), we have u|∂D×(,π) = .

From () we have also that{uNx}∞N=,{uNx}∞N= and{uNϕ}∞N= are bounded and there

exists a subsequence of{uN}, which is again denoted by{uN}, such thatuNx,uNx and uNϕ converge weakly inL() toux,ux anduϕ, respectively. Taking into account that uN,wj∈Cπ,F∈Hπ() and applying integration by parts in (), forN≥j, we obtain

LuN–F, (Lˆ)∗wj _L

()= .

Since the linear span of{wj}is dense on the spaceH˚π

,(), passing to the limit asN→ ∞,

we get

Lu–F, (Lˆ)∗η

L()=  ()

for everyη∈H˚π

,(). If we setλ=Lu–F, sinceC∞()⊂H˚,π (), from () we haveLˆλ= 

in the generalized functions sense. Moreover,

λL_()≤Cu_H˚π

()+FL() holds and, by usingu_H˚π

()≤CFϕL(), it can be seen that () holds. In the above expressions, byCwe denote generic constants which depend only on the given functions and Lebesgue measure of the domainD.

Now it remains to show thatu∈(A). Sinceu∈L() andF∈Hπ(), forF=LFˆ ∈

L(), from () we have

u,A∗η _L ()=

u,L∗(Lˆ)∗η _L ()=

Lu, (Lˆ)∗η _L ()=

F, (Lˆ)∗η_L

()= (F,η)L() for anyη∈C∞_ (), which implies thatF =Auin the generalized functions sense,i.e., u∈(A).

Moreover, from () we havePNAuN =PNF, wherePN is the orthogonal projector.

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i.e.,PNAuN→F=Austrongly inL() asN→ ∞. Then, sinceuN →uweakly inL()

asN→ ∞, we have (PNAuN,uN)L()→(Au,u)L()asN→ ∞. On the other hand, since the projection operatorPN is self-adjoint inL(),

(PNAuN,uN)L()=

AuN,PN∗uN _L

()= (AuN,PNuN)L()= (AuN,uN)L(),

thus (AuN,uN)L()→(Au,u)L()asN→ ∞, and sou∈(A). The proof of Theorem  is

complete.

Proof of Lemma SinceK=_r is constant, and hence

Kx=Kx =Kϕ= ,

to prove that condition () holds, we only need to show thatgsatisﬁes

gxcosϕ+gxsinϕ+gϕ  r>

 r +g

_.

By taking into account thatM=sup₍_x

,x)∈D(x



+x) andr> 

√

M, for the function

g(x,x,ϕ) =



M(xcosϕ+xsinϕ) given in () deﬁned onD×(, π), we obtain

gxcosϕ+gxsinϕ+gϕ 

r–

 r –g



= 

M

 +

r(–xsinϕ+xcosϕ)

– 

r –



M(xcosϕ+xsinϕ) 

≥ 

M

 – r

√

M

– 

r –



M

>  M

 – 

– 

M

= ,

and the proof is complete.

Proof of Lemma For

K(x,x,ϕ) =

xsinϕ–xcosϕ+

R– (xcosϕ+xsinϕ) / –

=(R

_{– (}_x

cosϕ+xsinϕ))/–xsinϕ+xcosϕ

R_–_x –x

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we have

Kx=  (R_–_x

–x)

–(xcosϕ+xsinϕ)cosϕ

(R_{– (}_x

cosϕ+xsinϕ))/

–sinϕR–x_–x_

+ x

–xsinϕ+xcosϕ ,

Kx=  (R_–_x

–x)

–(xcosϕ+xsinϕ)sinϕ

(R_{– (}_x

+cosϕR–x_–x_

+ x

–xsinϕ+xcosϕ ,

Kϕ=

–(xcosϕ+xsinϕ)

(R_–_x –x)

(–xsinϕ+xcosϕ)

(R_{– (}_x

+ 

and

–Kxsinϕ+Kxcosϕ

= 

(R_–_x –x)

R– (xcosϕ+xsinϕ)+ (xsinϕ–xcosϕ)

– (xsinϕ–xcosϕ)

=K.

SinceM=sup₍_x__,_x_₎_∈_D(x+x) andR>

√

M, for

g(x,x,ϕ) =



M(xcosϕ+xsinϕ) given in () deﬁned onD×(, π), we have

–Kϕg=

(xcosϕ+xsinϕ)

(R_–_x –x)

(–xsinϕ+xcosϕ)

(R_{– (}_x

+ 

× 

M(xcosϕ+xsinϕ) =(xcosϕ+xsinϕ)

_(–_x

sinϕ+xcosϕ+ (R– (xcosϕ+xsinϕ))/)

M(R_–_x

–x)(R– (xcosϕ+xsinϕ))/

≥.

Hence, to prove that condition () holds, we only need to show that

gxcosϕ+gxsinϕ+gϕK>g



holds. If we take into account again thatM=sup₍_x__,_x_₎_∈_D(x_ +x_) andR>√M, then we obtain

gxcosϕ+gxsinϕ+gϕK–g

= 

M

 + –xsinϕ+xcosϕ

xsinϕ–xcosϕ+ (R– (xcosϕ+xsinϕ))/

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– 

M(xcosϕ+xsinϕ) 

≥ 

M

(R_{– (}_x

xsinϕ–xcosϕ+ (R– (xcosϕ+xsinϕ))/

– 

M

> .

The proof is complete.

Competing interests

The author declares that he has no competing interests.

Received: 2 August 2013 Accepted: 19 August 2013 Published: 8 September 2013 References

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doi:10.1186/1687-2770-2013-202

Cite this article as:Ustaoglu:A generalization on the solvability of integral geometry problems along plane curves.