彭超權(quán),陶 婷,蔡明建
(中南民族大學(xué) 數(shù)學(xué)與統(tǒng)計(jì)學(xué)學(xué)院,武漢 430074)
可穿透腔體外有裂縫的正散射問題
彭超權(quán),陶 婷,蔡明建
(中南民族大學(xué) 數(shù)學(xué)與統(tǒng)計(jì)學(xué)學(xué)院,武漢 430074)
研究了時(shí)間調(diào)和的點(diǎn)源入射平面波通過腔體和裂縫的正散射問題,認(rèn)為散射體是由一個(gè)可穿透腔體和一個(gè)外部不可穿透的裂縫構(gòu)成,該問題歸結(jié)為對(duì)具有一定邊界條件的Helmholtz方程的求解.通過邊界積分方程的方法,利用位勢(shì)理論和Fredholm定理,證明了該問題解的存在唯一性.
邊界積分方程的方法;Helmholtz方程;Fredholm定理;腔體;裂縫
(1)
(2)
1995年,Kress研究了裂縫散射的正反散射問題,他利用邊界積分方程的方法,得到了解的存在唯一性[1].1997年,Monch考慮了具有Neumann邊界條件的裂縫散射問題,之后在2000年,Kirsch和Ritter通過遠(yuǎn)場(chǎng)信息對(duì)裂縫進(jìn)行了重構(gòu).2003年,Cakoni和Colton考慮了裂縫兩邊具有不同邊界條件的裂縫散射問題[2,3].2009年,Krutitskii研究了平面上一類裂縫的Helmholtz方程的邊界值問題.更多關(guān)于裂縫散射的問題大家可以參見文獻(xiàn)[4-8].通常腔體的散射問題可用有界障礙物內(nèi)的有界區(qū)域的Helmholtz方程來(lái)刻畫, 腔體外有障礙物的正散射問題在文獻(xiàn)[9]有詳細(xì)的討論,更多關(guān)于腔體的正反散射問題的研究可參見文獻(xiàn)[10-14], 由于反散射問題的研究需要以正散射問題作為堅(jiān)實(shí)的理論基礎(chǔ),本文將借助Kress的邊界積分方程的方法,在恰當(dāng)?shù)腟obolev空間中考慮腔體外有裂縫的正散射問題,將問題(1)、(2)轉(zhuǎn)化為一個(gè)邊界積分系統(tǒng),并證明邊界積分算子Fredholm性和單射性,從而得到解的存在唯一性及連續(xù)依賴性結(jié)論.
H-1/2(∑).
(3)
并且u滿足Sommerfeld衰減條件(2).
本文的主要結(jié)果是定理1和定理2.
定理1 問題(2)、(3)至多有一個(gè)解.
C(‖p‖H1/2(S1)+‖q‖H-1/2(S1)+‖f‖H1/2(Γ)).
(4)
定理1的證明 事實(shí)上,如果h=p=q=f=0,我們?nèi)裟茏C明u≡0,則結(jié)論成立.
(5)
利用邊界積分方程的方法來(lái)證明問題(2)、(3)解的存在性.由Green表示公式有:u(x)=
利用單雙層位勢(shì)穿過邊界S1的跳躍關(guān)系(參見文獻(xiàn)[16]的第三章),在區(qū)域D1上,考慮當(dāng)u從D1逼近邊界S1時(shí),有:
T1,S1S1u-,x∈S1,
(7)
在區(qū)域D上,考慮u從D逼近邊界S1,有:
K2,S2S1u+),x∈S1,
(8)
T2,S2S1u+),x∈S1,
(9)
在區(qū)域D上,考慮u從D逼近邊界S2,有:
K2,S2S2u+),x∈S2,
(10)
T2,S2S2u+),x∈S2,
(11)
在區(qū)域D2上,考慮當(dāng)u從D2逼近邊界S2時(shí),有:
T1,S2S2u-,x∈S2,
(12)
其中:
現(xiàn)在再來(lái)建立邊界積分系統(tǒng).
在邊界S1上定義:
[u]|Γ=(u+-u-)|Γ=c=0,
把c,d延拓到整個(gè)邊界S2:
由(7)、(10)式以及S1上的邊界條件,我們有:
(u+-u-)|S1=-(S1,S1S1+S2,S1S1)a+(K1,S1S1+
(13)
證明 參考文獻(xiàn)[3],利用Green表示公式即證.
定義r1=+S2,S1S1q-K2,S1S1p+p,(13)式即可寫成:
(14)
由(7)、(9)式以及S1上的邊界條件有:
(15)
(16)
由(9)、(11)式以及S2上的邊界條件,有:
(17)
(18)
為了表述清楚,我們令:
由(14)、(16)、(18)式,有邊界積分系統(tǒng):
(19)
其他的算子在積分系統(tǒng)(19)有連續(xù)的積分核.故A從H連續(xù)映射到H*.
引理4 算子A:H|→H*是具有零指標(biāo)的Fredholm算子,且具有平凡核.
證明 分為兩步:第一步,證明算子A:H|→H*是具有零指標(biāo)的Fredholm算子.第二步,證明KernA={0}.
第一步.參考文獻(xiàn)[15],我們知道存在相應(yīng)的緊算子:Li,S:H-1/2(S1)→H1/2(S1),Li,T:H1/2(S1)→H-1/2(S1),
J2,S:H-1/2(S2)→H1/2(S2),J2,T:H1/2(S2)→H-1/2(S2),
其中i=1,2,使得:
ψ∈H-1/2(S1),
(20)
ψ∈H1/2(S1),
(21)
ψ∈H-1/2(S2),
(22)
ψ∈H1/2(S2),
(23)
(24)
(25)
b)]=0.
(26)
(27)
構(gòu)造位勢(shì):
(28)
(29)
位勢(shì)v(x)和w(x)在D和D1滿足不同的Helmholtz方程:
由單雙層位勢(shì)在S1的跳躍關(guān)系,有:
2v+(x)|S1=S1,S1S1a-K1,S1S1b-b,2w-(x)|S1=
(30)
(31)
由(30)式和方程組(27)的第一個(gè)式子可知:
(v+-w-)|S1=0.
(32)
由(31)式和方程組(27)的第二個(gè)式子可知:
(33)
定理2的證明 由引理4知:A的逆算子A-1:H*→H存在且有界,再由位勢(shì)函數(shù)(6)式即可得證.
[1] Kress R.Fréchet differentiability of the far field operator for scattering from a crack[J].Journal of Inverse and Ⅲ-posed Problems Series, 2009,3(4):305-313.
[2] Cakoni F, Colton D L.The linear sampling method for cracks[J].Inverse Problems,2003(19): 279-295.
[3] Cakoni F, Colton D L.Qualitative method in inverse scattering theory[M].Berlin:Springer,2006:1-270.
[4] Kress R.Acoustic scattering,special theoretical tools[M].London: Academic Press, 2001:1-280.
[5] Yan G.Scattering problems by a partially coated crack[J].Nonlinear Analysis, 2008(68): 932-939.
[6] Yan G, Ye J.Boundary integral methods for scattering problems with cracks buried in a piecewise homogeneous medium[J].Mathematical Methods in the Applied Sciences, 2012(35): 84-96.
[7] Yan G, Yao M.Mathematical basis of scattering problems from penetrable obstacles and cracks[J].Journal of Mathematical Physics, 2010(51):123520.
[8] Yan G, Yao M.The method of boundary integral equations for a mixed scattering problem[J].Journal of Differential Equations, 2009(246):4618-4631.
[9] Liu X, Zhang B.Direct and inverse obstacle scattering problems in a piecewise homogeneous medium[J].SIAM Journal on Applied Mathematics, 2010,70(8): 3105-3120.
[10] Colton D, Kress R.Integral equation methods in scattering theory[M].New York:Wiley, 1983:1-271.
[11] Qin H H, Colton D.The inverse scattering problem for cavities[J].Applied Numerical Mathematics, 2012, 62(2):699-708.
[12] Liu X.The factorization method for cavities[J].Inverse Problems, 2014,30(1): 015006.
[13] Meng S, Haddar H, Cakoni F.The factorization method for cavity in an inhomogeneous medium[J].Inverse Problems, 2014,30(4): 045008 .
[14] Hsiao G C, Wendland W L.Boundary integral equations[J].Applied Mathematical Sciences, 2013,76(4):509-547.
[15] M?nch L.On the inverse acoustic scattering problem by an open arc: on the sound-hard case[J].Inverse Problems, 1997,13(5):1379-1392.
[16] Colton D L, Kress R.Integral equation methods in scattering theory[M].New York: Springer-Verlag, 1983:1-271.
The Direct Scattering Problem for a Penetrable Cavity and a Crack
PengChaoquan,TaoTing,CaiMingjian
(College of Mathematics and Statistics, South-Central University for Nationalities, Wuhan 430074, China)
In this paper, by using a time harmonic point source as the incident wave, we consider the scattering problem of a mixed scatterer composed of a penetrable cavity and an external impenetrable crack.The problem comes down to solving the Helmholtz equation with certain boundary conditions.By using the boundary integral equation method, based on the potential theory and Fredholm theorem, we prove that the scattering problem has a unique solution.
boundary integral equation method;Helmholtz equation;Fredholm theorem;cavity;crack
2016-07-30
彭超權(quán)(1979-),男,副教授,博士,研究方向:偏微分方程,E-mail:pcq1979@163.com
國(guó)家自然科學(xué)基金數(shù)學(xué)天元基金(11526196)
O175.25
A
1672-4321(2017)01-0123-05