Multidimensional inverse hyperbolic problem with impulse input and a single boundary measurement
Multidimensional inverse hyperbolic problem with impulse input and a single boundary measurement
复制标题
具有脉冲输入和单边界测量的多维反双曲问题
DOI：
10.1515/jiip.1999.7.6.573
复制
发表时间：
1999
影响因子：
1.3
通讯作者：
M. Yamamoto
中科院分区：
文献类型：
--
作者：
V. Romanov;M. Yamamoto
求系数q(x)> χ = （χ ι，…）的问题。， r c n)> η > 2，从单边界观测考虑双曲方程的低阶项。假设该系数在具有C^-分段光滑边界#Ω的有界域Ω内是未知的。在柱面ΘΩ χ [Ο， Ο]的合适有界部分上，假定给出了具有零初始数据且源位于超平面{(x, t) \ x\ = 0, t = 0}的双曲方程解的柯西数据。将乘数法应用于该反问题，在先验假设q的小或Ω的直径下，我们得到了一个条件稳定性估计。反问题的表述及主要结果近年来，双曲方程的反问题得到了广泛的研究。获得的主要结果与许多边界测量情况下的唯一性和稳定性问题有关（例如，参见[3,6,8]和其中的参考文献）。边界控制方法使我们能够证明Dirichlet-toNeumann映射在许多重要情况下的唯一性定理。对于单一的观察，几乎没有什么结果。其中之一与carleman型估计的应用有关，并且对数据的支持使用了非常强的限制，不能应用于最重要的逆问题*俄罗斯科学院西伯利亚分院数学研究所，Acad. Koptyug proprop。1、俄罗斯新西伯利亚630090；E-mail: romanov@math.nsc.ru t东京大学数学科学系，日本东京目黑小叶3-8-1，日本东京，153-8914。电子邮件：myama@ms.u-tokyo.ac.jp 574 V。G. Romanov和M. Yamamoto，其源位于期望系数未知的域外[2,4]。作为一篇最相关的论文，请参阅Rakesh[7]，其中讨论了球对称势的确定。对于具有内部源的单次观测的反双曲型问题的稳定性，我们使用Yamamoto[11]。其他结果是在未知系数对变量的某些成分具有可解析性的假设下得到的[9,10]。如果没有额外的假设（如分析性）或对数据支持的限制，即使是唯一性似乎也不知道。我们的结果在于将乘数法应用于求双曲方程下项系数的反问题。对于乘法器方法，我们可以参考Komornik [δ]。通过这种方法，我们得到了反问题的条件稳定性估计。这一估计是在先验假设下得出的，假设未知系数较小或该系数未知的域的直径较小。设u(x,t), χ = （xi，…），xn), η > 2，解Cauchy问题uti Δη + q(x)u = δ(χι)δ'（ί）， u\t<o = 0(1.1)，其中S(t)为Dirac函数。设K+ = {x G E | x\ > 0}，用Ω C R+表示有界域，边界为C分段光滑<9Ω，外单位法向ν - ν (χ)为ΟΩ at χ G ΟΩ。引入注释：GT = {(x,0 | x€Ω， x ！< t < t +χι}Σ0 ={(Μ)ιζ,t = xi}Στ={(Μ)|χ€Ω,t = t + Xl}圣= {(x, t)ΙΧΕΟΩ,x !<t<τ + χι}，其中Τ为正数。设问题（1.1）解的柯西数据在STr\ u(x,t) = /(M), ^（M) = ff(M)， OM）€st（1.2）上给出。反问题是：从数据（1.2）中求χ G Ω的q(x)。假设αίβιηΩ = 2r。假设Ω C£？（x°，r）其中B（x°，r）是半径为r的球，圆心为x°= (x？， 0，…， 0) G e，设K = tf (χ°，Γ0) = {（χ，ί） I |xi| < t < To |x x°|}，取T0 = Γ0（χ°，Γ） > Τ，使GT C K表示Ρ = P(x°，T0) = {x | |xi| < To |x x°|}Κ在空间r上的投影，因此[y]表示不超过y的最大整数。嵌入定理表明q G C(P), s = [(n + l)/2] -f 1，且存在常数Ck = Cfc(x°，T) >，使得逆双曲问题575，其中\q\k表示q在C(P)中的范数，而||<？考虑函数的集合Q(qo)：有一个固定的正数q0。本文的主要结果是以下两个定理。定理1.1。设4r <T，设uk为问题（1.1）的解，其中q = qk, gq (qo)， fk和gk为（1.2）中对于u - uk， k = 1,2的柯西数据。则存在正数C = C(T，£°，r), q$ = </o(T，£°，r)，使得对于任意#0 < <？以下估计< (ii/i)适用于qi <？2g <2(<？o)·定理1.2。让你*。为（1.1）中q = qk ζ q (Qo)> fk和PA的解：为（1.2）中u = u^, fc = 1,2的柯西数据。存在一个正数r* = r*(T, x°，ζ？o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o2. 定理的证明我们将使用下列引理来解柯西问题（1.1）。引理2.1。设q gq （qo）那么（1.1）的解可以表示为u(x, t) =±S(t \x, |) + u（x， ί）θο(x， ί \xi |)（2.1）其中u G H(K), S = [(n-f l)/2] + l， Q是Heaviside阶跃函数：#o(i) = 1，如果t > 0， Θ0（ί） = 0，如果t < 0。此外，1 /*l*il ti(x, |xi I + 0) = / 9(ξ, χ') de, x gp (2.2) 4 </ο, re' = (χ25···5 #π)>，并且存在一个正常数C* = C*（T, x°，go）使得\u(x， T)\ < q0C+, (x.t) gk .（2.3）我们在第3节中证明了这个引理。表达式（2.1）表示解的正则部分u（x,t）与u（x,t）重合，对于（x,t） G k, u G H（Sr）和du/dv G L(5r) 576 V。G. Romanov和M. Yamamoto利用迹定理，因为9Ω是分段C光滑的，并且u£H(K)具有s > 3/2。设Uk， k = 1,2为问题（1.1）的解，其中q - q^£q （qo）和/&，#fc为（1.2）中u - u^的数据。引入符号：ui-U2=v, q\-q2 = q, f i h = 7> 9ι~92=9，并使用关系：^ - Δ + ^) ^ = 0, （z,*）€GT, k = 1,2。(2.4)
A problem of finding a coefficient q(x)> χ = ( χ ι , . . . , r c n )> η > 2, of a lower-order term in a hyperbolic equation from a single boundary observation is considered. The coefficient is assumed to be unknown inside a bounded domain Ω with a C^-piecewise smooth boundary #Ω. On a suitable bounded part of the cylindrical surface ΘΩ χ [Ο,οο), the Cauchy data for a solution to the hyperbolic equation with zero initial data and a source located on the hyperplane { ( x , t ) \ x\ = 0, t = 0} are supposed to be given. Applying the multiplier method to this inverse problem, we obtain a conditional stability estimate under a priori assumptions on smallness of q or the diameter of Ω. 1. STATEMENT OF THE INVERSE PROBLEM AND MAIN RESULTS Inverse problems for hyperbolic equations are intensively studied for the last years. The main obtained results are related to uniqueness and stability questions for the case of many boundary measurements (see, for example, the books [3, 6, 8] and the references therein). The boundary control method allows us to prove a uniqueness theorem for many important cases when the Dirichlet-toNeumann map is used [1]. For a single observation, there are few results. One of them is related to application of Carleman-type estimates and uses very strong restrictions on supports of data and cannot be applied for the most important inverse problems *Sobolev Institute of Mathematics, Siberian Branch of Russian Academy of Sciences, Acad. Koptyug prosp., 4, Novosibirsk, 630090, Russia. E-mail: romanov@math.nsc.ru t Department of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914, Japan. E-mail: myama@ms.u-tokyo.ac.jp 574 V. G. Romanov and M. Yamamoto with sources located outside of the domain where a desired coefficient is unknown [2, 4]. As a most related paper, see Rakesh [7] where determination of spherically symmetric potentials is discussed. For the stability in an inverse hyperbolic problem by a single observation with the source located inside, we refer to Yamamoto [11]. Other results are obtained under the assumption on analyticity of unknown coefficients with respect to some components of variables [9, 10]. Without extra assumptions (such as analyticity) or a restriction for supports of data, even uniqueness seems not known. Our results consist in application of the multiplier method to an inverse problem of finding a coefficient of the lower term in a hyperbolic equation. For the multipler method, we can refer, for example, to Komornik [δ]. By this way we obtain a conditional stability estimate for the inverse problem. This estimate is derived under a priori assumptions on smallness of the unknown coefficient or smallness of the diameter of the domain where this coefficient is unknown. Let u(x,t), χ = (xi, . . . ,xn), η > 2, solve the Cauchy problem uti Δη + q(x)u = δ(χι)δ'(ί), u\t<o = 0 (1.1) where S(t) is the Dirac delta function. Set K+ = {x G E | x\ > 0} and denote by Ω C R+ a bounded domain with a C-piecewise smooth boundary <9Ω and the outward unit normal ν — ν (χ) to ΟΩ at χ G ΟΩ. Introduce the notations: GT = {(x,0 | x € Ω, X! < t <T + χι} Σ0 = {(Μ) ι ζ en, t = xi} Στ = {(Μ) | χ € Ω, t = T + Xl} ST = {(x,t) Ι Χ Ε Ο Ω , X! <t<τ + χι} where Τ is a positive fixed number. Let the Cauchy data of solutions to the problem (1.1) be given on STr\ u(x,t) = /(M), ^(M) = ff(M), OM) € ST. (1.2) The inverse problem is: find q(x) for χ G Ω from the data (1.2). Suppose that αίβιηΩ = 2r. Assume that Ω C £?(x°,r) where B(x°,r) is the ball of radius r centred at x° = (x?, 0 , . . . , 0) G E. We set K = tf (χ°,Γ0) = {(χ,ί) I |xi| < t < To |x x°|} and we choose T0 = Γ0(χ°,Γ) > Τ so that GT C K. Denote by Ρ = P(x°,T0) = {x | |xi| < To |x x°|} the projection of Κ on the space R. Henceforth [y] denotes the greatest integer not exceeding y. Let q G if+(P). The embedding theorem implies that q G C(P) with s = [(n + l)/2] -f 1 and there exist constants Ck = Cfc(x°,T) > 0 such that Inverse hyperbolic problem 575 where \q\k means the norm of q in C(P) while ||<?||n_j_2 IS the norm m Consider the set Q(qo) of functions: with a fixed positive number q0. The main results of the present paper are the following two theorems. Theorem 1.1. Let 4r <T and let uk be the solution to the problem (1.1) with q = qk G Q(qo), fk and gk be the Cauchy data in (1.2) for u — Uk, k = 1,2. Then there exist positive numbers C = C(T, £°,r), q$ = </o(T,£°,r) such that for arbitrary #0 < <?o the following estimate < ( ii/i holds for qi , <?2 G <2(<?o)· Theorem 1.2. Let u*. be the solution to the problem (1.1) with q = qk ζ Q(Qo)> fk and PA: be the Cauchy data in (1.2) for u = u^, fc = 1, 2. There exists a positive number r* = r*(T, x°, ζ?ο) such that: if r = i/2 diam Ω < r* and T > 4r, then we can taJce a positive number C = C(T, x°,f/o) satisfying the estimate The uniqueness in Q(go) follows from these theorems under the same restrictions. 2. PROOFS OF THE THEOREMS We shall use the following lemma on solutions to the Cauchy problem (1.1). Lemma 2.1. Let q G Q(qo). Then the solution to (1.1) can be represented in the form u(x, t) = ±S(t \x, |) + u(x, ί)θο(ί \xi |) (2.1) where u G H(K), s = [(n-f l)/2] + l, Q is the Heaviside step function: #o(i) = 1 if t > 0 and Θ0(ί) = 0 if t < 0. Moreover, 1 /*l*il ti(x, |xi I + 0) = / 9(ξ, χ') de, x G P (2.2) 4 </ο with re' = (χ25 · · · 5 #π)> and there exists a positive constant C* = C*(T, x°, go) such that \u(x,t)\ < q0C+, (x.t) G K. (2.3) We prove this lemma in Section 3. The representation (2.1) means that the regular part of the solution u(x,t) coincides with u(x, t) for (x, t) G K. Moreover, u G H(Sr) and du/dv G L(5r) 576 V. G. Romanov and M. Yamamoto by the trace theorem, because 9Ω is piecewise C smooth and u £ H(K] with s > 3/2. Let Uk, k = 1,2, be the solution to the problem (1.1) with q — q^ £ Q(qo) and /&, #fc be the data in (1.2) for u — u^. Introduce the notations: ui-U2=v, q\-q2 = q, f i h = 7> 9ι~92=9 and use the relations: ^ — Δ + ^ ) ^ = 0 , (z ,*)€GT , k = 1,2. (2.4)