Some Problems On The Interior Transmission Eigenvalue Of The Wave Field Scattering

In recent years,inverse problems of partial differential equations have been hot research top-ics in the fields of computational and applied mathematics,with the main difficulties of ill-posedness and nonlinearity.Among all the inverse problems,the inverse scattering problems for incident waves described by elliptic equations(Helmholtz equation or Maxwell equations)are of great importance.The basic descriptions of these problems are to determine the internal(boundary)property of the scattering media,using the information about the scattered wave outside of the media.The interior transmission problems are motivated by some recently developed approximate methods for solving inverse scattering problems of inhomogeneous medium.More precisely,for linear sampling methods which detect the obstacle boundary in terms of some blow-up property of the indicators,if the frequency for the incident wave is just the interior transmission eigenvalue,then the reconstruction algorithm will fail.Consequently,the effective realizations of such schemes require the frequency of the incident wave not to be the interior transmission eigenvalues.On the other hand,there are also close relations between interior transmission eigenvalue problems and the cloaking effect of the obstacles,namely,when the frequency of the incident acoustic or electromagnetic wave is the interior transmission eigenvalue,the scattered wave outside of the medium(obstacles)may be zero,that is,if we apply the scattered wave information corresponding to this frequency to detect the obstacles which are widely used in engineering communities,the media(obstacles)may be invisible.By this phenomenon,in practical engineering applications,the transmission eigenvalues(decided by the physical properties of the media)can be used to guide the design of stealth materials.In particular,we know that the real transmission eigenvalues can be determined by the scattering data,therefore real transmission eigenvalues carry some information about the properties of the material,which can be used to quantify anomalies with nonconstant reflection coefficient embedded in the homogeneous media.Therefore,the studies of transmission eigenvalue problems are of obvious engineering significance and physical background.Mathematically,the interior transmission problems can be described by a boundary value problem for a coupled elliptic equation in a bounded domain,composed of two physical equa-tions in the region(two Helmholtz equations for acoustic scattering,two Maxwell equations for electromagnetic scattering)and the transmission conditions on the boundary.Although the equations considered are standard,such kinds of problems cannot be solved by the operator theory for general elliptic equations,with the reason that this problem is neither elliptic nor self-adjoint in the framework of the operator theory.This feature leads to the existence of complex eigenvalues and the distribution of all eigenvalues in the complex plane have new characteristics.Due to these reasons,the researches on transmission eigenvalue problems have gotten much mathematical attentions in recent years.Currently,the researches on the trans-mission eigenvalue problems are mainly covered by some related work from D.Colton's group in Delware University,A.Kirsch's group in Karlsruhe University of Germany and H.Haddar's group in National Institute of Information and Automation Research Institute of France.How-ever,these work and some methods are developed case by case,it is very difficult to unify the research schemes.Based on the above summary on the existed work,we consider the interior transmission eigenvalue problems for several wave scattering models,with the research contents including the distribution of the transmission eigenvalues,the estimates on real transmission eigenvalues,characterizations of the existing region of transmission eigenvalues.This thesis consists of five chapters stated as follows.In Chapter 1,for wave scattering problems,we introduce the backgrounds and applica-tions of forward scattering problems as well as inverse scattering problems.Some important conclusions about the transmission eigenvalue problems and general models are given in this part,including how to prove the discreteness of the transmission eigenvalues by operator theory.The research contents and innovations of our work are also stated.In Chapter 2,we study a transmission eigenvalue model for acoustic wave with a cavity in the homogeneous media.The formation of the transmission eigenvalue problem with one cavity is derived from the wave scattering model.We prove the existence and uniqueness of the weak solution on the basis of the formation,together with the equivalence of the weak solution of interior transmission eigenvalue problem to the solution of the variational formation.For transmission eigenvalue problem with one cavity,the Faber-Krhn type inequality for the real eigenvalues are established.When the index of refraction is a positive constant,we establish the uniqueness result for recovering the index of refraction,stating that this constant refraction index can be determined uniquely by the minimum transmission eigenvalue.At the end of this chapter,we also prove that,under some reasonable assumptions,there will not exist any pure imaginary transmission eigenvalues.Moreover,we characterize the region in complex plane where there is no transmission eigenvalue.In Chapter 3,we consider the transmission eigenvalue problem of acoustic scattering with mixed boundary condition for inhomogeneous absorbing media,through the analysis of the equivalent form of four order differential equation.We exclude the existence of pure imaginary transmission eigenvalues under some restricted conditions.Using the perturbation theory of closed operator and the existence results of transmission eigenvalues for mixed boundary condition with inhomogeneous non-absorbing media,we prove that the transmission eigenvalue problem with mixed boundary condition for inhomogeneous absorbing media have infinitely many transmission eigenvalues with +? as the only possible accumulation point.At the end of this chapter,we also establish the quantitative analysis on the non-existence zone of transmission eigenvalues in the complex plane.In Chapter 4,an interior transmission eigenvalue problem motivated by the electromag-netic wave scattering for micro-structure media is studied.In this case,the index of refraction n(x)is a positive function with small oscillating period ?>0.For this model,due to the high oscillating of equation coefficient,the direct numerical calculations of transmission eigenvalue problems are not stable.We apply the homogenization scheme to establish a stable computing framework,which can lead to an approximate stable calculations for transmission eigenvalues.We study the limit behavior of the real transmission eigenvalues k? and the corresponding eigenfunction(EE?,E0?),when the oscillating period of the medium ??0.It is proved that the eigenvalue system with small periodical oscillation converges to a eigenvalue system of a homogeneous medium in certain mathematical sense.This theoretical result provides an ef-fective stable method to calculate the transmission eigenvalues of the Maxwell equation with oscillating periodic media.In Chapter 5,we summarize our full researches and give some prospects in the future.