2016-08-012024-05-18https://scholars.lib.ntu.edu.tw/handle/123456789/698967摘要：本三年期計畫主要是探討具不連續介質系統方程的卡樂門型估計，對於純量的二階橢圓方 程，我們對這個問題已經有些清楚的了解，但是對於系統方程仍在起步中。我們沒有辦法考 慮一般性的系統方程，這個計畫的第一目標是馬克思威爾方程。考慮馬克思威爾方程的好處 在於它具有座標轉換不變性，也就是說，馬克思威爾方程經由座標變換後仍是馬克思威爾方 程的形式，這對於推導不連續介質的卡樂門型估計是非常重要的第一步。除了馬克思威爾方 程，計畫中也希望將方法推廣至彈性方程，當然在這裡要先考慮同向性的物質。縱使如此， 彈性方程經由一般座標變換，係數會變成異向性，這大大增加了問題的困難度。卡樂門型估 計可以用來推導三球或三區域不等式，這個不等式可以用來處理一些反問題。<br> Abstract: In this proposal, we plan to derive the Carleman estimate for system of equations with jump discontinuous coefficients. Carleman estimates for second order elliptic equations with jump discontinuous coefficients have been successfully proved. For system of equations, we will first consider the Maxwell system. We assume that the coefficients are discontinuous and anisotropic. It should be noted that the Maxwell system is coordinates invariant. This property is vital in the derivation of the Carleman estimate when the coefficients have jump discontinuities. Besides of the Maxwell system, we also would like to extend the techniques to the isotropic elasticity system. For the elasticity system, we need to be cautious that the system may become anisotropic after a change of coordinates. This certainly increases the level of difficulty in proving the Carleman estimate for this case. Having derived the Carleman estimate, we can prove three-region inequalities. These inequalities have many applications in the study of inverse problems.卡樂門型估計馬克思威爾方程不連續介質3 區域不等式。Carleman estimatesMaxwell equationsDiscontinuous media3-region inequalities.