反演多邊形區域之反問題研究(2/3)

In this research project, the applicant proposes to study the invers problems ofpartial differential equations with polygonal inclusions. The inverse problems arethe mathematical subjects concerning on the recovery of the unknownparameters in the differential operators using the partial information of thecorresponding solutions. Recently, the inverse problems attract more researchinterests in both the theoretical analysis aspects and the numerical computations.Personally, my research interests are focus on the inverse problems in theobjective to determinate certain geometrical shapes. The mathematical setup ismainly relied on the classical Calderón’s problems under two principalassumptions, the piecewise constant conductivity function and a singlemeasurement. Here, the inclusions are characterized by the piecewise constantconductivity function and they are supposed to be two dimensional convexpolygons. I have already obtained a stability estimation of this inverse problem inthe collaboration with H. Liu. However, this stability estimation is far away to theoptimal one. By introducing a new analysis tool, a significant improvement of thisresult shall be the first task in this research project.The improvement of the existing result is only the departure of the researchproject. The mathematical arguments using in the stability estimation provide astrong potential to extend the studies into a larger perspective. The secondmission in this research project is to answer the question whether a similarstability estimation holds if the concave angular corners were presented.Furthermore, the scenario where the angular corners are replaced by smoothboundaries with extremely high curvature should be taken into consideration inthis research project.