Executive Summary : | The Research Proposal contains two parts that are related to each other as they find their connection with the solution of semi linear elliptic problems arising from geometry : The research proposal contains 4 problems on Magnetic Poincare inequality and 2 problems on Fractional Hardy inequality. The first 2 problems are open problems for the last 2 decades in the work of the Physicist/ Mathematician Lieb et.al published in Annals of Mathematics in 2003. PI feels that he has an idea to solve the problem which he had gathered from his recent experience working on Poincare type inequality. The problems are mainly related to Magnetic Poincare inequality on punctured domains. The result Lieb et.al. concerns with the subcritical case of the Sobolev embedding and the critical case was left as a challenging problem which is our problem 1 and 2 in this research proposal. Problem 3 and 4 are concerned with the fractional analog of the result by Lieb et. al. Second part of the research proposal deals with the celebrated Hardy type inequality. In particular we are interested in establishing fractional boundary Hardy inequality. The subcritical case is established by Dyda and many other authors later with different approaches. Solving the critical can is broadly the main objective of Problem 5 and 6. If completed, the proposal will provide more deep insight into both Magnetic Poincare and fractional Hardy inequality and the results will be of interest to the Physicist as the magnetic Poincare inequality finds its application in the matter of Bose gases, theory of many body problems among many others. |