A Glimpse of Shape Optimization Problems

In this mini review, we give a glimpse of a branch of geometric analysis known as shape optimization problems. We introduce isoperimetric problems as a special class of shape optimization problems. We include a brief history of the isoperimetric problems and give a brief survey of the kind of shape optimization problems that we (with our collaborators) have worked on. We discuss the key ideas used in proving these results in the Euclidean case. Without getting into the technicalities, we mention how we generalized the results which were known in the Euclidean case to other geometric spaces. We also describe how we extended these results from the linear setting to a non-linear one. We describe briefly the difficulties faced in proving these generalized versions and how we overcame these difficulties.