A Symmetry Problem in Elliptic PDE theory (M.Sc. Thesis Defence)
Chennai Mathematical Institute.
Given a PDE, we can ask whether the solutions are symmetric if the data is symmetric. For e.g. we can ask whether the eigenfunctions of the laplacian (with Dirichlet boundary conditions) are radially symmetric in a ball. As it turns out, only the first eigenfunction is radially symmetric. So now we ask the question, whether the first eigenfunction is symmetric about a plane, if the domain is symmetric about a plane. This turns out to be true and in this talk we will prove a generalization of this.