Some Inequalities for Eigenfunctions and Eigenvalues of Certain Elliptic Operators

We present some geometric and analytic inequalities related to solutions of certain elliptic PDEs. In chapter 1, we introduced the symmetrization techniques which will be used through the paper. In chapter 2, we prove a family of isoperimetric inequalities for bodies of revolution which arise in connection with the problem of finding the extremals in some Hardy-Sobolev inequality. In particular, we are able to prove that the inequalities are sharp and that a characterization of the equality case is available, yielding the best constant. In chapter 3, we prove two stability type estimates which involve the symmetrized normalized first eigenfunction of certain elliptic operators (modelled on the Laplacian). In particular, we prove that the L^infty-distance of the symmetrized first eigenfunction from the first eigenfunction of a suitable symmetrized problem can be controlled using the distance between the first eigenvalues of such problems. In chapter 4, we prove a generalization of the classical Faber-Krahn inequality for the principal weighted eigenvalue of p-Laplace operator plus an indefinite potential.