Eigenfunctions of the Laplace-Beltrami operator, and isoperimetric and isocapacitary inequalities

We discuss some methods of geometric nature in the study of qualitative and quantitative aspects of eigenvalue problems for the Laplace operator, and some of its generalizations. Among various issues of relevance on this topic, two questions are focused. On the one hand, information on the spectrum of the Laplacian, and, in particular, on its discreteness, is provided. On the other hand, criteria for the regularity of eigenfunctions, and specifically their integrability and boundedness, are illustrated.