Analysis tools (PDE's Theory, Functional Analysis, Measure Theory, Harmonic Analysis,...) are often used in Differential Geometry to solve natural problems concerning minimal or constant mean curvature surfaces. Conversely, the use of Geometry as a tool to solve analytical problems is less frequent. In this talk I would like to show that minimal and constant mean curvature surfaces can be used in order to some overdetermined elliptic problems in domains of Rn (and also in domains of other geometric spaces), i.e. solutions to an elliptic differential equation as $\Delta u = f(u)$ with two boundary conditions. In particular, I will show some results obtained in collaboration with F. Schlenk, A. Ros and F. Morabito.

Se demuestra la existencia de dominios no compactos y no triviales del espacio euclideo para los cuales la derivada normal de la primera función propia del Laplaciano (con condición de Dirichlet cero en el borde) es constante al borde. Estos dominios son obtenidos por perturbación de un cilindro y dan un contraejemplo a una conjectura de Berestycki-Caffarelli-Nirenberg.

We prove the existence of extremal domains with small prescribed volume for the first eigenvalue of Laplace-Beltrami operator in some Riemannian manifold. These domains are close to geodesic spheres of small radius centered at a nondegenerate critical point of the scalar curvature.