Thesis title: Energy instability and overdetermined elliptic problems in cones and cylinders: an approach via domain variations
In this thesis, we study semilinear elliptic problems of the type $- \Delta u = f(u)$ in domains that are constrained to be inside a fixed unbounded open set $\mathcal C$, with appropriate boundary conditions. Our aim is to understand how the geometry of $\mathcal C$ selects domains in which positive solutions of the equation have special properties, mainly related to notions of symmetry. Our arguments are primarily based on analyzing how the energy of a positive solution in a domain varies when the domain moves inside $\mathcal C$. We first consider the case where $\mathcal C$ is generic. We show how to define an energy functional $T$ when the equation possesses more than one solution and compute the domain derivative of $T$. In the case when $\mathcal C$ is a cone or a cylinder, we show that some special domains may be unstable as critical points to the energy shape functional. This opens room for the search for nonsymmetric domains with the same special properties, to be found, for example, by local minimization of the energy functional. This is done by analyzing the sign of the second derivative of the energy functional to understand the stability/instability of its critical domains. Furthermore, we show that in a special class of domains, namely bounded cylinders, solutions other than the one-dimensional ones do exist, under fairly general assumptions on the nonlinearity $f$. This is accomplished by means of bifurcation theory and Morse index comparison.