An Energy Approach to Relative Overdetermined Problems – Part I

We consider partial differential equations of the type $- \Delta u = f(u)$ in domains $\Omega$ 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 have constant normal derivatives on the part of $\partial \Omega$ inside $\mathcal C$. Our arguments are primarily based on analyzing how the energy of a positive solution in a domain varies when the domain moves inside $\mathcal C$. After discussing how to overcome the issue of defining the energy when there are multiple solutions, we present a variational formulation of the overdetermined problem, in terms of the derivative of the energy. We then discuss particular classes of $\mathcal C$ where there are special domains that, being symmetric in some sense, would be expected to be minimizers for the energy. We will show that symmetry is not always the best in terms of energy. Time permitting, we will discuss related symmetry-breaking results whose proofs are based on bifurcation theory.