Minimizers for boundary reactions: renormalized energy, location of singularities, and applications

Abstract

The Casten-Holland and Matano theorem for interior reactions states that no nonconstant stable solutions exist in convex domains $\Omega$ of $\mathbb{R}^n$ under zero Neumann boundary conditions. In this paper we establish that the analogous statement fails for boundary reactions when $n=2$ (that is, for harmonic functions in $\Omega$ with a Neumann reaction term on its boundary $\partial\Omega$). For instance, nonconstant stable solutions exist when $\Omega$ is a square, or a smooth strictly convex approximation of it. In regular polygons of many sides, which approach the circle, we can prove the existence of as many nonconstant stable solutions as wished. Instead, in the circle such stable solutions do not exist. More importantly, we can predict the existence or not of nonconstant stable solutions, as well as the location of its boundary "vortices" $(p,q)$, through the properties of a real function defined on $\partial\Omega\times\partial\Omega$ (the renormalized energy) which depends only on the conformal structure of the domain $\Omega$. This requires the development of a new Ginzburg-Landau theory for real-valued functions and the analysis of the half-Laplacian on the real line.