Convergence of interfaces in boundary reactions

Abstract

In this paper we study the singular limit for critical points of boundary reactions \begin{equation*} (-\Delta)^{\frac{1}{2}}u = \frac{1}{\varepsilon}(u-u^3) \quad \text{in } U \subset \textbf{R}^n . \end{equation*} We show the existence of a $(n-1)$-rectifiable energy concentration set. Furthermore, we show that the limit of the energy measures can be associated to a stationary, $(n-1)$-rectifiable varifold supported in the concentration set. This is analogous to a result of Hutchinson and Tonegawa for phase transitions.