# The Neumann problem in an irregular domain

### Łukasz Bolikowski

University of Warsaw, Poland### Maria Gokieli

University of Warsaw, Poland### Nicolas Varchon

University of Warsaw, Poland

## Abstract

We consider ‘patterns’ stability for the reaction-diffusion equation with Neumann boundary conditions in an irregular domain in ℝ_N_, *N* ≥ 2, the model example being two convex regions connected by a small ‘hole’ in their boundaries. By patterns we mean solutions having an interface, i.e. a transition layer between two constants. It is well known that in 1D domains and in many 2D domains, patterns are unstable for this equation. We show that, unlike the 1D case, but as in 2D dumbbell domains, stable patterns exist. In a more general way, we prove invariance of stability properties for steady states when a sequence of domains Ω_n_ converges to our limit domain in the sense of Mosco. We illustrate the theoretical results by numerical simulations of evolving and persisting interfaces.

## Cite this article

Łukasz Bolikowski, Maria Gokieli, Nicolas Varchon, The Neumann problem in an irregular domain. Interfaces Free Bound. 12 (2010), no. 4, pp. 443–462

DOI 10.4171/IFB/241