# Contact angle selection for interfaces in growing domains

## TL;DR

This paper investigates how a growing bistable region influences the contact angle of interfaces in an Allen-Cahn model, revealing a mechanism for angle selection during domain expansion.

## Contribution

It introduces a novel analysis of contact angle selection in a directional quenching scenario with expanding bistable regions, overcoming technical challenges with Fredholm properties.

## Key findings

- Contact angle depends on the growth of the bistable region.
- Fredholm properties are established in weighted spaces.
- The analysis applies near symmetric, perpendicular contact configurations.

## Abstract

We study interfaces in an Allen-Cahn equation, separating two metastable states. Our focus is on a directional quenching scenario, where a parameter renders the system bistable in a half plane and monostable in its complement, with the region of bistability expanding at a fixed speed. We show that the growth mechanism selects a contact angle between the boundary of the region of bistability and the interface separating the two metastable states. Technically, we focus on a perturbative setting in a vicinity of a symmetric situation with perpendicular contact. The main difficulty stems from the lack of Fredholm properties for the linearization in translation invariant norms. We overcome those difficulties establishing Fredholm properties in weighted spaces and farfield-core decompositions to compensate for negative Fredholm indices.