# Singular Stochastic Allen-Cahn equations with dynamic boundary conditions

## TL;DR

This paper establishes well-posedness for a class of stochastic Allen-Cahn equations with dynamic boundary conditions, allowing for singular drifts and no growth restrictions, relevant for physics applications.

## Contribution

It introduces a novel well-posedness result for stochastic Allen-Cahn equations with singular, monotone drifts and dynamic boundary conditions, without growth restrictions.

## Key findings

- Proved existence and uniqueness of solutions.
- Handled singular nonlinearities with maximal monotone operators.
- Developed a vanishing viscosity approach for boundary dynamics.

## Abstract

We prove a well-posedness result for stochastic Allen-Cahn type equations in a bounded domain coupled with generic boundary conditions. The (nonlinear) flux at the boundary aims at describing the interactions with the hard walls and is motivated by some recent literature in physics. The singular character of the drift part allows for a large class of maximal monotone operators, generalizing the usual double-well potentials. One of the main novelties of the paper is the absence of any growth condition on the drift term of the evolution, neither on the domain nor on the boundary. A well-posedness result for variational solutions of the system is presented using a priori estimates as well as monotonicity and compactness techniques. A vanishing viscosity argument for the dynamic on the boundary is also presented.