# Weak formulation for singular diffusion equation with dynamic boundary condition

## TL;DR

This paper introduces a weak formulation for a singular diffusion equation with dynamic boundary conditions, enabling better analysis of solutions through a reformulation involving convex energy subdifferentials.

## Contribution

It presents a novel weak formulation based on an evolution equation with subdifferential of convex energy, linking weak solutions to classical PDE solutions.

## Key findings

- Validation of the weak formulation's adequacy
- Establishment of common properties with regular PDE solutions
- Theoretical framework for singular diffusion equations with dynamic boundaries

## Abstract

In this paper, we propose a weak formulation of the singular diffusion equation subject to the dynamic boundary condition. The weak formulation is based on a reformulation method by an evolution equation including the subdifferential of a governing convex energy. Under suitable assumptions, the principal results of this study are stated in forms of Main Theorems A and B, which are respectively to verify: the adequacy of the weak formulation; the common property between the weak solutions and those in regular problems of standard PDEs.