## Fundamental Lemma of calculus of variations

The functional derivative in the Cahn-Hilliard equation can be calculated using this rule

\frac{\delta F}{\delta c} = \frac{\partial f}{\partial c} - \nabla \cdot \frac{\partial f}{\partial\nabla c}

## Weak form of the ACInterface Kernel

The term $$L\nabla(\kappa\nabla\eta)$$$is multiplied with the test function $$\psi$$$ and integrated, yielding

\left(L\nabla(\kappa\nabla\eta),\psi\right) = \left(\nabla \cdot \underbrace{(\kappa\nabla\eta)}_{\equiv \mathbf{a}},\underbrace{L\psi}_{\equiv b}\right)

we moved the $$L$$$over to the right and identify a vector term $$\mathbf{a}$$$ and a scalar term $$b$$\$. Then we use the third equality in the Product rule section to obtain

-\mathbf{a}\cdot \nabla b + \nabla (\mathbf{a}b).

The last term is converted into a surface integral using the Divergence theorem, tielding

- \left( \kappa\nabla\eta,\nabla(L\psi) \right) + \left< L\kappa \nabla\eta \cdot \vec n, \psi\right>