Homogenization of Dirichlet boundary data

To solve a linear Dirichlet problem with boundary value $g$, choose a sufficiently regular extension $G$ whose trace is $g$ and write $v=w+G$. Then $w$ has zero trace, while moving the known term $LG$ to the right-hand side turns the equation $Lv=f$ into the homogeneous-boundary problem $Lw=f-LG$.