  The unsigned p-Willmore functional introduced in \cite{mondino2011}
generalizes important geometric functionals which measure the area and Willmore
energy of immersed surfaces. Presently, techniques from \cite{dziuk2008} are
adapted to compute the first variation of this functional as a weak-form system
of equations, which are subsequently used to develop a model for the p-Willmore
flow of closed surfaces in $\mathbb{R}^3$. This model is amenable to
constraints on surface area and enclosed volume, and is shown to decrease the
p-Willmore energy monotonically over time. In addition, a penalty-based
regularization procedure is formulated to prevent artificial mesh degeneration
along the flow; inspired by a conformality condition derived in
\cite{kamberov1996}, this procedure encourages angle-preservation in a closed
and oriented surface immersion as it evolves. Following this, a finite-element
discretization of both systems is discussed, and an application to mesh editing
is presented.
