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.
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