  We propose a novel weak solution theory for the Mullins-Sekerka equation
primarily motivated from a gradient flow perspective. Previous existence
results on weak solutions due to Luckhaus and Sturzenhecker (Calc. Var. PDE 3,
1995) or R\"oger (SIAM J. Math. Anal. 37, 2005) left open the inclusion of both
a sharp energy dissipation principle and a weak formulation of the contact
angle at the intersection of the interface and the domain boundary. To
incorporate these, we introduce a functional framework encoding a weak solution
concept for Mullins-Sekerka flow essentially relying only on (i) a single sharp
energy dissipation inequality in the spirit of De~Giorgi, and (ii) a weak
formulation for an arbitrary fixed contact angle through a distributional
representation of the first variation of the underlying capillary energy. Both
ingredients are intrinsic to the interface of the evolving phase indicator and
an explicit distributional PDE formulation with potentials can be derived from
them. Existence of weak solutions is established via subsequential limit points
of the naturally associated minimizing movements scheme. Smooth solutions are
consistent with the classical Mullins-Sekerka flow, and even further, we expect
our solution concept to be amenable, at least in principle, to the recently
developed relative entropy approach for curvature driven interface evolution.
