Existence of Weak Solutions for the Mullins--Sekerka Flow

The long-time existence of solutions for the Mullins--Sekerka problem in a new weak formulation is proved. Using a variational approach introduced by Luckhaus and Sturzenhecker [Calc. Var. Partial Differential Equations, 3 (1995), pp. 253--271], time-discrete solutions are constructed, satisfying approximate Gibbs--Thomson laws in a BV-formulation. But since the passage to a limit allows a loss of surface area for the phase interfaces, convergence in this setting is in general not true. We consider the surface measure of the phase interfaces and use the theory of varifolds to obtain a rigorous passage to a limit in a suitable weak formulation of the Gibbs--Thomson law.