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A weak formulation for a class of parabolic free boundary problems (FBP) isproposed that does not involve the notion of a free boundary but reduces to aFBP when classical solutions exist. It is aimed at hydrodynamic limits (HDL) ofparticle systems with selection in circumstances where the macroscopic modeldoes not possess (or is hard to prove to possess) a regular free boundary inthe classical sense. The formulation involves the macroscopic density ofparticles and a measure that accounts for selection. It consists of a secondorder parabolic equation satisfied by the density and driven by the measure,coupled with a complementarity condition satisfied by the density-measure pair.The approach is applied to an injection-branching-selection particle system ofdiffusion on $\R$ under arbitrarily varying injection and removal rates, forwhich the corresponding FBP is not in general known to be classically solvable.The HDL is characterized as the unique solution to the weak formulation. Theproof of convergence is based on PDE uniqueness, which in turn relies on thebarrier method.

A weak formulation of free boundary problems and its application to hydrodynamic limits of particle systems with selection