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The Poisson equation involving surface measures

Abstract We prove the (optimal) -regularity of weak solutions to the equation in a domain with Dirichlet boundary conditions, where is a compact (Lipschitz) manifold and We also discuss optimality and necessity of the assumptions on Q and Γ. Our findings can be applied to study the regularity of solutions for several free boundary problems, in particular the biharmonic Alt–Caffarelli problem.