Reduced measures on the boundary

We study the existence of solutions of the nonlinear problem(0.1)-Δu+g(u)=0inΩ,u=μon∂Ω,where μ is a bounded measure and g:R→R is a nondecreasing continuous function with g(t)=0, ∀t⩽0. Problem (0.1) admits a solution for every μ∈L1(∂Ω), but this need not be the case when μ is a general bounded measure. We introduce a concept of reduced measure μ* (in the spirit of Brezis et al. (Ann. Math. Stud., to appear)); this is the “closest” measure to μ for which (0.1) admits a solution.