Reduced measures on the boundary

Brezis, Haïm;Ponce, Augusto
(2005) International Journal of Functional Analysis, Operator Theory & Applications — Vol. 229, n° 1, p. 95-120 (2005)

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We study the existence of solutions of the nonlinear problem {-Delta u + g(u) = 0 in Omega, u = mu on partial derivative Omega, where mu is a and g : R -> R is a nondecreasing continuous function with g (t) = 0, for all t <= 0. Problem (0.1) admits a solution for every mu epsilon L-1 (partial derivative Omega), but this need not be the case when p is a general bounded measure. We introduce a concept of reduced measure mu* (in the spirit of Brezis et al. (Ann. Math. Stud., to appear)); this is the "closest" measure to mu for which (0.1) admits a solution.
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Brezis, H., & Ponce, A. (2005). Reduced measures on the boundary. International Journal of Functional Analysis, Operator Theory & Applications, 229(1), 95-120. https://doi.org/10.1016/j.jfa.2004.12.001 (Original work published 2005)