Relaxation in an L-optimization problem

Hitoshi Ishii*, Paola Loreti

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    5 Citations (Scopus)


    Let Ω be an open bounded subset of ℝn and f a continuous function on Ω̄ satisfying f(x) > 0 for all x ∈ Ω̄. We consider the maximization problem for the integral ∫Ω f(x)u(x) dx over all Lipschitz continuous functions u subject to the Dirichlet boundary condition u = 0 on ∂Ω and to the gradient constraint of the form H(Du(x)) ≤ 1, and prove that the supremum is 'achieved' by the viscosity solution of Ĥ(Du(x)) = 1 in Ω and u = 0 on ∂Ω, where Ĥ denotes the convex envelope of H. This result is applied to an asymptotic problem, as p → ∞, for quasi-minimizers of the integral ∫Ω [1/pH(Du(x))p - f(x)u(x)] dx. An asymptotic problem as k → ∞ for inf ∫Ω [k dist(Du(x), K) - f(x)u(x)] dx is also considered, where the infimum is taken all over u ∈ W0 1,1(Ω) and the set K is given by {ξ

    Original languageEnglish
    Pages (from-to)599-615
    Number of pages17
    JournalRoyal Society of Edinburgh - Proceedings A
    Issue number3
    Publication statusPublished - 2003

    ASJC Scopus subject areas

    • Mathematics(all)


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