### Abstract

Let Ω be an open bounded subset of ℝ^{n} 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 ∈ W_{0}
^{1,1}(Ω) and the set K is given by {ξ

Original language | English |
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Pages (from-to) | 599-615 |

Number of pages | 17 |

Journal | Royal Society of Edinburgh - Proceedings A |

Volume | 133 |

Issue number | 3 |

Publication status | Published - 2003 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

^{∞}-optimization problem.

*Royal Society of Edinburgh - Proceedings A*,

*133*(3), 599-615.