Relaxation in an L∞-optimization problem
Relaxation in an L∞-optimization problem
复制标题
L∞ 优化问题中的松弛
DOI：
10.1017/s0308210500002559
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发表时间：
2003
期刊：
影响因子：
--
通讯作者：
P. Loreti
中科院分区：
文献类型：
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作者：
H. Ishii;P. Loreti
设Ω是Rn的有界开子集，f是Ω̄上满足f(X)>；0的连续函数，对所有x个∈Ω̄。考虑了∫Ω上的积分∂Ωf(X)u(X)dx在Dirichlet边界条件u=0和形式H(Du(X))≤1的梯度约束下的极大化问题，证明了上确界是通过Ĥ(Du(X))=1在Ω中的粘性解和u=0在∂Ω上的粘性解来实现的，其中Ĥ表示H的凸包络.这一结果被应用于一个渐近问题，如p→∞，对于积分的拟极小点，还考虑了一个渐近问题k→∞，其中下确界被遍历，集合K由{ξ|H(ξ)≤1})给出。
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 ∫Ωf(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 An asymptotic problem as k → ∞ for inf is also considered, where the infimum is taken all over and the set K is given by {ξ | H(ξ) ≤ 1}.