2018/05/31 by João Vítor da Silva, Joao Vitor da Silva, Leandro M. Del Pezzo +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Combinatorics #Geometric Analysis and Curvature Flows #Mathematical analysis #Mathematical physics #Mathematics #Nabla symbol #Nonlinear Partial Differential Equations #Omega #Physics #Quantum mechanics #Subsequence #math.AP #msc:35B65 #msc:35J20 #msc:35J60 #msc:35J66
paper · pdf · doi:10.1016/j.jde.2019.06.007
published as J. Differential Equations 267 (2019), no. 10, 5870-5900 · 23 pages, 2 figures
arxiv created 2018/08/28 · openalex publication_date 2019/06/25 · arxiv updated 2020/10/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
In this manuscript we study the following optimization problem with volume constraint: min\(1)/(p)∫Ω |∇ v|pdx- ∫∂ Ω gv dS \colon v ∈ W1, p (Ω), and |\v>0\| ≤ α\. Here g is acontinuous function and α is a fixed constant such that 0< α< |Ω|. Under the assumption that ∫∂ Ω g(x)dS >0 we prove that a minimizer exists and satisfies \ -Δp up = 0 · in · \up>0\ ∪ \up<0\,
|∇ up|p-2(∂ up)/(∂ ν) = g · on · ∂ Ω ∩∂(\up>0\ ∪ \up<0\ ) ,
|\up>0\| = α. · · . Next, we analyze the limit as p→ ∞. We obtain that any sequence of weak solutions converges, up to a subsequence, limpj → ∞ upj(x)=u∞(x), uniformly in Ω, and uniform limits, u_∞, are solutions to the maximization problem with volume constraint max\ ∫∂ Ω gv dS \colon v ∈ W1, ∞ (Ω), ‖∇ v‖L∞(Ω)≤ 1 and |\v>0\| ≤ α\. Furthermore, we obtain the limit equation that is verified by u_∞ in the viscosity sense. Finally, it turns out that such a limit variational problem is connected to the Monge-Kantorovich mass transfer problem with the involved measures are supported on ∂ Ω and along the limiting free boundary, ∂ \u∞ ≠ 0\.