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• W2953724892 abstract "In this manuscript we study the following optimization problem with volume constraint:min{1p∫Ω|∇v|pdx−∫∂ΩgvdHN−1:v∈W1,p(Ω), and LN({v>0})≤α}. Here Ω⊂RN is a bounded and smooth domain, g is a continuous function and α is a fixed constant such that 0<α<LN(Ω). Under the assumption that ∫∂Ωg(x)dHN−1>0 we prove that a minimizer exists and satisfies{−Δpup=0in {up>0}∪{up<0},|∇up|p−2∂up∂η=gon ∂Ω∩∂({up>0}∪{up<0}),LN({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 constraintmax⁡{∫∂ΩgvdHN−1:v∈W1,∞(Ω),‖∇v‖L∞(Ω)≤1 and LN({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}. Furthermore, we show some explicit examples of solutions for certain configurations of the domain and data." @default.
• W2953724892 created "2019-07-12" @default.
• W2953724892 creator A5016268671 @default.
• W2953724892 creator A5041065153 @default.
• W2953724892 creator A5051603007 @default.
• W2953724892 date "2019-11-01" @default.
• W2953724892 modified "2023-10-09" @default.
• W2953724892 title "An optimization problem with volume constraint with applications to optimal mass transport" @default.
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• W2953724892 doi "https://doi.org/10.1016/j.jde.2019.06.007" @default.
• W2953724892 hasPublicationYear "2019" @default.
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