An optimization problem with volume constraint with applications to optimal mass transport

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:267

Article

da Silva, Joao Vitor^1,2  Del Pezzo, Leandro M.^5,6  Rossi, Julio D.^3,4 

[1] Univ Brasilia, Inst Ciencias Exatas, Dept Matemat, Campus Univ Darcy Ribeiro, BR-70910900 Brasilia, DF, Brazil

[2] CONICET Argentine, Inst Invest Matemat Luis A Santalo IMAS, Ciudad Univ,Pabellon 1 1428 Av Cantilo S-N, Buenos Aires, DF, Argentina

[3] Consejo Nacl Invest Cient & Tecn, Ciudad Univ Pabellon 1,C1428EGA, Buenos Aires, DF, Argentina

[4] Univ Buenos Aires, FCEyN, Dept Math, Ciudad Univ Pabellon 1,C1428EGA, Buenos Aires, DF, Argentina

[5] Consejo Nacl Invest Cient & Tecn, Av Figueroa Alcorta 735,C1428BCW, Buenos Aires, DF, Argentina

[6] UTDT, Dept Matemat & Estadist, Av Figueroa Alcorta 735,C1428BCW, Buenos Aires, DF, Argentina

关键词: Optimization problems;    Volume constraint;    Neumann boundary condition;    Infinity-Laplace operator;    Monge-Kantorovich problem;   

DOI  :  10.1016/j.jde.2019.06.007

来源: Elsevier

【 摘 要 】

In this manuscript we study the following optimization problem with volume constraint: min {1/p integral(Omega) vertical bar del v vertical bar(P) dx - integral(partial derivative Omega) g v d HN-1: v is an element of W-1,W-p (Omega), and L-N({v >0}) >= alpha}. Here Omega subset of R-N is a bounded and smooth domain, alpha is a continuous function and a is a fixed constant such that 0 < alpha < L-N (Omega). Under the assumption that integral(partial derivative Omega) g(x)d HN-1 > 0 we prove that a minimizer exists and a satisfies {-Delta(p)u(p) = 0 in {u(p) > 0} boolean OR {u(p) < 0}, vertical bar del u(p)vertical bar(p-2) partial derivative u(p)/partial derivative eta = g on partial derivative Omega boolean AND partial derivative({u(p) > 0} boolean OR {u(p) <0}), L-N ({u(p) >0}) = alpha. Next, we analyze the limit as p -> infinity. We obtain that any sequence of weak solutions converges, up to a subsequence, lim(pj ->infinity) u(pj) (x)= u(infinity) (x), uniformly in (Omega) over bar, and uniform limits, u(infinity), are solutions to the maximization problem with volume constraint max {integral(partial derivative Omega) g v d HN-1: v is an element of W-1,W-infinity (Omega), parallel to del v parallel to(L infinity(Omega)) <= 1 and L-N ({v > 0) <= alpha}. Furthermore, we obtain the limit equation that is verified by u(infinity) 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 partial derivative Omega and along the limiting free boundary, partial derivative{u(infinity)not equal 0}. Furthermore, we show some explicit examples of solutions for certain configurations of the domain and data. (C) 2019 Elsevier Inc. All rights reserved.

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