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MASS TRANSPORT PROBLEMS OBTAINED AS LIMITS OF p−LAPLACIAN TYPE PROBLEMS WITH SPATIAL DEPENDENCE
"... Abstract. We consider the following problem: given a bounded convex domain Ω ⊂ RN we consider the limit as p → ∞ of solutions to −div (b−pp Dup−2Du) = f+ − f−, in Ω, b−pp Dup−2 ∂u∂η = 0, on ∂Ω. Under appropriate assumptions on the coefficients bp, that in particular verify that limp→ ∞ bp = b ..."
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Abstract. We consider the following problem: given a bounded convex domain Ω ⊂ RN we consider the limit as p → ∞ of solutions to −div (b−pp Dup−2Du) = f+ − f−, in Ω, b−pp Dup−2 ∂u∂η = 0, on ∂Ω. Under appropriate assumptions on the coefficients bp, that in particular verify that limp→ ∞ bp = b uniformly in Ω, we prove that there is a uniform limit of upj (along a sequence pj →∞) and that this limit is a Kantorovich potential for the optimal mass transport problem of f+ to f− with cost c(x, y) given by the formula c(x, y) = infσ(0)=x,σ(1)=y σ b ds.
Local regularity results for value functions of tugofwar with noise and running payoff
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THE BEHAVIOR OF SOLUTIONS TO AN ELLIPTIC EQUATION INVOLVING A p−LAPLACIAN AND A q−LAPLACIAN FOR LARGE p.
"... Abstract. In this paper we study the behavior as p→ ∞ of solutions up,q to −∆pu−∆qu = 0 in a bounded smooth domain Ω with a Lipschitz Dirichlet boundary datum u = g on ∂Ω. We find that there is a uniform limit of a subsequence of solutions, that is, there is pj → ∞ such that upj,q → u ∞ uniformly i ..."
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Abstract. In this paper we study the behavior as p→ ∞ of solutions up,q to −∆pu−∆qu = 0 in a bounded smooth domain Ω with a Lipschitz Dirichlet boundary datum u = g on ∂Ω. We find that there is a uniform limit of a subsequence of solutions, that is, there is pj → ∞ such that upj,q → u ∞ uniformly in Ω and we prove that this limit u ∞ is a solution to a variational problem, that, when the Lipschitz constant of the boundary datum is less or equal than one, is given by the minimization of the Lqnorm of the gradient with a pointwise constraint on the gradient. In addition, we show that the limit is a viscosity solution to a limit PDE problem that involves the q−Laplacian and the ∞−Laplacian.
DOI 10.1515/anona20130022  Adv. Nonlinear Anal. 2014; 3 (3):133–140 Research Article
"... Mass transport problems obtained as limits of 푝Laplacian type problems with spatial dependence Abstract:We consider the following problem: given a bounded convex domain 훺 ⊂ ℝ 푁 we consider the limit as 푝 → ∞ of solutions to {{{{{ − div(푏 −푝 푝 퐷푢푝−2퐷푢) = 푓+ − 푓 − in 훺,푏−푝 푝 퐷푢푝−2 휕푢휕 휂 = 0 ..."
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Mass transport problems obtained as limits of 푝Laplacian type problems with spatial dependence Abstract:We consider the following problem: given a bounded convex domain 훺 ⊂ ℝ 푁 we consider the limit as 푝 → ∞ of solutions to {{{{{ − div(푏 −푝 푝 퐷푢푝−2퐷푢) = 푓+ − 푓 − in 훺,푏−푝 푝 퐷푢푝−2 휕푢휕 휂 = 0 on 휕훺. Under appropriate assumptions on the coecients 푏 푝 that in particular verify that lim푝→ ∞ 푏 푝 = 푏 uniformly in 훺, we prove that there is a uniform limit of 푢푝 푗 (along a sequence 푝 푗 → ∞) and that this limit is a Kantorovich potential for the optimal mass transport problem of 푓+ to 푓 − with cost 푐(푥, 푦) given by the formula 푐(푥, 푦) = inf휎(0)=푥, 휎(1)= 푦 ∫ 휎 푏 푑푠. Keywords:Mass transport, Monge–Kantorovich problems, 푝Laplacian equation