Existence of optimal maps in the reflector-type problems

Wilfrid Gangbo; Vladimir Oliker

Displaying similar documents to “Existence of optimal maps in the reflector-type problems”

Long-term planning short-term planning in the asymptotical location problem

Alessio Brancolini, Giuseppe Buttazzo, Filippo Santambrogio, Eugene Stepanov (2008)

ESAIM: Control, Optimisation and Calculus of Variations

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Given the probability measure over the given region $Ω \subset ℝ^{n}$ , we consider the optimal location of a set composed by points in in order to minimize the average distance $Σ \mapsto \int_{Ω} dist (x, Σ) d ν$ (the classical optimal facility location problem). The paper compares two strategies to find optimal configurations: the long-term one which consists in placing all points at once in an optimal position, and the short-term one which consists in placing the points one by one adding at each step at most one point and...

Structure of approximate solutions of variational problems with extended-valued convex integrands

Alexander J. Zaslavski (2008)

ESAIM: Control, Optimisation and Calculus of Variations

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In this work we study the structure of approximate solutions of autonomous variational problems with a lower semicontinuous strictly convex integrand : × $\to$ $\cup$ ${\infty}$ , where is the -dimensional Euclidean space. We obtain a full description of the structure of the approximate solutions which is independent of the length of the interval, for all sufficiently large intervals.

Lower semicontinuity of multiple µ-quasiconvex integrals

Ilaria Fragalà (2010)

ESAIM: Control, Optimisation and Calculus of Variations

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Lower semicontinuity results are obtained for multiple integrals of the kind $\int_{ℝ^{n}} f (x, \nabla_{μ} u) d μ$ , where is a given positive measure on $ℝ^{n}$ , and the vector-valued function belongs to the Sobolev space $H_{μ}^{1, p} (ℝ^{n}, ℝ^{m})$ associated with . The proofs are essentially based on blow-up techniques, and a significant role is played therein by the concepts of tangent space and of tangent measures to . More precisely, for fully general , a notion of quasiconvexity for along the tangent bundle to , turns out to be necessary for...

Generalized Characterization of the Convex Envelope of a Function

Fethi Kadhi (2010)

RAIRO - Operations Research

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We investigate the minima of functionals of the form $\int_{[a, b]} g (\dot{u} (s)) d s$ where is strictly convex. The admissible functions $u : [a, b] ⟶ ℝ$ are not necessarily convex and satisfy $u \leq f$ on , , , is a fixed function on . We show that the minimum is attained by $\bar{f}$ , the convex envelope of .