Displaying similar documents to “Equivalent formulation and numerical analysis of a fire confinement problem”

The geometrical quantity in damped wave equations on a square

Pascal Hébrard, Emmanuel Humbert (2006)

ESAIM: Control, Optimisation and Calculus of Variations

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The energy in a square membrane subject to constant viscous damping on a subset ω Ω decays exponentially in time as soon as satisfies a geometrical condition known as the “Bardos-Lebeau-Rauch” condition. The rate τ ( ω ) of this decay satisfies τ ( ω ) = 2 min ( - μ ( ω ) , g ( ω ) ) (see Lebeau [ (1996) 73–109]). Here μ ( ω ) denotes the spectral abscissa of the damped wave equation operator and  g ( ω ) is a number called the geometrical quantity of and defined as follows. A ray in is the trajectory generated by the free...

Steinhaus chessboard theorem

Władysław Kulpa, Lesƚaw Socha, Marian Turzański (2000)

Acta Universitatis Carolinae. Mathematica et Physica

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Existence of optimal maps in the reflector-type problems

Wilfrid Gangbo, Vladimir Oliker (2007)

ESAIM: Control, Optimisation and Calculus of Variations

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In this paper, we consider probability measures and on a -dimensional sphere in 𝐑 d + 1 , d 1 , and cost functions of the form c ( 𝐱 , 𝐲 ) = l ( | 𝐱 - 𝐲 | 2 2 ) that generalize those arising in geometric optics where l ( t ) = - log t . We prove that if and vanish on ( d - 1 ) -rectifiable sets, if lim t 0 + l ( t ) = + , and g ( t ) : = t ( 2 - t ) ( l ' ( t ) ) 2 is monotone then there exists a unique optimal map that transports onto ν , where optimality is measured against Furthermore, inf 𝐱 | T o 𝐱 - 𝐱 | > 0 . Our approach is based on direct variational arguments. In the special case when l ( t ) = - log t , existence...