, Signalons qu'il s'agit là d'une limitation uniquement dû au modèle physique : en effet, lorsque I s'annule, (II.1) peut-être satisfaite pour différents profils de u. Dans cette seconde partie de manuscrit, la surface S sera donc reconstruite par résolution de l'équation simplifiée (II.2). Comme nous le verrons dès le chapitre 5, il s'agit d'une équation de Hamilton-Jacobi du premier ordre, mal posée en l'état : non unicité flagrante, existence de solutions généralement non classiques, etc. Nous y montrerons en particulier comment le formalisme des solutions de viscosité permet, Ce faisant, si u satisfait à (II.2), elle satisfait en particulier à (II.1)
, corresponds to the abbreviation of the Latin locution exampli gratia, which means for instance. etc. corresponds to the abbreviation of the Latin locution et cetera
, ? If X is a set and A, B two subsets of X, B A is the set of x ? X such that x ? B and x / ? B
, X n are n sets, and x ? X 1 × · · · × X n , for all integer j from 1 to n, we denote by x j or x(j) the j th component of x. Numbers ? N and R respectively correspond to the sets of natural numbers
, ? We denote by the usual order on N or R. In addition, we define N
? We add to R two symbols, ?? and +?, and we extend the usual ordering relation on R admitting that ?? x +? for all x ,
Concretely, if a, b ? R, we define [a, b] = {x ? R | a x b}, [a, +?[ = {x ? R | x a}, ]a, b[ = {x ? R | a < x < b} ,
, ? If A is a non-empty subset (or family of elements) of R lower (resp. upper) bounded, we denote by inf A (resp. sup A) the infimum (resp. supremum) of A, and we also denote it by min A (resp. max A) when there exists a an element of A such that a = inf A (resp. a = sup A). When A is not lower (resp. upper) bounded
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