Geometrical Optics and Related Topics by Serge Alinhac (auth.), Ferruccio Colombini, Nicolas Lerner

By Serge Alinhac (auth.), Ferruccio Colombini, Nicolas Lerner (eds.)

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T. a suitable Riemann-type metric, uniformlyequivalent to the standard Ll distance. 1); • a dense subset V PL C V of piecewise Lipschitz functions; F. Colombini et al. ), Geometrical Optics and Related Topics © Birkhäuser Boston 1997 32 P. Baiti and A. Bressan • for each u E V PL , a space Tu. of first order generalized tangent vectors (v,O, with weighted norm II(v,~)IIu.. If, say, u is piecewise Lipschitz with N jumps, then Tu. ::: Ll x ]RN. e. e. ; "((a) = u, "((b) = v}. We remark that this weighted distance does not fit within the standard framework of Riemann or Finsler manifolds [8, p.

Colombini et al. ), Geometrical Optics and Related Topics © Birkhäuser Boston 1997 32 P. Baiti and A. Bressan • for each u E V PL , a space Tu. of first order generalized tangent vectors (v,O, with weighted norm II(v,~)IIu.. If, say, u is piecewise Lipschitz with N jumps, then Tu. ::: Ll x ]RN. e. e. ; "((a) = u, "((b) = v}. We remark that this weighted distance does not fit within the standard framework of Riemann or Finsler manifolds [8, p. 362]' because our tangent spaces Tu. are defined not for all u in some open set U C Ll, but only for u in a dense subset with empty interior.

Jv < PN < M, 41 Weighted path length We now approximate each u 8m with a piecewise constant function u m . The restriction of the original path "I to the subinterval [Om-I, Om] is then replaced by a new path "I' defined as follows. 34) "I '('13)'= u m . 1 + u m-l . 35) "1'('13) ~ um·X. , p;] +u m- l ·X xfm. lulpi,+OO[ Clearly, "I' is piecewise elementary, with "1'( -M) = u m- l , "I'(M) = u m. We now perform a suitable parameter rescaling 0 f-+ '13(0) mapping [Om-l,Om] onto [-M,M] and define the path "Iv(O) ~ "1'('13(0)).

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