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The wave equation -- hyperbolicity by the energy method

Rewrite (hyper.1) as a first order system
or equivalently,
The essential ingredient here is the existence of a positive symmetrizer, H > 0 ,
so that multiplication by H on the left gives
Multiplying by tex2html_wrap_inline10995 we are led to
and the real part of both sides are in fact perfect derivatives, for by the symmetry of H,
and similarly, by the symmetry of tex2html_wrap_inline10999, we have
Hence, by integration over the tex2html_wrap_inline11001-period we end up with energy conservation, asserting
We note that the positivity of H was not used in the proof and is assumed just for the sake of making (u,Hu) an admissible convex ``energy norm.''

Eitan Tadmor
Thu Jan 22 19:07:34 PST 1998