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3.4: Systems of Second Order

( \newcommand{\kernel}{\mathrm{null}\,}\)

Here we consider the system
nk,l=1Akl(x,u,u)uxkxl+lower order terms=0,
where Akl are (m×m) matrices and u=(u1,,um)T. We assume Akl=Alk, which is no restriction of generality provided $u\in C^2$ is satisfied.
As in the previous sections, the classification follows from the question whether or not we can calculate formally the solution from the differential equations, if sufficiently many data are given on an initial manifold. Let the initial manifold S be given by χ(x)=0 and assume that χ0. The mapping x=x(λ), see previous sections, leads to
nk,l=1Aklχxkχxlvλnλn=terms known on S,
where v(λ)=u(x(λ)).

The characteristic equation is here
det(nk,l=1Aklχxkχxl)=0.
If there is a solution χ with χ0, then it is possible that second derivatives are not continuous in a neighborhood of S.

Definition. The system is called elliptic if
det(nk,l=1Aklζkζl)0
for all ζR, ζ0.

Contributors and Attributions


This page titled 3.4: Systems of Second Order is shared under a not declared license and was authored, remixed, and/or curated by Erich Miersemann.

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