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Mathematics LibreTexts

3.4: Systems of Second Order

Here we consider the system
\begin{equation}
\label{systwo}
\sum_{k,l=1}^nA^{kl}(x,u,\nabla u)u_{x_kx_l}+\mbox{lower order terms}=0,
\end{equation}
where \(A^{kl}\) are \((m\times m)\) matrices and \(u=(u_1,\ldots,u_m)^T\). We assume \(A^{kl}=A^{lk}\), 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 \(\mathcal{S}\) be given by \(\chi(x)=0\) and assume that \(\nabla\chi\not=0\). The mapping \(x=x(\lambda)\), see previous sections, leads to
$$
\sum_{k,l=1}^nA^{kl}\chi_{x_k}\chi_{x_l}v_{\lambda_n\lambda_n}=\mbox{terms known on}\ \mathcal{S},
$$
where \(v(\lambda)=u(x(\lambda))\).

The characteristic equation is here
$$
\det\left(\sum_{k,l=1}^nA^{kl}\chi_{x_k}\chi_{x_l}\right)=0.
$$
If there is a solution \(\chi\) with \(\nabla\chi\not=0\), then it is possible that second derivatives are not continuous in a neighborhood of \(\mathcal{S}\).

Definition. The system is called elliptic if
$$
\det\left(\sum_{k,l=1}^nA^{kl}\zeta_k\zeta_l\right)\not=0
$$
for all \(\zeta\in\mathbb{R}\), \(\zeta\not=0\).

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