3.4: Sistemas de Segundo Orden

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Aquí consideramos el sistema
\ begin {ecuación}
\ label {systwo}
\ suma_ {k, l=1} ^na^ {kl} (x, u,\ nabla u) u_ {x_kx_l} +\ mbox {términos de orden inferior} =0,
\ end {ecuación}
donde$A^{kl}$ están$(m\times m)$ las matrices y$u=(u_1,\ldots,u_m)^T$. Asumimos$A^{kl}=A^{lk}$, que no es restricción de generalidad siempre que se satisfagan $u\ en C^2$.
Al igual que en los apartados anteriores, la clasificación se desprende de la pregunta de si podemos o no calcular formalmente la solución a partir de las ecuaciones diferenciales, si se dan bastantes datos sobre un colector inicial. Que el colector inicial$\mathcal{S}$ sea dado por$\chi(x)=0$ y asuma eso$\nabla\chi\not=0$. El mapeo$x=x(\lambda)$, ver secciones anteriores, lleva a
$$
\ sum_ {k, l=1} ^na^ {kl}\ chi_ {x_k}\ chi_ {x_l} v_ {\ lambda_n\ lambda_n} =\ mbox {términos conocidos en}\\ mathcal {S},
$$
donde$v(\lambda)=u(x(\lambda))$.

La ecuación característica está aquí
$$
\ det\ left (\ suma_ {k, l=1} ^Na^ {kl}\ chi_ {x_k}\ chi_ {x_l}\ right) =0.
$$
Si hay una solución$\chi$ con$\nabla\chi\not=0$, entonces es posible que los segundos derivados no sean continuos en un barrio de$\mathcal{S}$.

Definición. El sistema se llama elíptico si
$$
\ det\ left (\ sum_ {k, l=1} ^na^ {kl}\ zeta_k\ zeta_l\ right)\ not=0
$$
para todos$\zeta\in\mathbb{R}$,$\zeta\not=0$.

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