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| Partial Differential Equations of Higher Order and Symmetric Positive Systems |
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Citation： |
Xu Zhengfan.Partial Differential Equations of Higher Order and Symmetric Positive Systems[J].Chinese Annals of Mathematics B,1982,3(5):663~678 |
| Page view： 795
Net amount： 866 |
Authors： |
Xu Zhengfan; |
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| Abstract： |
In this paper we study the relation between symmetric positive systems and equations of higher order. The main result is:
Theorem 1. An equation of second order $L\phi =f$
can be transformed into a symmetric positive system by introducing new unknown functions
$u_i=\sum\limits_{j=0}^n {\alpha_ij \varphi _j(i=0,1,\cdots,n),\varphi_0=\varphi_2,\varphi_j=\partial \varphi /\partial x_j}$
iff there exists L_1 of order 1 such that
$Re(L_1 \varphi \cdot \bar {L\varphi})=\sum\limits_{i=1}^n{\frac{\partial}{\partial x_i}}+B(\varphi,\varphi)$,
where P_i(\varphi,\varphi)(i=1,2,\cdots,n),B(\varphi,\varphi) are differential quadarlic forms and B(\varphi,\varphi) is positive definite.
This Theorem can be extended into equations of higher order.
Some examples of deducing equations of higher order into symmetric positive systems are given.
Finally, we give a counter example which shows that a boundary problem of a symmetric positive system deduced from an equation of higher order is admissible, but its corresponding bounbary problem of the original equation is not well-posed. |
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