夹逼准则与椭圆型偏微分方程的上下解方法

Squeeze Theorem and the Upper and Lower Solutions of Defferential Equation

引进微分方程上下解的概念,应用极限夹逼准则的思想,以椭圆型偏微分方程为例,用上解与下解来夹逼,证明了半线性椭圆偏微分方程边值问题解的存在性。这种证明是构造性的证明,它比单纯的存在性证明(如不动点定理)来得优越。因为我们不仅证明出解的存在,而且能够通过计算机进行逐次迭代,把这个解按任意事先要求的精度把它估算出来。

Applying the idea of limit squeeze thearem, the concept of upper and lower solutions for differential equations is introdued in this paper. Take elliptic partial differential equations as an example, we use the upper and lower solutions to squeeze and prove the existence of the boundary value problems for the semi-linear elliptic differential equation. This is the proof of tectonic better than pure existence （ such as the fixed point theorem）. Because we not only proves the existence of the solution, and through the solution for successive iteration, but also estimate the solution according to the solution of the preci- sion requirement at any prior.