格林公式及其在微分方程中的应用

Green’s formula and its application indifferential equations

格林公式沟通了被积函数在积分区域上的积分和边界积分的关系。在一维函数中,格林公式即为定积分的分部积分法,在高维函数中,分部积分法与格林公式不再相同。首先给出一维分部积分法并加以证明,其次给出二维格林公式及其证明并利用高维函数分部积分法证明一般形式的格林公式,最后给出格林公式在微分方程变分问题中的一些应用。

Green’s formula communicates the relationship between the integral of the integrand in the integral region and the boundary integral.In one-dimensional function,Green’s formula is the partial integration method of definite integral.In higher dimensional function,the partial integration method is no longer the same as Green’s formula.Firstly,the one-dimensional partial integration method is given and proved.Secondly,the two-dimensional Green’s formula and its proof are given,and the general Green’s formula is proved by using the high-dimensional partial integration method.Finally,some applications of the Green’s formula in the variational problems of differential equations are given.