Posts Tagged 'integral representation formulas'

Reviewing the regularity theory of elliptics PDEs via the Laplace equation. Part III. (representation formulas)

This is the third of a series of posts dealing with the regularity theory of elliptic equations. My motivation in writing these is outlined in the first post. The previous post is here.

Let us recall Green’s identity, if ${u,v}$ are any functions smooth in ${\bar{\Omega}}$ and ${\Omega}$ is a bounded domain with smooth boundary we have

$\displaystyle \int_\Omega u\Delta v - v \Delta u dx = \int_{\partial \Omega}u\frac{\partial v}{\partial \nu}-v\frac{\partial u}{\partial \nu}dS$

this identity can be obtained with a couple of integration by parts involving the vector fields ${u \nabla v}$ and ${v \nabla u}$.

Lets rewrite the identity as

$\displaystyle \int_\Omega u\Delta vdx = -\int_\Omega v \Delta u dx + \int_{\partial \Omega}u\frac{\partial v}{\partial \nu}-v\frac{\partial u}{\partial \nu}dS$

thus, at least formally, if somehow we could find for every ${x \in \Omega}$ a function ${v_x(y)}$ such that

$\displaystyle \Delta v_x(y)=\delta_x(y)$

$\displaystyle v_x(y) \equiv 0 \mbox{ on } \partial \Omega$

then Green’s identity applied to both ${u}$ and ${v_x}$ in ${\Omega}$ would give us an integral representation formula for harmonic functions