- #1
yungman
- 5,741
- 294
Homework Statement
Suppose u is harmonic (\nabla^2 u = 0) and v=0 \;\hbox{ on } \;\Gamma where \Gamma is the boundary of a simple or multiply connected region and \Omega is the region bounded by \Gamma.
Using Green's identities, show:
\int \int_{\Omega} \nabla u \cdot \nabla v \; dx dy = 0
Homework Equations
Green's identities:
\int \int_{\Omega}\; (u\nabla^2 v \;+\; \nabla u \cdot \nabla v) \; dx dy \;= \;\int_{\Gamma} \; u\frac{\partial v}{\partial n} \; ds
\int \int_{\Omega} \;(u\nabla^2 v \;-\; v\nabla^2 u )\; dx dy \;=\; \int_{\Gamma} \;(u\frac{\partial v}{\partial n} - v\frac{\partial u}{\partial n}) \;ds
\frac{\partial u}{\partial n} = \nabla u \; \cdot \; \widehat{n}
The Attempt at a Solution
I use
\int \int_{\Omega}\; (u\nabla^2 v \;+\; \nabla u \cdot \nabla v) \; dx dy \;= \;\int_{\Gamma} \; u\frac{\partial v}{\partial n} \; ds
If v=0 \hbox { on } \Gamma \;\;\Rightarrow\;\; v \hbox { is a constant = 0 } \;\;\Rightarrow\;\; v= 0 \;\hbox{ on } \;\Omega.
\int \int_{\Omega}\; (u\nabla^2 v \;+\; \nabla u \cdot \nabla v) \; dx dy \;= \int \int_{\Omega}\; \nabla u \cdot \nabla v) \; dx dy \;= \;\int_{\Gamma} \; u\frac{\partial v}{\partial n} \; ds = 0
Is this the right way?
Last edited:
