- #1
yungman
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This is to solve Dirichlet problem using Green's identities. The book gave some examples.
My question is: Why the book keep talking v is harmonic(periodic) function. What is the difference whether v is harmonic function or not as long as v has continuous first and second derivatives.?
Green's identity:
\int\int_{\Omega}(u\nabla^2v + \nabla u \cdot \nabla v)dx dy = \int_{\Gamma} u\frac{\partial v}{\partial n} ds
If we let u=1:
\int\int_{\Omega} \nabla^2 v dx dy = \int_{\Gamma} \frac{\partial v}{\partial n} ds
For Dirichlet problem, \nabla^2 v = 0, Therefore:
\int_{\Gamma} \frac{\partial v}{\partial n} ds = 0
I have no issue with the math portion. It will be the same even though v is not harmonic as long as v has continuous first and second derivatives.
As long as \nabla^2 v = 0, the result is the same. Why the book keep mentioning v being harmonic function in a few example.
My question is: Why the book keep talking v is harmonic(periodic) function. What is the difference whether v is harmonic function or not as long as v has continuous first and second derivatives.?
Green's identity:
\int\int_{\Omega}(u\nabla^2v + \nabla u \cdot \nabla v)dx dy = \int_{\Gamma} u\frac{\partial v}{\partial n} ds
If we let u=1:
\int\int_{\Omega} \nabla^2 v dx dy = \int_{\Gamma} \frac{\partial v}{\partial n} ds
For Dirichlet problem, \nabla^2 v = 0, Therefore:
\int_{\Gamma} \frac{\partial v}{\partial n} ds = 0
I have no issue with the math portion. It will be the same even though v is not harmonic as long as v has continuous first and second derivatives.
As long as \nabla^2 v = 0, the result is the same. Why the book keep mentioning v being harmonic function in a few example.
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