This is to solve Dirichlet problem using Green's identities. The book gave some examples.(adsbygoogle = window.adsbygoogle || []).push({});

My question is: Why the book keep talking [itex]v[/itex] is harmonic(periodic) function. What is the difference whether [itex]v[/itex] is harmonic function or not as long as [itex]v[/itex] has continuous first and second derivatives.?

Green's identity:

[tex]\int\int_{\Omega}(u\nabla^2v + \nabla u \cdot \nabla v)dx dy = \int_{\Gamma} u\frac{\partial v}{\partial n} ds[/tex]

If we let u=1:

[tex]\int\int_{\Omega} \nabla^2 v dx dy = \int_{\Gamma} \frac{\partial v}{\partial n} ds[/tex]

For Dirichlet problem, [itex]\nabla^2 v = 0 [/itex], Therefore:

[tex] \int_{\Gamma} \frac{\partial v}{\partial n} ds = 0 [/tex]

I have no issue with the math portion. It will be the same even though [itex]v[/itex] is not harmonic as long as [itex]v[/itex] has continuous first and second derivatives.

As long as [itex]\nabla^2 v = 0 [/itex], the result is the same. Why the book keep mentioning [itex] v[/itex] being harmonic function in a few example.

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# Solving Dirichlet problem using Green's identities.

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