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Consider $\Delta u =f(x) , x \in \Omega $ and $\nabla u\cdot n +\alpha u = g(x) , x\in \partial\Omega $, where $n$ is outward normal. Can anyone help me to define a bilinear form for this PDE and find out whether it is coercive or not? I am not quite familiar how to do it.

So far my progress:

$$\int_\Omega \Delta u .v= \int fv $$

$$-\int_\Omega \nabla u. \nabla v +\int\nabla.( \nabla u v) =\int fv$$

Using the given relation it can be further written as

$$-\int\nabla u .\nabla v -\int \alpha u.v = \int- g\alpha v +\int fv$$

i.e., bilinear form is $$B[u,v]=\int \nabla u \nabla v +\alpha \int uv $$

That means if $\alpha \ge 1 $ then it is coercive.

Am I right?

Thank you.

Gonçalo
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1 Answers1

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I am assuming you are familiar with the setting of your equation i.e. Hilbert spaces. If this is the case, then you can easily show $B[u,u] \geq C\|u\|^2$ by choosing $v = u$ in your bilinear term, using Friedrich's inequality and the assumption that $\alpha > 0$.

uli.xu
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