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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 its coercive or not ? I am not quite familier how to do it . Thank you 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

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

ie . Bilinear form is $B[u,v]=\int \nabla u \nabla v +\alpha \int uv $

that means if $\alpha \ge 1 $ then its coercive . Am i right ? Thank you

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1 Answer 1

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 $\alpha > 0$.

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