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I am trying to solve the Helmholtz equation with Robin boundary conditions with complex coefficients and the weak formulation

$$ \iint_\limits\Omega\nabla p_0(x,y)\nabla\left(\overline{v(x,y)}\right)dxdy-\frac{w^2}{c^2}\iint\limits_{\Omega}p_0(x,y)\left(\overline{v(x,y)}\right)dxdy\\ +ik\int\limits_{\partial\Gamma_R}p_0(x,y)\left(\overline{v(x,y)}\right)dx=0, \quad \forall v\in V $$

but I do not know how can I prove the existence and uniqueness of the solution because the coercitive in the Lax-Milgram theorem is a big problem.

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    $\begingroup$ I think it would be more helpful to ask on math.stackexchange.com and state explicitly the equations you're are trying to solve. $\endgroup$ – Pedro H. N. Vieira Apr 6 '18 at 20:27
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Here are two options:

  1. Prove the inf-sup condition.
  2. Use the Fredholm alternative and the result that says "only the trivial function can satisfy a second-order PDE with zero right hand side, zero Dirichlet condition, and zero normal derivative on the same part of boundary".

The PhD thesis of Prof. Melenk contains a lot of information regarding the Helmholtz equation:

www.asc.tuwien.ac.at/~melenk/publications/diss.ps.gz

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