Helmholtz equation

Strong form

\[\begin{split} \begin{align*} &\text{Find}~u(\textbf{x}): \Omega \to \mathbb{R}~\text{such that} \\ &(\mathbb{EVP} | \mathbb{BVP})_u\begin{cases} \nabla^2 u + k^2 u = f & \forall\textbf{x}\in\Omega \\ u=u_{\text{D}} & \forall \textbf{x}\in\partial\Omega_{\text{D}} \\ \textbf{n}\cdot\nabla{u} = u_{\text{N}} & \forall\textbf{x}\in\partial\Omega_{\text{N}}=\partial\Omega/\partial\Omega_{\text{D}} \end{cases} \\ &\text{given} \\ &\mathbb{S}_u \begin{cases} \Omega\subset\mathbb{R}^d & \text{domain} \\ u_{\text{D}}(\textbf{x})~,~\partial\Omega_{\text{D}} & \text{Dirichlet boundary condition}\\ u_{\text{N}}(\textbf{x})~,~\partial\Omega_{\text{N}} & \text{Neumann boundary condition}\\ f(\textbf{x}) & \text{forcing}\\ \end{cases} \end{align*} \end{split}\]

Weak form

\[\begin{split} \begin{aligned} & f=0~\text{and}~\partial\Omega_{\text{N}}=\varnothing \implies \begin{array}{ll} \text{Find}~u\in V~\text{and}~\lambda\in\mathbb{C}~\text{such that}\\ a(u, v)=\lambda b(u,v) \quad\forall v\in V \\ \text{where }\\ a(u, v) = \int_\Omega\text{d}\Omega~\nabla v\cdot\nabla u \\ b(u,v) = \int_\Omega\text{d}\Omega~vu \\ \lambda = k^2 \\ \end{array} \\ \\ & f\neq0~\text{or}~\partial\Omega_{\text{N}}\neq\varnothing \implies \begin{array}{ll} \text{Find}~u\in V~\text{such that} \\ F(u, v)=\int_\Omega\text{d}\Omega~-\nabla v\cdot\nabla u + k^2vu - vf + \int_{\partial\Omega_{\text{N}}}\text{d}\Gamma~vu_{\text{N}}=0 \\ \forall v\in V \end{array} \end{aligned} \end{split}\]