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Let $\Omega$ be an open and connect subset of $\mathbb{R}^2$,we denote by $\partial \Omega$ its boundary the latter is supposed to be smooth ($\mathcal{C}^\infty)$, its outword normal vector is denoted by $n$. Let $f: \Omega \mapsto \mathbb{R}$ such that $f(x)\geq \alpha > 0$. Now, let $A : \mathbb{H}^{1/2}(\partial \Omega)\mapsto \mathbb{H}^{-1/2}(\partial \Omega) $. Such that $A(\varphi)=\frac{\partial u }{n} $, with $u$ is the unique solution in $\mathbb{H}^1(\Omega)$ of

$div(f\nabla u)=0$ and $u_{|\partial\Omega}=\varphi$.

Can I say that $A$ is a pseudo-differential Operator ?

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Finally, The answer is yes. In deed the result remains true if we remplace $div(f \nabla .)$by any other second order elliptic opertor. Morover, $A$ is of order one. The proof can be found here : https://arxiv.org/pdf/1212.6785.pdf

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