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I'm considering the uniqueness problem for a constant coefficient differential operator $A$ on compact domain $\Omega$ with given boundary information such that we have

\begin{equation}\label{overdetermined problem of constant coefficient} \left\{\begin{aligned} Au&=0&\hspace{5pt}&\text{in}&\hspace{5pt}&\Omega\subset\mathbb{R}^n\\ \partial_\nu^ku&=f_k&\hspace{5pt}&\text{on}&\hspace{5pt}&\partial\Omega \end{aligned}\right. \end{equation}

Whene can we have uniqueness and existence of solutions? For example, I know when $A=\Delta$ and $k=0$ there exists one unique solution. I'm wondering if we can have some similar results for any constant coefficient differential operator.

By the way, I'm not sure whether I need to substitude 'for' by 'of' in the phrase 'the cauchy problem for a constant coefficient differential operator $A$ on compact domain $\Omega$' or not. Please let me know if I were wrong.

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    $\begingroup$ The problem you are asking about is not a Cauchy problem since the condition are specified on the whole boundary of the domain $\partial\Omega$, while for a true Cauchy problem the behaviour at a particular part of the boundary is not required. Typically the conditions $\partial_\nu^k u =f_k$ are stated only for (possibly even a compact subset of) an hypersurface, but not on the whole boundary of the domain of definition. $\endgroup$ Commented Jun 17, 2024 at 13:24
  • $\begingroup$ @DanieleTampieri thank you for pointing that out. I've changed it. $\endgroup$ Commented Jun 18, 2024 at 5:48

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