I read in the book the following argument. Consider $\Omega$ be an open set in $\mathbb{R}^n$ and $u$ be the solution of Schrödinger equation $iu_t+\Delta u = 0$ in $\Omega \times [0,T]$. We assume the Dirichlet boundary condition $u|_{\partial\Omega} = 0$ and $\frac{\partial u}{\partial \nu} = 0$ on $\Gamma$ where $\Gamma \subset \partial \Omega$ and $\nu$ be the unit normal vector.

Then they said that using Holmgren's uniqueness theorem and unique continuation of Schrodinger equation to imply $u=0$ on $\Omega$. I do not understand this argument. Can anyone help me to explain it?


I do not bother to write down all technical details. Here a sketch of the idea (at least in the static case):

You need to extend your domain $\Omega$ to a slightly bigger domain $\tilde{\Omega}$ near a point $x \in \Gamma$ and consider the zero extension of a solution $u$ in $\tilde{\Omega}$. Let us denote this extension by $\tilde{u}$. Then you have that $\tilde{u}$ solves the same equation in $\tilde{\Omega}$. (Here you apply the boundary conditions to see that $\tilde{u} \in H^2(\tilde{\Omega})$.) Clearly $\tilde{u}$ vanish in a neighbourhood contained in $\tilde{\Omega}\setminus \Omega$. This implies by the weak UCP that $\tilde{u}$ must be identically zero. Thus also $u$.

The weak UCP follows quite easily from Holmgren's uniqueness theorem by considering the set \begin{equation}\{\, x \in \Omega\,|\,u = 0 \text{ in a neighbourhood of $x$ } \}\end{equation} for a solution $u$ of $\Delta u = 0$ in $\Omega$. It is a nonempty, open and closed set. To see that it is closed you need to apply Holmgren's uniqueness theorem.

I believe that the above should also remain true for $H^1$ weak solutions. It is also valid for a quite general class of elliptic second order PDEs (in the above one could just use directly analyticity of harmonic functions!).

I guess with the time domain you extend by zero in $\tilde{\Omega} \times [0,T]$ anyway, and then use the weak UCP for $i\partial_t + \Delta$? (I must admit that I am only familiar with the UCP for elliptic second order PDEs.) Is this any helpful?


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