On the Schiffer and Berenstein conjectures with high-frequency for convex domains in the plane

In this paper, by introducing two-point stationary-phase amplitude defect, we provide a partial positive answer to the Schiffer and Berenstein conjectures in $\mathbb{R}^2$. More precisely, assuming that a bounded uniformly convex domain $\Omega \subset \mathbb{R}^2$ has a connected boundary of class $C^{2,\epsilon}$ with $\epsilon \in (0,1)$, we show that if, for some nonzero constant $c_D$, the overdetermined elliptic problem \begin{equation} -\Delta u = \alpha u \ \text{ in } \ \Omega, \qquad u = 0 \ \text{ on } \ \partial\Omega, \qquad \frac{\partial u}{\partial \nu} = c_{D} \ \text{ on } \ \partial\Omega \nonumber \end{equation} admits a nontrivial solution corresponding to a large eigenvalue $\alpha$, then the domain $\Omega$ must be a disk. Similarly, we establish that if a domain $\Omega \subset \mathbb{R}^2$ has a connected Lipschitz boundary and the problem \begin{equation} -\Delta u = \alpha u \ \text{ in } \ \Omega, \qquad \frac{\partial u}{\partial \nu} = 0 \ \text{ on } \ \partial\Omega, \qquad u = c_{N} \ \text{ on } \ \partial\Omega \nonumber \end{equation} has a nontrivial solution corresponding to a large eigenvalue $\alpha$, then $\Omega$ is a disk as well.