Qualitative properties of eigenfunctions in domains with small holes

In this paper we study qualitative properties of the eigenvalues and eigenfunctions of $-\Delta$ with Dirichlet boundary condition in a smooth bounded domain $\Omega$ with a small circular hole. In the literature, this is known as a "singular perturbation", in contrast with the "regular perturbation" case. Denoting by $\Omega_\epsilon:=\Omega\setminus B(P,\epsilon)$ where $B(P,\epsilon)$ is the ball centered at $P$ and radius $\epsilon$, for $P\in\Omega$ and $\epsilon$ small enough we investigate 1) quantitative estimates for the eigenfunctions of $-\Delta$ in $\Omega_\epsilon$; 2) the simplicity of the eigenvalues of $-\Delta$ in $\Omega_\epsilon$; 3) the behavior of nodal sets of the eigenfunctions of $-\Delta$ in $\Omega_\epsilon$. A key ingredient in our analysis consists of pointwise estimates on the so-called $u$-capacitary potential firstly introduced in \cite{afhl}.