| Abstract: |
| This work concentrates on the eigenvalues and eigenfunctions of the Dirichlet Laplacian in a bounded domain with a small hole $B(P,\varepsilon)$ removed.
We derive pointwise estimates of the $u-$capacity potential $V_{\varepsilon}$ associated with $B(P,\varepsilon)$ in dimensions two and higher.
Using these estimates, we obtain a uniform estimate for $u_\varepsilon-u-V_{\varepsilon}$ when the eigenvalue is simple. When the eigenvalue is multiple, we also derive the bifurcation of $\lambda_\varepsilon$, which was previously obtained by Abatangelo, L\`ena and Musolino in 2022 and 2024. Finally, in dimension two, we apply the uniform estimate of $u_\varepsilon$ to study the number of intersection points between the nodal line $\mathcal{N}_\varepsilon$ and $\partial B(P,\varepsilon)$. |
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