Assume we have a complete orthogonal system (with respect to the L2 norm) on a domain $D$ (for e.g.D$, given by the eigenfunctions of the Laplacian on $D$. For example, the set ${e^{int}}$ on $[-\pi, \pi]$, or the spherical harmonics on the unit ball)ball. Now consider a domain $D'$, which is "close" to D in some sense (say, the L2 distance between the boundary of $D$ and is close to the boundary of $D'$ is upper bounded)in some suitable norm). What can be said about

Are the basis eigenfunctions of the Laplacian on $D'$ ? Is there any sense D$ close, in which some sense, to the functions in eigenfunctions of the Laplacian on $D'$? Does knowing the basis of $D$ are "close" to the functions in help approximate the basis of $D'$ ? Any known results along these or similar lines appreciated.

Assume we have a complete orthogonal system (with respect to the L2 norm) on a domain $D$ (for e.g., the set ${e^{int}}$ on $[-\pi, \pi]$, or the spherical harmonics on the unit ball). Now consider a domain $D'$, which is "close" to D in some sense (say, the L2 distance between the boundary of $D$ and the boundary of $D'$ is upper bounded). What can be said about the basis of $D'$ ? Is there any sense in which the functions in the basis of $D$ are "close" to the functions in the basis of $D'$ ? Any known results along these or similar lines appreciated.