Potential and scattering theory on wildly perturbed domains

Layer potential techniques in spectral analysis. Part I: Complete asymptotic expansions for eigenvalues of the Laplacian in domains with small inclusions

“…Rauch and Taylor [31] have shown that the spectrum of a bounded domain does not change after imposing Dirichlet conditions on a compact subset of capacity zero. After that, many people have studied the asymptotic expansion of the eigenvalues for the case of small holes with the Dirichlet or the Neumann boundary condition.…”

Asymptotic analysis of two elliptic equations with oscillating terms

“…When lim-= 0, the following result has been proved via different E methods in [2], [9], [18], [19] : Of course, C D dépends on the size r e of the inclusions and, for example, if N is greater or equal to 3, the change of variables x = r e y in (2) shows the existence of a critical size rl = e N//^"2) such that : In [4], [20], a particular case of the third above situation is studied, by means of asymptotic expansions, that is the case : r z = ke (0 < k < 1/2) : THEOREM 1.2 : Suppose r E = ks (0 <c k < 1/2 ), then the séquence …”

The 𝑟-stability of hypersurfaces with zero Gauss-Kronecker curvature

“…We will consider hypersurfaces with H n = 0 on a subset of capacity zero (see below the definition of capacity). It is known (see [8] §2, p. 35) that submanifolds of codimension d ≥ 2 have capacity zero. In section 2 we will give definitions and develop some facts about capacity.…”