Multiple eigenvalues for the Steklov problem in a domain with a small hole. A functional analytic approach

Let $alpha in ,]0,1[$. Let $Omega ^{o}$ be a bounded open domain of ${mathbb{R}}^{n}$ of class $C^{1,alpha }$. Let $ u _{Omega ^{o} }$ denote the outward unit normal to $partial Omega ^{o}$. We assume that the Steklov problem $ Delta u=0 $ in $Omega ^{o}$, $rac{partial u}{ partial u _{ Omega ^{o} } }=lambda u$ on $partial Omega ^{o}$ has a multiple eigenvalue $ ilde{lambda }$ of multiplicity $r$. Then we consider an annular domain $Omega (epsilon )$ obtained by removing from $Omega ^{o}$ a small cavity of class $C^{1,alpha }$ and size $epsilon >0$, and we show that under appropriate assumptions each elementary symmetric function of $r$ eigenvalues of the Steklov problem $ Delta u=0 $ in $Omega (epsilon )$, $rac{partial u}{partial u _{ Omega (epsilon ) } }=lambda u$ on $partial Omega (epsilon )$ which converge to $ ilde{lambda }$ as $epsilon $ tend to zero, equals real a analytic function defined in an open neighborhood of $(0,0)$ in ${mathbb{R}}^{2}$ and computed at the point $(epsilon , delta _{2,n} epsilon log epsilon )$ for $epsilon >0$ small enough. Here $ u _{Omega (epsilon ) }$ denotes the outward unit normal to $partial Omega (epsilon )$, and $delta _{2,2}equiv 1$ and $delta _{2,n}equiv 0$ if $ngeq 3$. Such a result is an extension to multiple eigenvalues of a previous result obtained for simple eigenvalues in collaboration with S.~Gryshchuk.