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

摘要

Let alpha is an element of ]0, 1[. Let Omega degrees be a bounded open domain of R-n of class C-1,C-alpha . Let nu Omega degrees denote the outward unit normal to as partial derivative Omega degrees. We assume that the Steklov problem Delta u = 0 in Omega degrees, partial derivative u/partial derivative nu Omega degrees = lambda u on partial derivative Omega degrees has a multiple eigenvalue (lambda) over bar of multiplicity r. Then we consider an annular domain Omega (epsilon) obtained by removing from Omega degrees a small cavity of class C-1,C-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), partial derivative u/partial derivative nu Omega(epsilon) = lambda u on partial derivative Omega(epsilon) which converge to (lambda) over bar as epsilon tend to zero, equals real a analytic function defined in an open neighborhood of (0, 0) in R-2 and computed at the point (epsilon, delta(2,n)epsilon log epsilon) for epsilon 0 small enough. Here nu Omega(epsilon) denotes the outward unit normal to a partial derivative Omega(epsilon), and delta(2,2) (math) 1 and delta(2)(,n) (math) 0 if n = 3. Such a result is an extension to multiple eigenvalues of a previous result obtained for simple eigenvalues in collaboration with S. Gryshchuk.