Biharmonic and elasticity eigenvalue problems in singularly perturbed domains

Domain perturbation theory for the eigenvalues of the Laplace operator on families of bounded, Lipschitz domains of R

N is nowadays a well-understood, yet complicated subject. For the biharmonic operator or the elasticity operator, the situation is more involved, mainly due to two additional hurdles: 1) boundary conditions are very sensitive to the variation of the curvature of the boundary; 2) standard techniques, such as the separation of variables, are not available. After a review of the main results and counterexamples for the Laplace operator and the biharmonic operator, I will focus on three specific singular perturbations where spectral continuity fails: the dumbbell domain (Neumann b.c); a Lipschitz domain, whose boundary is locally defined as the graph of a fast oscillating smooth function (Intermediate b.c.); thin annuli in R2 (Neumann b.c).

I will conclude with a discussion on the spectral convergence of the Reissner-Mindlin system on thin domains of RN .

Based on joint projects with J.M. Arrieta (Madrid), D. Buoso (Piemonte Orientale), P.D. Lamberti (Padova), and L. Provenzano (Sapienza Roma).

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