Pier Domenico Lamberti (OTDE-Workshop, III-2026)

The presentation was delivered at the Workshop on Operator Theory in Function Spaces and its Applications, dedicated to the anniversary of Professor Victor Burenkov, (September 28–30, 2026, held online)." Speaker: Pier Domenico Lamberti. Title: «Spectral Stability of Maxwell's Equations under Shape Perturbations ». Abstract: We study the spectral problem for stationary Maxwell's equations in a cavity with homogeneous boundary conditions, focusing on the dependence of the eigenvalues on perturbations of the cavity's shape. First, for families of diffeomorphic domains, we establish the analytic dependence of the eigenvalues on the domain perturbation, recover Hirakawa's formula for the shape derivatives, and derive a Rellich-Pohozaev-type identity. Second, we consider families of domains with oscillatory boundaries within the framework of Burenkov's atlas classes, providing a setting for studying spectral stability beyond smooth diffeomorphic perturbations. A key ingredient in this analysis is the validity of suitable uniform Gaffney inequalities for the corresponding family of domains. Official page of the Workshop on the website of the SJMI VSC RAS: https://smath.ru/activities/upcoming/19688/en/ The Workshop is organized by: -Vladikavkaz Scientific Center of the RAS (Southern Mathematical Institute; North-Caucasus Center for Mathematical Research*); -Lomonosov Moscow State University (Faculty of Mechanics and Mathematics, Department of General Control Problems); -Peoples' Friendship University of Russia named after Patrice Lumumba (Institute of Mathematics of Peoples' Friendship University of Russia). *Workshop is supported by the Ministry of Science and Higher Education of Russian Federation, project № 075-02-2026-738.