Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma
The regularity of solutions of the Dirichlet problem for the Laplace operator in corner domains is limited by the existence of harmonic functions that are zero on the boundary of some tangent cones. This is the prototypal example of the seminal Kondratev theory [1967] for elliptic boundary value problems in conical domains. As a consequence, the Sobolev norms of solutions blow up when the associated quantity \( s-n/p \) tends to some limiting values. I will revisit these quite classical facts. Then I will address a more paradoxical situation in which families of domains are defined, depending continuously of a small parameter \( \varepsilon \), and so that in the limit \( \varepsilon\to 0 \) the regularity of the boundary drops, creating a sort of "pseudo-corner''. We will see that, though being finite for each positive \( \varepsilon \), some Sobolev norms of solutions blow up in a quantified way as \( \varepsilon\to 0 \). These new results are based upon the recent work [Costabel, Dallariva, Dauge, Musolino; 2024].
simone.creo@uniroma1.it