Seminario
Aula Seminari, Palazzina RM004
Abstract:
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 Kondrat'ev 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 $\varepsilon\to0$ 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
$\varepsilon\to0$. These new results are based upon the recent work
[Costabel, Dallariva, Dauge, Musolino; 2024].