Málstofa í stærðfræði - Gianmarco Brocchi

Fyrirlesari: Gianmarco Brocchi (Háskóla Íslands)
Dags.: Föstudagurinn 29. nóvember, kl. 11:40
Staðsetning: Stofa V-156 í VR-II.

Titill: Functional Calculus and what it can do for you

Ágrip: Functional calculus allows us to apply (a class of) functions to an
operator. For the Laplacian on R^d, this can be achieved via the Fourier trans-
form, and it is useful in solving Boundary Value Problems (BVPs): the Fourier
transform provides us with a representation of the solution and a characterisa-
tion of the trace space of the solutions. What happens when the boundary of
our domain becomes rough and symmetries are lost? Can we still find a way
to describe solutions and trace space when Fourier methods break down? Ulti-
mately: how do solutions (and these methods) depend on small perturbations
of the boundary?
In this talk I will introduce the “first order approach” for divergence form
equations −divA∇u = 0, which relates harmonic extensions from the real line
and holomorphic functions. This relation works in higher dimensions as well,
and allows us to rewrite our problem in a suitable way so that holomorphic
functional calculus can be applied.
We will take a closer look at degenerate BVPs: when the coefficient A(x)
of our divergence form equation lacks uniform boundedness and accretivity,
and can exhibit singularities. Current state-of-the-art results can handle sin-
gularities characterised by scalar Muckenhoupt weights. These results have
been recently extended on manifolds satisfying some curvature assumption.
But even on flat euclidean space, anisotropic degenerate coefficients have been
out of reach, due to the lack of off-diagonal estimates. Is there another way to
handle more general matrix-degenerate coefficients? How far can we push the
new methods before the theory falls apart?
New results are from joint works with Andreas Rosén at Chalmers.