Boundary value problems for first-order elliptic operators with compact and noncompact boundary.
Lashi Bandara (Brunel)
The index theorem for compact manifolds with boundary, established by
Atiyah-Patodi-Singer in the mid-70s, is considered one of the most
significant mathematical achievements of the 20th century. An important
and curious fact is that local boundary conditions are topologically
obstructed for index formulae and non-local boundary conditions lie at
the heart of this theorem. Consequently, this has inspired the study of
boundary value problems for first-order elliptic differential operators
by many different schools, with a class of induced operators adapted to
the boundary taking centre stage in formulating and understanding
non-local boundary conditions.
That being said, much of this analysis has been confined to the
situation when adapted boundary operators can be chosen self-adjoint.
Dirac-type operators are the quintessential example. Nevertheless,
natural geometric operators such as the Rarita-Schwinger operator on
3/2-spiniors, arising from physics in the study of the so-called Delta
baryon, falls outside of this class. Analytically, this requires
analysis beyond self-adjoint operators. In recent work with Bär, the
compact boundary case is handled for general first-order elliptic
operators, using spectral theory to choose adapted boundary operators to
be invertible bi-sectorial. The Fourier circle methods present in the
self-adjoint analysis are replaced by the bounded holomorphic
functional calculus, coupled with pseudo-differential operator theory
and semi-group techniques. This allows for a full understanding of the
maximal domain of the interior operator as a bounded surjection to a
space on the boundary of mixed Sobolev regularity, constructed from
spectral projectors associated to the adapted boundary operator.
Regularity and Fredholm extensions are also studied.
For the noncompact case, a preliminary trace theorem as well as
regularity theory are handed by resorting to the case with compact
boundary. This necessitates deforming the coefficients of the interior
operator in a compact neighbourhood. Therefore, even for Dirac-type
operators, allowing for fully general symbols in the compact boundary
case is paramount. Under slightly stronger geometric assumptions near
the noncompact boundary (automatic for the compact case) and when the
interior operator admits a self-adjoint adapted boundary operator, an
upgraded trace theorem mirroring the compact setting is obtained.
Importantly, there is no spectral assumptions other than
self-adjointness on the adapted boundary operator. This, in particular,
means that the spectrum of this operator can be the entire real line.
Again, the primarily tool that is used in the analysis is the bounded
holomorphic functional calculus.
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