This talk is devoted to a functional analytic approach to the
subelliptic oblique derivative problem for second-order, elliptic
differential operators
with a complex parameter, and prove an existence and uniqueness theorem of
the homogeneous oblique derivative problem in the framework of $L^{p}$
Sobolev spaces
when the complex parameter tends to infinity. As an application of the main
theorem, we prove generation theorems of analytic semigroups for this
subelliptic oblique derivative problem in the $L^{p}$ topology and in the
topology of uniform convergence. These rather surprising results (elliptic
estimates
for a degenerate problem) work, since we are considering the homogeneous
boundary condition. We make use of Agmon's technique of treating a spectral
parameter
as a second-order elliptic differential operator of an extra variable on the
unit circle and relating the old problem to a new one with the additional
variable.
Seminar on Analysis at University of Tsukuba