Following the previous note, today we discuss a similar Rayleigh-type quotient for the conformal Killing operator on manifolds with boundary. We also prove that
whenever admits no non-zero conformal Killing vector fields, the following holds
where the infimum is taken over all smooth vector fields on with .
is no longer available, we use a new approach in order to estimate from below. To this purpose, we make use of a Riemannian version for the Korn inequality recently proved by S. Dain [here].
First, in view of Corollary 1.2 in Dain’s paper, the following inequality holds
for some positive constant independent of . This helps us to conclude that
as in Dahl et al’ paper. Therefore, we can argue by contradiction by assuming that there exists a sequence of vector fields such that
Since is compact, it follows that is uniformly bounded in . Therefore, it is standard to conclude that there exists some such that
- weakly in ) and
- strongly in .
Consequently, . In particular, . For any smooth vector field , using the divergence theorem, we obtain
We further have
Hence, we have proved that
Following the argument in Dahl et al.’ paper, one can prove that the limit in the preceding equality vanishes showing that
for any test vector field . Hence, solves
in the sense of distributions. Finally, to show that is a conformal Killing vector field, we again use the divergence theorem as follows
which obviously shows that .