In the previous note, I showed a Rayleigh-type quotient for the conformal Killing operator on manifolds
with boundary
, i.e. the following result holds:
Whenever
admits no non-zero conformal Killing vector fields, the following holds
where the infimum is taken over all smooth vector fields
on
with
.
Today, I am going to prove a slightly stronger version of the above inequality, namely, when some terms on the boundary take part in. Precisely, we shall prove
Whenever
admits no non-zero conformal Killing vector fields, the following holds
where the infimum is taken over all smooth vector fields
on
with
.
However, a proof for this new inequality remains the same. To do so, we first make use of some Sobolev embeddings as follows: