3
$\begingroup$

[This question is looking at the paper

  • Yau, S.-T., On The Ricci Curvature of a Compact Kähler Manifold and the Complex Monge-Ampére Equation, I, Comm. Pure Appl. Math., 31 (1978) 339-411, doi:10.1002/cpa.3160310304, (pdf)]

My problem arises from (2.43) $$ \Delta \varphi=f $$ where $$ -m \leqq f \leqq C_{1} \exp \{C \sup \varphi\} \exp \left\{-\inf _{M} \varphi\right\}. $$

The paper already has a estimation of $\sup\varphi$ then the paper gets: $$ \sup _{M}|\nabla \varphi| \leqq C_{6}\left(\exp \{-C \inf \varphi\}+\int_{M}|\varphi|\right) $$ How does this step use the Schauder estimate? Why is there a $\int_{M}|\varphi|$ on the right handside, should it be $\sup|\varphi|$?

I'm so confused about this step, thanks for helping.

$\endgroup$
2
  • $\begingroup$ Can you remind us of the name of the paper or give a link? $\endgroup$ Mar 16 at 7:11
  • $\begingroup$ sorry, On The Ricci Curvature of a Compact Kahler Manifold and the Complex Monge-Ampere Equation, I* ,jasonpayne.webs.com/Math5339/… $\endgroup$ Mar 16 at 10:22

1 Answer 1

1
$\begingroup$

There are many non-equivalent formulations of the Schauder estimates; the version you are suggesting is the most common. For the version he needs, Yau gives a precise page & theorem reference (p.156, formula 5.5.23) to Morrey's book. In the case of the Euclidean Laplacian you can also see p.69-70 of Gilbarg & Trudinger.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.