MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Hello, this question might sound a little vague, but I still dare to state , and I am basically requesting for some reference:

Let us consider the orientation-preserving homeomorphic solutions $f: D \to D $ of the Beltrami equation $f_\bar{z}=\mu. f_z, ||\mu||_{L^{\infty}(D)}\le k\le 1, D$ is the unit disk in the complex plane $C$, $f_z, f_\bar{z} $ are the partial derivatives of $f$ w.r.t $z, \bar{z} $ respectively. I am looking for the known results on the sufficient , or, necessary and sufficient condition on the Beltrami coefficient $\mu$ such that any solution of the equation is at least $C^1(\bar{D})$, but I would be happier to get $C^k(\bar{D}), 1\le k \le \infty $, or even real-analytic upto the boundary. By $C^k(\bar{D})$, I mean the $k$th derivative exists and is continuous upto the boundary. Holder continuity of solutions is also I would be interested in. ( Please note that I stated 'any' solution , since we know that any two solutions differ by a Mobius transformation of $D$.

In general, I am looking for results for the boundary regularity of the solution to the Beltrami equation.

I would highly appreciate if you state any known results on this topic , thanks in advance !

share|cite|improve this question
IMHO, either this or… should be closed, and answers/comments left on the remaining one. – Yemon Choi Jun 21 '12 at 2:20

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.