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edited Nov 14 2011 at 22:40 |
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 !
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edited Nov 3 2011 at 14:25 |
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edited Nov 1 2011 at 13:48 |
Hello, this question might sound a little vague, but I still dare to state :
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 $C$. 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 !
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5
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edited Oct 31 2011 at 16:22 |
Hello, this question might sound a little vague, but I still dare to state :
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 $C$. 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 contonuous 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 !
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4
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edited Oct 31 2011 at 15:41 |
Hello, this question might sound a little vague, but I still dare to state :
Let us consider the orientation-preserving homeomorphic solutions $f: D \to D $ of the Beltrami equation $f_\bar{z}=\mu. f_z$f_z, $||\mu||_{L^{\infty}(D)}\le k\le 1, D$ is the unit disk in the complex plane $C$. 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 contonuous 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 !
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3
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edited Oct 31 2011 at 3:07 |
Hello, this question might sound a little vague, but I still dare to state :
Let us consider the homeomorphic solutions $f: D \to D $ of the Beltrami equation $f_\bar{z}=\mu. f_z$, $D$ is the unit disk in the complex plane $C$. 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 contonuous 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 !
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2
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edited Oct 29 2011 at 21:00 |
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asked Oct 28 2011 at 0:26 |
Boundary regularity of the solution to the Beltrami equation
Hello, this question might sound a little vague, but I still dare to state :
Let us consider the homeomorphic solutions $f: D \to D $ of the Beltrami equation $f_\bar{z}=\mu. f_z$, $D$ is the unit disk in the complex plane $C$. 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 contonuous 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 $D$.
I would highly appreciate if you state any known results on this topic , thanks in advance !
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