HI everyone,
I need to know the latest researches and problems that still unsolved in area of Complex differential equations in The Unit Disc, with applying Nevanlinna Theory.
Best regards
Dear Mohamed,
See for example Solutions of complex linear differential equations in the unit disc by Saleh Omran,
MATHEMATICAL COMMUNICATIONS 205
Math. Commun. 16(2011), 205-214.
Complex oscillation of differential polynomials generated by
analytic solutions of differential equations in the unit disc by
Ting-Bin Cao, Lei-Min Li, Jin Tu and Hong-Yan Xu,
and
G.G. Gundersen, E.M. Steinbart, S. Wang,
The possible orders of solutions of linear differential equations with polynomial coeficients, Trans. Amer. Math. Soc. 350(3) (1998),1225-1247.
There are also a major breakthrough concerning the initial Schoen Conjecture related to Complex differential equations (for hypebolic harmonic) in The Unit Disc.
Dear Mohamed,
The equation which describes harmonic hyperbolic mappings (wrt. hyperbolic distance) is an important complex non-linerar equation.
In [1], Markovic shows that every quasisymmetric homeomorphism of the circle $\partial \mathbb{H}^2$ admits a harmonic hyperbolic quasiconformal extension to the hyperbolic plane $\mathbb{H}^2$. This proves the Schoen Conjecture. The Proof is based on:
Theorem 1.1. For every $K > 1$ there exists a constant $C = C(K)$ with
the following properties. We let $g : \mathbb{H}^2 \rightarrow \mathbb{H}^2$ be any K-quasiconformal
homeomorphism and assume that there exists a harmonic quasiconformal
homeomorphism $f : \mathbb{H}^2 \rightarrow \mathbb{H}^2$ that agrees with g on the boundary $\partial \mathbb{H}^2$.
Then
$\sup_{z\in \mathbb{H}^2} d(h(x),f(x)) \leq C$,
where $d(\cdot, \cdot)$ denotes the hyperbolic distance.
See
[1] V. Markovic, Harmonic maps and the Schoen Conjecture, March 7, 2015, http://www.its.caltech.edu/~markovic/M-schoen.pdf
best,MM
- Badges
- Science topic
- Similar topics
- Mathematics