  As long ago as 1962 Nitsche conjectured that a harmonic homeomorphism $h
\colon A(r,R) \to A(r_*, R_*)$ between planar annuli exists if and only if
$\frac{R_*}{r_*} \ge {1/2} (\frac{R}{r} + \frac{r}{R})$. We prove this
conjecture when the domain annulus is not too wide; explicitly, when $\log
\frac{R}{r} \le {3/2}$. For general $A(r,R)$ the conjecture is proved under
additional assumption that either $h$ or its normal derivative have vanishing
average on the inner boundary circle. This is the case for the critical Nitsche
mapping which yields equality in the above inequality. The Nitsche mapping
represents so-called free evolution of circles of the annulus $A(r,R)$. It will
be shown on the other hand that forced harmonic evolution results in greater
ratio $\frac{R_*}{r_*}$. To this end, we introduce the underlying differential
operators for the circular means of the forced evolution and use them to obtain
sharp lower bounds of $\frac{R_*}{r_*}$.

Complex Variables;Analysis of PDEs