We establish an equivalence between the
Nirenberg problem on the circle and the
boundary of holomorphic immersions of the disk into the plane. More precisely we
study the nonlocal Liouville-type equation
(1)
where
stands for the
fractional Laplacian and
is a bounded function. The equation (1) can actually be interpreted as the prescribed
curvature equation for a curve in conformal parametrization. Thanks to this
geometric interpretation we perform a subtle blow-up and quantization analysis of
(1). We also show a relation between (1) and the analogous equation in
,
(2)
with
bounded on
.
Keywords
nonlocal Liouville equation, Nirenberg problem, fractional
harmonic maps, blow-up analysis of solutions, regularity of
solutions, conformal variational problems, quasiconformal
mappings in the plane