  Riemann-Hilbert problems are jump problems for holomorphic functions along
given interfaces. They arise in various contexts, e.g. in the asymptotic study
of certain nonlinear partial differential equations and in the asymptotic
analysis of orthogonal polynomials. Matrix-valued Riemann-Hilbert problems were
considered by Deift et al. in the 1990s with a noncommutative adaptation of the
steepest descent method. For orthogonal polynomials on the line or on the
circle with respect to exponentially varying weights, this led to a strong
asymptotic expansion in the given parameters. For orthogonal polynomials with
respect to planar exponentially varying weights, the corresponding asymptotics
was obtained by Hedenmalm and Wennman (2017), using a technically involved
construction of an invariant foliation for the orthogonality. Planar orthogonal
polynomials are characterized in terms of a matrix dbar-problem (Its,
Takhtajan), which we call a soft Riemann-Hilbert problem. Here, we use this
perspective to offer a simplified approach based not on foliations but instead
on the ad hoc insertion of an algebraic ansatz for the Cauchy potential in the
soft Riemann-Hilbert problem. This allows the problem to decompose into a
hierarchy of scalar Riemann-Hilbert problems along the interface, which appears
as the free boundary for an obstacle problem. Inspired by microlocal analysis,
the method allows for control of the solution in such a way that for
real-analytic weights, the asymptotics holds in the $L^2$ sense with error
$O(e^{-\delta\sqrt{m}})$ in a fixed neighborhood of the closed exterior of the
interface, for some constant $\delta>0$, where $m\to+\infty$. Here, $m$ is the
degree of the polynomial, and in terms of pointwise asymptotics, the expansion
dominates the error term in the exterior domain and across the interface a
distance proportional to $m^{-\frac14}$.
