Higher-order error bounds in the two-dimensional asymptotic expansion

Establish universal control of the higher-order error terms in the two-dimensional far-field asymptotic expansion for proper solutions of the Bernoulli one-phase problem with a compactly supported defect.

Background

For proper solutions in dimension two, the paper proves an expansion of the form u(x)=xd+k(u)clog∣x∣+E(x)u(x)=x_d+k(u)clog|x|+E(x) with bounded error, and it quantitatively controls the logarithmic coefficient under suitable assumptions on the defect.

The result does not provide universal bounds on the higher-order error term across the class of proper solutions. Such bounds would strengthen the asymptotic theory and make the two-dimensional expansion quantitatively comparable to the higher-dimensional expansion, where a universal decay rate is obtained.

References

Note that we lack universal control on the higher order error terms in the asymptotic expansion in $d=2$, it remains open whether it can be achieved.

— Solutions of the Bernoulli one-phase problem with a defect  (2609.17066 - Feldman et al., 15 Sep 2026) in Section 5, subsection “Asymptotics of general solutions”