Fractional-regularity stability estimate for heterogeneous Helmholtz solutions

Establish, for fractional regularity exponents $s,s'>0$, the uniform estimate $k |u_{'}|_{H^{1+s'}(D)} \lesssim (k/')^{-s'} \|f\|_{L^2(\Omega)}$ for the highly heterogeneous Helmholtz scattering problem when the periodic coefficients have only fractional regularity, thereby replacing the Lipschitz-based estimate used to prove continuity of the stability constant with respect to the heterogeneity scale.

Background

The proof of continuity of the heterogeneous stability constant with respect to the microscopic scale uses Lipschitz continuity of the periodic coefficients. The authors discuss replacing this argument with a fractional Sobolev estimate involving coefficient regularity Ws,∞W^{s,\infty} and solution regularity H1+s′(D)H^{1+s'}(D).

The required estimate would need to hold uniformly for nearby microscopic scales and would permit the analysis to be extended to less regular coefficients. The paper states that the estimate appears plausible but is not established because its proof introduces technical difficulties.

References

In this case, we would need to show a bound of the form

k |u_{\varepsilon'}|{H{1+s'}(D)} \lesssim \left (\frac{k}{\varepsilon'}\right ){-s'} |f|{L2(\Omega)}

uniformly for $\varepsilon' \in (-\delta,+\delta)$. Whereas it feels likely that such bound holds true, actually establishing it seems to lead to technicalities that we prefer avoiding here.

— Frequency-explicit convergence analysis of a multiscale finite element method for highly heterogeneous scattering problems  (2609.04930 - Chaumont-Frelet et al., 4 Sep 2026) in Remark following Lemma 2.1, Section 2.1 (uniform stability estimates)