Exponential stability bound for the heterogeneous Helmholtz equation

Let $Omega y$ be a bounded Lipschitz domain in $R^d$, let $a=1$, and let $c\in L^{\infty}(\Omega)$ satisfy

$$0<c_{\min}\leq c\leq c_{\max}<\infty.$$

For $omega\geq omega_0>0$, consider the heterogeneous Helmholtz solution $u$ with volume data $f$ and boundary data $g$. Exponential stability conjecture. The stability estimate

$$\left(\int_{\Omega}|\nabla u|^2+\left(\frac{\omega}{c}\right)^2|u|^2\right)^{1/2}\leq C_{\operatorname{stab}}\left(\|f\|_{L^2(\Omega)}^2+\|g\|_{L^2(\Gamma)}^2\right)^{1/2}$$

holds, with

$$C_{\operatorname{stab}}\leq C_1\exp(C_2\omega),$$

where $C_1,C_2>0$ depend on $c_{\min}$, $c_{\max}$, and $\Omega$. The conjecture proposes an explicit upper bound for the stability constant that remains controlled for arbitrary bounded measurable wave-speed coefficients, rather than growing with the number of coefficient jumps.