<p style='text-indent:20px;'>In this article, we discuss quantitative Runge approximation properties for the acoustic Helmholtz equation and prove stability improvement results in the high frequency limit for an associated partial data inverse problem modelled on [<xref ref-type="bibr" rid="b3">3</xref>,<xref ref-type="bibr" rid="b35">35</xref>]. The results rely on quantitative unique continuation estimates in suitable function spaces with explicit frequency dependence. We contrast the frequency dependence of interior Runge approximation results from non-convex and convex sets.</p>

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We show that the singular set Σ in the classical obstacle problem can be locally covered by a C ∞ hypersurface, up to an "exceptional" set E, which has Hausdorff dimension at most n − 2 (countable, in the n = 2 case). Outside this exceptional set, the solution admits a polynomial expansion of arbitrarily large order. We also prove that Σ \ E is extremely unstable with respect to monotone perturbations of the boundary datum. We apply this result to the planar Hele-Shaw flow, showing that the free boundary can have singular points for at most countable many times.