Boundary regularity of harmonic functions in $C^1$ slit domains
We establish precise upper and lower estimates for harmonic functions vanishing on the slit of a $C^1$ slit domain, with no assumption that the slit lies in a hyperplane. The classical $\sqrt{d}$ growth near the edge, $d$ being the distance to the slit, persists in this generality, up to an explicit factor \[\exp\Big( \pm C \int_ρ^r ω(s)\, \frac{ds}{s} \Big)\] determined by the $C^1$-modulus of continuity $ω$ of the slit, where $ρ< r$ are the two scales being compared. The upper estimates allow a right-hand side and non-zero boundary data. The correction factors remain bounded above and below by positive constants as $ρ\to0$ precisely when $ω$ satisfies the Dini condition. Moduli of continuity beyond the Dini regime, as is the case for the logarithmic moduli arising at singular sets in relevant free boundary problems, were not covered by the previous $C^{1,α}$ theory. Previously, the $\sqrt{d}$ growth was known for slits contained in a hyperplane, which additionally have a $C^{1,α}$ edge (De Silva, Savin). For Lipschitz slits, there are boundary Harnack principles, but no growth rate is identified precisely. The main technical ingredient is a change of coordinates flattening a Lipschitz slit domain onto the model half-hyperplane slit, with quantitative estimates up to second order. The construction is geometric and does not use the equation, so we expect it to be useful for other boundary regularity problems.