Uniform regularity and weight estimates for the Poisson problems when rounding off the conical points

We establish uniform solvability estimates for the Poisson problems associated to a suitably bounded family Ω n n∈ℑ of domains in ℝ d. The main example is that of a suitable sequence of smooth domains that “converges” to a domain with conical points by rounding off the conical points. We give full details for the case of a straight polygonal domain approximated by a sequence of smooth domains rounding off its corners. The method of proof relies on a conformal modification of the metric, with respect to which the union of our domains becomes a manifold with boundary and relative bounded geometry.