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The validity of Weyl's law for the Steklov problem on domains with Lipschitzboundaries is a well-known open question in spectral geometry. We answer thisquestion in two dimensions and show that Weyl's law holds for an even largerclass of surfaces with rough boundaries. This class includes domains withinterior cusps as well as 'slow' exterior cusps. Moreover, the condition on thespeed of exterior cusps cannot be improved, which makes our result in a senseoptimal. The proof is based on the methods of Suslina and Agranovich combinedwith some observations about the boundary behaviour of conformal mappings.

Weyl's law for the Steklov problem on surfaces with rough boundary