ABSTRACT
We will study removable singularities for the Cauchy-Riemann, Laplace, and minimal surface equations. As these equations are in the divergence form https://www.w3.org/1998/Math/MathML"> d i v [ ϕ ( D u ) ] = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429096679/ee3e4f1b-65eb-4f80-b900-911f329ff879/content/eq1191.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , the integration by parts theorem established in the previous chapter is a natural tool. We define removable sets by means of Hausdorff measures, mostly without any topological restrictions. The results are established by short and simple arguments, which rely on the relationship between weak and strong solutions of partial differential equations. A few basic facts about distributions and weak solutions are stated without proofs. We made no attempt to survey the long history concerning removable singularities.
