Abstract

During ‘Analysis of PDEs’ in Michaelmas term we proved the Cauchy-Kovalevskaya theorem. The theorem asserts the local existence of a solution for a certain class of first order systems of PDEs. The hypothesis of the theorem requires all given functions within the system are analytic. The analyticicty is a key component of the hypothesis as the celebrated example due to Lewy shows that smoothness is not enough to guarantee these systems have solutions. This is done by construction of a linear PDE which has no solutions at all. In this talk, a statement of the Cauchy-Kovalevskaya theorem will be given and the statement of Lewy’s example and a proof (or at least a sketch, depending on time) that the example is in fact a PDE with no solutions will also be given.