Abstract: In the late 50s and early 60s, De Giorgi, Nash and then Moser showed that weak solutions of real second order elliptic PDE in divergence form were in fact H\”older continuous, with an exponent depending only on the ellipticity parameter.
This major discovery had a big impact on the theory of nonlinear equations and opened up the study of boundary value
problems for these divergence form PDE when the coefficients are not assumed to be smooth.
Although this theory does not carry over to the case of complex coefficients, a recently discovered
condition ({\it p-ellipticity}) does allow us to to boost the regularity of solutions and solve certain natural
boundary value problems.