A classical problem in the regularity theory of geometric PDEs concerns the study of regular solutions on a punctured disc. For some cases such as the minimal surface equation or other much more general elliptic geometric equations in divergence form, isolated singularities are automatically removable. This is also true for suitable concepts of generalized solutions of many other elliptic PDEs.
Nonetheless, some geometric PDEs like the elliptic Monge-Ampère equation describing graphs of constant curvature
$u_{xx}u_{yy} - u_{xy}^2 = K(1+u_x^2+u_y^2)^2$, $K>0$
admit non-removable conical singularities. That is, they admit solutions on a punctured disc that extend continuously but not C¹-smoothly to the puncture. For them, the usual regularity theory of generalized solutions is not applicable.
In this talk we shall explain how to deal with geometric PDEs in presence of non-removable isolated singularities. Using methods from complex analysis, geometry and elliptic PDEs, we will provide a local classification result in terms of the limit normal cone at the singularity. In the global case, we shall expose different classification results for solutions of special geometric PDEs in the presence of an arbitrary number of conical singularities.