A Proof Of The C\u3csup\u3eP\u3csup\u3e′\u3c/sup\u3e\u3c/sup\u3e-Regularity Conjecture In The Plane

We establish a new oscillation estimate for solutions of nonlinear partial differential equations of elliptic, degenerate type. This new tool yields a precise control on the growth rate of solutions near their set of critical points, where ellipticity degenerates. As a consequence, we are able to prove the planar counterpart of the longstanding conjecture that solutions of the degenerate p-Poisson equation with a bounded source are locally of class Cp′=C1, [Formula presented] this regularity is optimal