# A proof of the $C^{p^\prime}$-regularity conjecture in the plane

## TL;DR

This paper proves the $C^{p'}$-regularity conjecture for solutions of degenerate elliptic PDEs in the plane, establishing optimal regularity near critical points using a new oscillation estimate.

## Contribution

It introduces a novel oscillation estimate that enables the proof of the $C^{p'}$-regularity conjecture for degenerate $p$-Poisson equations in the plane, confirming optimal regularity.

## Key findings

- Solutions are locally of class $C^{p'}$ near critical points.
- The new oscillation estimate controls solution growth near degeneracy.
- The result confirms the conjecture's optimal regularity in the planar case.

## Abstract

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 $C^{p^\prime}=C^{1,\frac{1}{p-1}}$; this regularity is optimal.