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arXiv · 1901.03827

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

Damião J. Araújo·Eduardo V. Teixeira·José Miguel Urbano
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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.

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Damião J. Araújo, Eduardo V. Teixeira, José Miguel Urbano. 2019-01-12. A proof of the $C^{p^\prime}$-regularity conjecture in the plane. https://doi.org/10.1016/j.aim.2017.06.027

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