A note on the optimal boundary regularity for the planar generalized $p$-Poisson equation

Saikatul Haque

Published 2018-03-21Version 1

In this note, we establish sharp regularity for solutions to the following generalized $p$- Poisson equation $$-\ div\ \big(\langle A\nabla u,\nabla u\rangle^{\frac{p-2}{2}}A\nabla u\big)=-\ div\ \mathbf{h}+f.$$ in the plane (i.e. in $\mathbb{R}^n=\mathbb{R}^2$) for $p>2$ in the presence of Dirichlet as well as Neumann boundary conditions and with $\mathbf{h}\in C^{1-n/q}$, $f\in L^q$, $2=n<q\leq\infty$. The regularity assumptions on the principal part $A$ as well as that on the Dirichlet/Neumann conditions are exactly the same as in the linear case and therefore sharp (see Remark 2.5 below). Our main results Theorem 2.3 and Theorem 2.4 should be thought of as the boundary analogues of the sharp interior regularity result established in the recent interesting paper [1] in the case of \begin{equation}\label{e0} -\ div\ (|\nabla u|^{p-2} \nabla u) =f \end{equation} for more general variable coefficient operators and with an additional divergence term.