A Radó theorem for p-harmonic functions

Abstract

Let A be a nonlinear differential operator on an open set $\mathcal {X}\subset \mathbb {R}^n$ and $\mathcal {S}$ a closed subset of $\mathcal {X}$. Given a class $\mathcal {F}$ of functions in $\mathcal {X}$, the set $\mathcal {S}$ is said to be removable for $\mathcal {F}$ relative to A if any weak solution of $A (u) = 0$ in $\mathcal {X}{\setminus } \mathcal {S}$ of class $\mathcal {F}$ satisfies this equation weakly in all of $\mathcal {X}$. For the most extensively studied classes $\mathcal {F}$, we show conditions on $\mathcal {S}$ which guarantee that $\mathcal {S}$ is removable for $\mathcal {F}$ relative to A.