Zero Lebesgue measure sets as removable sets for degenerate fully nonlinear elliptic PDEs

Abstract

Our main result in this note can be stated as follows: Assume $$E\subset B_{1}$$ and 0.1 $$\begin{aligned} F(D^2u(x),\nabla u(x), u(x),x) \le \psi (x)\ \text { in } B_{1}{\setminus }E\end{aligned}$$ holds in the $$C-$$ viscosity sense where $$|E|=0$$ and F is a degenerate elliptic operator. This way, (0.1) holds in the whole unit ball $$B_{1}$$ (i.e, E is removable for (0.1)) provided 0.2 $$\begin{aligned} \mathcal {M}_{\lambda , \Lambda }^{-}(D^2u) -\gamma |\nabla u| \le f \ \text { in } B_{1} \end{aligned}$$ where $$f\in L^{n}(B_{1})$$ . Zeroth order term can appear in (0.2) provided u is bounded in $$B_{1}$$ . This extends a result due to Caffarelli et al. proven in (Commun Pure Appl Math 66(1):109–143, 2013) where a second order linear uniformly elliptic PDE with bounded RHS appeared in place of (0.2).