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

  • J. Ederson M. Braga
  • Diego Moreira
Article
  • 62 Downloads

Abstract

Our main result in this note can be stated as follows: Assume \(E\subset B_{1}\) and
$$\begin{aligned} F(D^2u(x),\nabla u(x), u(x),x) \le \psi (x)\ \text { in } B_{1}{\setminus }E\end{aligned}$$
(0.1)
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
$$\begin{aligned} \mathcal {M}_{\lambda , \Lambda }^{-}(D^2u) -\gamma |\nabla u| \le f \ \text { in } B_{1} \end{aligned}$$
(0.2)
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).

Keywords

Fully nonlinear Degenerate equations Removable sets 

Mathematics Subject Classification

35B65 35J25 35J60 35J62 35R35 

Notes

Acknowledgements

The authors would like to thank Professor Yanyan Li for nice discussions about the topic treated in this paper. Also, the authors would like to thank both referees for careful reading and nice suggestions and comments on this paper. The first author is supported by CAPES-Brazil. The second author is partially supported by CNPq-Brazil.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade Federal do CearáFortalezaBrazil

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