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

  • J. Ederson M. Braga
  • Diego Moreira


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


Fully nonlinear Degenerate equations Removable sets 

Mathematics Subject Classification

35B65 35J25 35J60 35J62 35R35 



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.


  1. 1.
    Amendola, M.E., Galise, G., Vitolo, A.: Riesz capacity, maximum principle, and removable sets of fully nonlinear second-order elliptic operators. Differ. Integral Equ. 26(7–8), 845–866 (2013)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Braga, J.E.M, Figalli, A., Moreira, D.: Optimal regularity for the convex envelope and semiconvex functions related to supersolutions of fully nonlinear elliptic equations. Preprint. Google Scholar
  3. 3.
    Caffarelli, L.A.: Interior a priori estimates for solutions of fully nonlinear equations. Ann. Math. (2) 130(1), 189–213 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Caffarelli, L.A., Cabré, X.: Fully Nonlinear Elliptic Equations, American Mathematical Society Colloquium Publications, vol. 43. American Mathematical Society, Providence 1995. vi+104 pp. ISBN: 0-8218-0437-5Google Scholar
  5. 5.
    Calderón, A.P., Zygmund, A.: Local properties of solutions of elliptic partial differential equations. Studia Math. 20, 171–225 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cannarsa, P., Sinestrari, C.: Semiconcave Functions, Hamilton-Jacobi Equations, and optimal control. Progress in Nonlinear Differential Equations and their Applications, vol. 58. Birkhäuser Boston Inc, Boston 2004. xiv+304 pp. ISBN: 0-8176-4084-3Google Scholar
  7. 7.
    Caffarelli, Luis, Li, Yan, Yan, Nirenberg, Louis: Some remarks on singular solutions of nonlinear elliptic equations. I. J. Fixed Point Theory Appl. 5(2), 353–395 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Caffarelli, L., Li, Y.Y., Nirenberg, L.: Some remarks on singular solutions of nonlinear elliptic equations. II. Symmetry and monotonicity via moving planes. Advances in Geometric Analysis, Adv. Lect. Math. (ALM), vol. 21, pp. 97–105. Int. Press, Somerville (2012)Google Scholar
  9. 9.
    Caffarelli, L.A., Li, Y., Nirenberg, L.: Some remarks on singular solutions of nonlinear elliptic equations III: viscosity solutions including parabolic operators. Commun. Pure Appl. Math. 66(1), 109–143 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Caffarelli, L.A., Crandall, M.G., Kocan, M., Swiȩch, A.: On viscosity solutions of fully nonlinear equations with measurable ingredients. Commun. Pure Appl. Math. 49(4), 365–397 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Evans, L.C., Gariepy, R.: Measure Theory and Fine Properties of Functions, Revised edn. CRC Press Inc, Boca Raton (2015)zbMATHGoogle Scholar
  12. 12.
    Galise, Giulio, Vitolo, Antonio: Removable singularities for degenerate elliptic Pucci operators. Adv. Differ. Equ. 22(1–2), 77–100 (2017)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Gutiérrez, C.E.: The Monge-Ampère Equation. Progress in Nonlinear Differential Equations and Their Applications, vol. 44. Birkäuser, Boston (2001)Google Scholar
  14. 14.
    FR, Harvey, HB, Lawson: Removable singularities for nonlinear subequations. Indiana Univ. Math. J. 63(5), 1525–1552 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Labutin, D.A.: Removable singularities for fully nonlinear elliptic equations. Arch. Ration. Mech. Anal 155(3), 201–214 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Labutin, Denis A.: Isolated singularities for fully nonlinear elliptic equations. J. Differ. Equ. 177(1), 49–76 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Labutin, Denis A.: Potential estimates for a class of fully nonlinear elliptic equations. Duke Math. J. 111(1), 1–49 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lions, P.L.: A remark on Bony maximum principle. Proc. Am. Math. Soc. 88(3), 503–508 (1983)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade Federal do CearáFortalezaBrazil

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