Entire solutions to a class of fully nonlinear elliptic equations

Ovidiu Savin

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2008)

  • Volume: 7, Issue: 3, page 369-405
  • ISSN: 0391-173X

Abstract

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We study nonlinear elliptic equations of the form F ( D 2 u ) = f ( u ) where the main assumption on F and f is that there exists a one dimensional solution which solves the equation in all the directions ξ ∈ ℝ n . We show that entire monotone solutions u are one dimensional if their 0 level set is assumed to be Lipschitz, flat or bounded from one side by a hyperplane.

How to cite

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Savin, Ovidiu. "Entire solutions to a class of fully nonlinear elliptic equations." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 7.3 (2008): 369-405. <http://eudml.org/doc/272267>.

@article{Savin2008,
abstract = {We study nonlinear elliptic equations of the form $F(D^2u)=f(u)$ where the main assumption on $F$ and $f$ is that there exists a one dimensional solution which solves the equation in all the directions $\xi \in \mathbb \{R\}^n$. We show that entire monotone solutions $u$ are one dimensional if their $0$ level set is assumed to be Lipschitz, flat or bounded from one side by a hyperplane.},
author = {Savin, Ovidiu},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {nonlinear elliptic equations; global solutions; monotone solutions},
language = {eng},
number = {3},
pages = {369-405},
publisher = {Scuola Normale Superiore, Pisa},
title = {Entire solutions to a class of fully nonlinear elliptic equations},
url = {http://eudml.org/doc/272267},
volume = {7},
year = {2008},
}

TY - JOUR
AU - Savin, Ovidiu
TI - Entire solutions to a class of fully nonlinear elliptic equations
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2008
PB - Scuola Normale Superiore, Pisa
VL - 7
IS - 3
SP - 369
EP - 405
AB - We study nonlinear elliptic equations of the form $F(D^2u)=f(u)$ where the main assumption on $F$ and $f$ is that there exists a one dimensional solution which solves the equation in all the directions $\xi \in \mathbb {R}^n$. We show that entire monotone solutions $u$ are one dimensional if their $0$ level set is assumed to be Lipschitz, flat or bounded from one side by a hyperplane.
LA - eng
KW - nonlinear elliptic equations; global solutions; monotone solutions
UR - http://eudml.org/doc/272267
ER -

References

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  3. [3] H. Berestycki, F. Hamel and R. MonneauR., One-dimensional symmetry of bounded entire solutions of some elliptic equations, Duke Math. J. 103 (2000), 375–396. Zbl0954.35056MR1763653
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  7. [7] L. Caffarelli and L. Wang, A Harnack inequality approach to the interior regularity of elliptic equations, Indiana Univ. Math. J.42 (1993), 145–157. Zbl0810.35023MR1218709
  8. [8] E. De Giorgi, Convergence problems for functional and operators, Proc. Int. Meeting on Recent Methods in Nonlinear Analysis (1978), 131–188. Zbl0405.49001MR533166
  9. [9] D. De Silva and O. Savin, Symmetry of global solutions to fully nonlinear equations in 2D, Indiana Univ. Math. J., to appear. Zbl1165.35021
  10. [10] N. Ghoussoub and C. GuiC., On a conjecture of De Giorgi and some related problems, Math. Ann. 311 (1998), 481–491. Zbl0918.35046MR1637919
  11. [11] L. Modica, Γ -convergence to minimal surfaces problem and global solutions of Δ u = 2 ( u 3 - u ) , Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis, Rome, 1978, 223–244, Pitagora, Bologna, 1979. Zbl0408.49041MR533169
  12. [12] O. Savin, Regularity of flat level sets for phase transitions, Ann. of Math., to appear. Zbl1180.35499
  13. [13] O. Savin, Small perturbation solutions for elliptic equations, Comm. Partial Differential Equations32 (2007) 557–578. Zbl1221.35154MR2334822
  14. [14] E. Valdinoci, B. Sciunzi and O. Savin, “Flat Level Set Regularity of p -Laplace Phase Transitions”, Mem. Amer. Math. Soc., Vol. 182, 2006. Zbl1138.35029MR2228294

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