Pointwise estimates and rigidity results for entire solutions of nonlinear elliptic pde's
ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 2, pp. 616-627.

We prove pointwise gradient bounds for entire solutions of pde's of the form      ℒu(x) = ψ(x, u(x), ∇u(x)), where ℒ is an elliptic operator (possibly singular or degenerate). Thus, we obtain some Liouville type rigidity results. Some classical results of J. Serrin are also recovered as particular cases of our approach.

DOI : 10.1051/cocv/2012024
Classification : 35J60, 35B45
Mots-clés : gradient bounds, P-function estimates, rigidity results 
@article{COCV_2013__19_2_616_0,
     author = {Farina, Alberto and Valdinoci, Enrico},
     title = {Pointwise estimates and rigidity results for entire solutions of nonlinear elliptic pde's},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {616--627},
     publisher = {EDP-Sciences},
     volume = {19},
     number = {2},
     year = {2013},
     doi = {10.1051/cocv/2012024},
     mrnumber = {3049726},
     zbl = {1273.35126},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2012024/}
}
TY  - JOUR
AU  - Farina, Alberto
AU  - Valdinoci, Enrico
TI  - Pointwise estimates and rigidity results for entire solutions of nonlinear elliptic pde's
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2013
SP  - 616
EP  - 627
VL  - 19
IS  - 2
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2012024/
DO  - 10.1051/cocv/2012024
LA  - en
ID  - COCV_2013__19_2_616_0
ER  - 
%0 Journal Article
%A Farina, Alberto
%A Valdinoci, Enrico
%T Pointwise estimates and rigidity results for entire solutions of nonlinear elliptic pde's
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2013
%P 616-627
%V 19
%N 2
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2012024/
%R 10.1051/cocv/2012024
%G en
%F COCV_2013__19_2_616_0
Farina, Alberto; Valdinoci, Enrico. Pointwise estimates and rigidity results for entire solutions of nonlinear elliptic pde's. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 2, pp. 616-627. doi : 10.1051/cocv/2012024. http://www.numdam.org/articles/10.1051/cocv/2012024/

[1] S. Bernstein, Über ein geometrisches theorem und seine anwendung auf die partiellen differentialgleichungen vom elliptischen Typus. Math. Z. 26 (1927) 551-558. | JFM | MR

[2] L. Caffarelli, N. Garofalo and F. Segala, A gradient bound for entire solutions of quasi-linear equations and its consequences. Commun. Pure Appl. Math. 47 (1994) 1457-1473. | MR | Zbl

[3] D. Castellaneta, A. Farina and E. Valdinoci, A pointwise gradient estimate for solutions of singular and degenerate PDEs in possibly unbounded domains with nonnegative mean curvature. Commun. Pure Appl. Anal. 11 (2012) 1983-2003. | MR | Zbl

[4] D.G. De Figueiredo and P. Ubilla, Superlinear systems of second-order ODE's. Nonlinear Anal. 68 (2008) 1765-1773. | MR | Zbl

[5] D.G. De Figueiredo, J. Sánchez and P. Ubilla, Quasilinear equations with dependence on the gradient. Nonlinear Anal. 71 (2009) 4862-4868. | MR | Zbl

[6] E. Dibenedetto, C1 + α local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7 (1983) 827-850. | MR | Zbl

[7] A. Farina, Liouville-type theorems for elliptic problems, in Handbook of differential equations: stationary partial differential equations, Elsevier/North-Holland, Amsterdam. Handb. Differ. Equ. 4 (2007) 61-116. | MR | Zbl

[8] A. Farina and E. Valdinoci, Flattening results for elliptic PDEs in unbounded domains with applications to overdetermined problems. Arch. Ration. Mech. Anal. 195 (2010) 1025-1058. | MR | Zbl

[9] A. Farina and E. Valdinoci, A pointwise gradient estimate in possibly unbounded domains with nonnegative mean curvature. Adv. Math. 225 (2010) 2808-2827. | MR | Zbl

[10] A. Farina and E. Valdinoci, A pointwise gradient bound for elliptic equations on compact manifolds with nonnegative Ricci curvature. Discrete Contin. Dyn. Syst. 30 (2011) 1139-1144. | MR | Zbl

[11] D. Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin (2001). Reprint of the 1998 edition. | MR | Zbl

[12] P. Hartman, Ordinary differential equations, Society for Industrial and Applied Mathematics SIAM, Philadelphia, PA. Classics Appl. Math. 38 (2002). Corrected reprint of the second (1982) edition [Birkhäuser, Boston, MA, MR0658490 (83e:34002)]. With a foreword by Peter Bates. | MR | Zbl

[13] L. Modica, A gradient bound and a Liouville theorem for nonlinear Poisson equations. Commun. Pure Appl. Math. 38 (1985) 679-684. | MR | Zbl

[14] L.E. Payne, Some remarks on maximum principles. J. Anal. Math. 30 (1976) 421-433. | MR | Zbl

[15] J. Serrin, Entire solutions of nonlinear Poisson equations. Proc. London Math. Soc. 24 (1972) 348-366. | MR | Zbl

[16] R.P. Sperb, Maximum principles and their applications, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York. Math. Sci. Eng. 157 (1981). | MR | Zbl

[17] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations. J. Differ. Equ. 51 (1984) 126-150. | MR | Zbl

Cité par Sources :