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.
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] Über ein geometrisches theorem und seine anwendung auf die partiellen differentialgleichungen vom elliptischen Typus. Math. Z. 26 (1927) 551-558. | JFM | MR
,[2] A gradient bound for entire solutions of quasi-linear equations and its consequences. Commun. Pure Appl. Math. 47 (1994) 1457-1473. | MR | Zbl
, and ,[3] 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
, and ,[4] Superlinear systems of second-order ODE's. Nonlinear Anal. 68 (2008) 1765-1773. | MR | Zbl
and ,[5] Quasilinear equations with dependence on the gradient. Nonlinear Anal. 71 (2009) 4862-4868. | MR | Zbl
, and ,[6] C1 + α local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7 (1983) 827-850. | MR | Zbl
,[7] 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] Flattening results for elliptic PDEs in unbounded domains with applications to overdetermined problems. Arch. Ration. Mech. Anal. 195 (2010) 1025-1058. | MR | Zbl
and ,[9] A pointwise gradient estimate in possibly unbounded domains with nonnegative mean curvature. Adv. Math. 225 (2010) 2808-2827. | MR | Zbl
and ,[10] A pointwise gradient bound for elliptic equations on compact manifolds with nonnegative Ricci curvature. Discrete Contin. Dyn. Syst. 30 (2011) 1139-1144. | MR | Zbl
and ,[11] Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin (2001). Reprint of the 1998 edition. | MR | Zbl
and ,[12] 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] A gradient bound and a Liouville theorem for nonlinear Poisson equations. Commun. Pure Appl. Math. 38 (1985) 679-684. | MR | Zbl
,[14] Some remarks on maximum principles. J. Anal. Math. 30 (1976) 421-433. | MR | Zbl
,[15] Entire solutions of nonlinear Poisson equations. Proc. London Math. Soc. 24 (1972) 348-366. | MR | Zbl
,[16] Maximum principles and their applications, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York. Math. Sci. Eng. 157 (1981). | MR | Zbl
,[17] Regularity for a more general class of quasilinear elliptic equations. J. Differ. Equ. 51 (1984) 126-150. | MR | Zbl
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