Pointwise estimates and rigidity results for entire solutions of nonlinear elliptic pde’s

Alberto Farina; Enrico Valdinoci

ESAIM: Control, Optimisation and Calculus of Variations (2013)

  • Volume: 19, Issue: 2, page 616-627
  • ISSN: 1292-8119

Abstract

top
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.

How to cite

top

Farina, Alberto, and Valdinoci, Enrico. "Pointwise estimates and rigidity results for entire solutions of nonlinear elliptic pde’s." ESAIM: Control, Optimisation and Calculus of Variations 19.2 (2013): 616-627. <http://eudml.org/doc/272759>.

@article{Farina2013,
abstract = {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.},
author = {Farina, Alberto, Valdinoci, Enrico},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {gradient bounds; P-function estimates; rigidity results; -function estimates},
language = {eng},
number = {2},
pages = {616-627},
publisher = {EDP-Sciences},
title = {Pointwise estimates and rigidity results for entire solutions of nonlinear elliptic pde’s},
url = {http://eudml.org/doc/272759},
volume = {19},
year = {2013},
}

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
PB - EDP-Sciences
VL - 19
IS - 2
SP - 616
EP - 627
AB - 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.
LA - eng
KW - gradient bounds; P-function estimates; rigidity results; -function estimates
UR - http://eudml.org/doc/272759
ER -

References

top
  1. [1] S. Bernstein, Über ein geometrisches theorem und seine anwendung auf die partiellen differentialgleichungen vom elliptischen Typus. Math. Z.26 (1927) 551–558. Zbl53.0670.01MR1544873JFM53.0670.01
  2. [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. Zbl0819.35016MR1296785
  3. [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. Zbl1266.35097MR2911121
  4. [4] D.G. De Figueiredo and P. Ubilla, Superlinear systems of second-order ODE’s. Nonlinear Anal.68 (2008) 1765–1773. Zbl1137.34320MR2388850
  5. [5] D.G. De Figueiredo, J. Sánchez and P. Ubilla, Quasilinear equations with dependence on the gradient. Nonlinear Anal.71 (2009) 4862–4868. Zbl1178.34027MR2548718
  6. [6] E. DiBenedetto, C1 + α local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal.7 (1983) 827–850. Zbl0539.35027MR709038
  7. [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. Zbl1191.35128MR2569331
  8. [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. Zbl1236.35058MR2591980
  9. [9] A. Farina and E. Valdinoci, A pointwise gradient estimate in possibly unbounded domains with nonnegative mean curvature. Adv. Math.225 (2010) 2808–2827. Zbl1200.35131MR2680184
  10. [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. Zbl1230.53037MR2812957
  11. [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. Zbl1042.35002MR1814364
  12. [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. Zbl1009.34001MR1929104
  13. [13] L. Modica, A gradient bound and a Liouville theorem for nonlinear Poisson equations. Commun. Pure Appl. Math.38 (1985) 679–684. Zbl0612.35051MR803255
  14. [14] L.E. Payne, Some remarks on maximum principles. J. Anal. Math.30 (1976) 421–433. Zbl0334.35029MR454338
  15. [15] J. Serrin, Entire solutions of nonlinear Poisson equations. Proc. London Math. Soc.24 (1972) 348–366. Zbl0229.35035MR289961
  16. [16] R.P. Sperb, Maximum principles and their applications, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York. Math. Sci. Eng. 157 (1981). Zbl0454.35001MR615561
  17. [17] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations. J. Differ. Equ.51 (1984) 126–150. Zbl0488.35017MR727034

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.