1D symmetry for solutions of semilinear and quasilinear elliptic equations

Farina, Alberto; Valdinoci, Enrico

doi:10.1090/S0002-9947-2010-05021-4

$1$D symmetry for solutions of semilinear and quasilinear elliptic equations
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by Alberto Farina and Enrico Valdinoci

Trans. Amer. Math. Soc. 363 (2011), 579-609

DOI: https://doi.org/10.1090/S0002-9947-2010-05021-4

Published electronically: September 21, 2010

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Abstract:

Several new $1$D results for solutions of possibly singular or degenerate elliptic equations, inspired by a conjecture of De Giorgi, are provided. In particular, $1$D symmetry is proven under the assumption that either the profiles at infinity are $2$D, or that one level set is a complete graph, or that the solution is minimal or, more generally, $Q$-minimal.

References

Giovanni Alberti, Luigi Ambrosio, and Xavier Cabré, On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property, Acta Appl. Math. 65 (2001), no. 1-3, 9–33. Special issue dedicated to Antonio Avantaggiati on the occasion of his 70th birthday. MR 1843784, DOI 10.1023/A:1010602715526
Luigi Ambrosio and Xavier Cabré, Entire solutions of semilinear elliptic equations in $\mathbf R^3$ and a conjecture of De Giorgi, J. Amer. Math. Soc. 13 (2000), no. 4, 725–739. MR 1775735, DOI 10.1090/S0894-0347-00-00345-3
Martin T. Barlow, Richard F. Bass, and Changfeng Gui, The Liouville property and a conjecture of De Giorgi, Comm. Pure Appl. Math. 53 (2000), no. 8, 1007–1038. MR 1755949, DOI 10.1002/1097-0312(200008)53:8<1007::AID-CPA3>3.3.CO;2-L
Henri Berestycki, Luis Caffarelli, and Louis Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 1-2, 69–94 (1998). Dedicated to Ennio De Giorgi. MR 1655510
H. Berestycki, F. Hamel, and R. Monneau, One-dimensional symmetry of bounded entire solutions of some elliptic equations, Duke Math. J. 103 (2000), no. 3, 375–396. MR 1763653, DOI 10.1215/S0012-7094-00-10331-6
Guy Bouchitté, Singular perturbations of variational problems arising from a two-phase transition model, Appl. Math. Optim. 21 (1990), no. 3, 289–314. MR 1036589, DOI 10.1007/BF01445167
Lucio Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincaré C Anal. Non Linéaire 15 (1998), no. 4, 493–516 (English, with English and French summaries). MR 1632933, DOI 10.1016/S0294-1449(98)80032-2
Ennio De Giorgi, Convergence problems for functionals and operators, Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978) Pitagora, Bologna, 1979, pp. 131–188. MR 533166
Donatella Danielli and Nicola Garofalo, Properties of entire solutions of non-uniformly elliptic equations arising in geometry and in phase transitions, Calc. Var. Partial Differential Equations 15 (2002), no. 4, 451–491. MR 1942128, DOI 10.1007/s005260100133
E. DiBenedetto, $C^{1+\alpha }$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983), no. 8, 827–850. MR 709038, DOI 10.1016/0362-546X(83)90061-5
Manuel del Pino, Mike Kowalczyk, and Juncheng Wei, On De Giorgi Conjecture in Dimension $N \geq 9$, Preprint (2008), http://eprintweb.org/S/article/math/0806.3141.
Alberto Farina, Symmetry for solutions of semilinear elliptic equations in $\mathbf R^N$ and related conjectures, Ricerche Mat. 48 (1999), no. suppl., 129–154. Papers in memory of Ennio De Giorgi (Italian). MR 1765681
Alberto Farina, Monotonicity and one-dimensional symmetry for the solutions of $\Delta u+f(u)=0$ in ${\Bbb R}^N$ with possibly discontinuous nonlinearity, Adv. Math. Sci. Appl. 11 (2001), no. 2, 811–834. MR 1907468
Alberto Farina, Rigidity and one-dimensional symmetry for semilinear elliptic equations in the whole of $\Bbb R^N$ and in half spaces, Adv. Math. Sci. Appl. 13 (2003), no. 1, 65–82. MR 2002396
Alberto Farina, Liouville-type theorems for elliptic problems, Handbook of differential equations: stationary partial differential equations. Vol. IV, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2007, pp. 61–116. MR 2569331, DOI 10.1016/S1874-5733(07)80005-2
Alberto Farina, Berardino Sciunzi, and Enrico Valdinoci, Bernstein and De Giorgi type problems: new results via a geometric approach, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 7 (2008), no. 4, 741–791. MR 2483642
Alberto Farina and Enrico Valdinoci, Geometry of quasiminimal phase transitions, Calc. Var. Partial Differential Equations 33 (2008), no. 1, 1–35. MR 2413100, DOI 10.1007/s00526-007-0146-1
N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems, Math. Ann. 311 (1998), no. 3, 481–491. MR 1637919, DOI 10.1007/s002080050196
Enrico Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics, vol. 80, Birkhäuser Verlag, Basel, 1984. MR 775682, DOI 10.1007/978-1-4684-9486-0
Juha Heinonen, Tero Kilpeläinen, and Olli Martio, Nonlinear potential theory of degenerate elliptic equations, Dover Publications, Inc., Mineola, NY, 2006. Unabridged republication of the 1993 original. MR 2305115
David Jerison and Régis Monneau, Towards a counter-example to a conjecture of De Giorgi in high dimensions, Ann. Mat. Pura Appl. (4) 183 (2004), no. 4, 439–467. MR 2140525, DOI 10.1007/s10231-002-0068-7
Patrizia Pucci, James Serrin, and Henghui Zou, A strong maximum principle and a compact support principle for singular elliptic inequalities, J. Math. Pures Appl. (9) 78 (1999), no. 8, 769–789. MR 1715341, DOI 10.1016/S0021-7824(99)00030-6
Arshak Petrosyan and Enrico Valdinoci, Density estimates for a degenerate/singular phase-transition model, SIAM J. Math. Anal. 36 (2005), no. 4, 1057–1079. MR 2139200, DOI 10.1137/S0036141003437678
Vasile Ovidiu Savin, Phase Transitions: Regularity of Flat Level Sets, Ph.D. thesis, University of Texas at Austin, 2003.
Berardino Sciunzi and Enrico Valdinoci, Mean curvature properties for $p$-Laplace phase transitions, J. Eur. Math. Soc. (JEMS) 7 (2005), no. 3, 319–359. MR 2156604, DOI 10.4171/JEMS/31
Peter Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), no. 1, 126–150. MR 727034, DOI 10.1016/0022-0396(84)90105-0
J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), no. 3, 191–202. MR 768629, DOI 10.1007/BF01449041
Enrico Valdinoci, Berardino Sciunzi, and Vasile Ovidiu Savin, Flat level set regularity of $p$-Laplace phase transitions, Mem. Amer. Math. Soc. 182 (2006), no. 858, vi+144. MR 2228294, DOI 10.1090/memo/0858