Allen-Cahn equation for the truncated Laplacian: Unusual phenomena

Isabeau Birindelli; Giulio Galise; Isabeau Birindelli; Giulio Galise

doi:10.3934/mine.2020034

Mathematics in Engineering

2020, Volume 2, Issue 4: 722-733. doi: 10.3934/mine.2020034

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Allen-Cahn equation for the truncated Laplacian: Unusual phenomena

Isabeau Birindelli ^,,
Giulio Galise

Dipartimento di Matematica Guido Castelnuovo, Sapienza Università di Roma, P.le Aldo Moro 2, I-00185 Roma, Italy

Received: 11 February 2020 Accepted: 04 June 2020 Published: 01 July 2020

We study entire viscosity solutions of the Allen-Cahn type equation for the truncated Laplacian that are either one dimensional or radial, in order to shed some light on a possible extension of the Gibbons conjecture in this degenerate elliptic setting.
- fully nonlinear degenerate elliptic equations,
- entire solutions,
- one dimensional symmetry
Citation: Isabeau Birindelli, Giulio Galise. Allen-Cahn equation for the truncated Laplacian: Unusual phenomena[J]. Mathematics in Engineering, 2020, 2(4): 722-733. doi: 10.3934/mine.2020034

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Abstract

We study entire viscosity solutions of the Allen-Cahn type equation for the truncated Laplacian that are either one dimensional or radial, in order to shed some light on a possible extension of the Gibbons conjecture in this degenerate elliptic setting.

References

[1]	Aronson DG, Weinberger HF (1978) Multidimensional nonlinear diffusion arising in population genetics. Adv Math 30: 33-76. doi: 10.1016/0001-8708(78)90130-5
[2]	Ambrosio L, Cabré X (2000) Entire solutions of semilinear elliptic equations in $\mathbb{R}^3$ and a conjecture of De Giorgi. J Am Math Soc 13: 725-739. doi: 10.1090/S0894-0347-00-00345-3
[3]	Barlow MT, Bass RF, Gui C (2000) The Liouville property and a conjecture of De Giorgi. Commun Pure Appl Math 53: 1007-1038. doi: 10.1002/1097-0312(200008)53:8<1007::AID-CPA3>3.0.CO;2-U
[4]	Berestycki H, Caffarelli LA, Nirenberg L (1997) Monotonicity for elliptic equations in unbounded Lipschitz domains. Commun Pure Appl Math 50: 1089-1111. doi: 10.1002/(SICI)1097-0312(199711)50:11<1089::AID-CPA2>3.0.CO;2-6
[5]	Berestycki H, Hamel F, Monneau R (2000) One-dimensional symmetry of bounded entire solutions of some elliptic equations. Duke Math J 103: 375-396. doi: 10.1215/S0012-7094-00-10331-6
[6]	Berestycki H, Hamel F, Nadirashvili N (2005) The speed of propagation for KPP type problems. I: Periodic framework. J Eur Math Soc 7: 173-213.
[7]	Berestycki H, Nirenberg L (1991) On the method of moving planes and the sliding method. Bol Soc Bras Mat 22: 1-37. doi: 10.1007/BF01244896
[8]	Berestycki H, Nirenberg L, Varadhan S (1994) The principle eigenvalue and maximum principle for second order elliptic operators in general domains. Commun Pure Appl Math 47: 47-92. doi: 10.1002/cpa.3160470105
[9]	Birindelli I, Galise G, Ishii H (2018) A family of degenerate elliptic operators: Maximum principle and its consequences. Ann Inst H Poincaré Anal Non Linéaire 35: 417-441.
[10]	Birindelli I, Galise G, Ishii H (2020) Existence through convexity for the truncated Laplacians. DOI:10.1007/s00208-019-01953-x.
[11]	Birindelli I, Galise G, Ishii H (2020) Towards a reversed Faber-Krahn inequality for the truncated Laplacian. Rev Mat Iberoam 3: 723-740.
[12]	Birindelli I, Galise G, Ishii H (2020) Positivity sets of supersolutions of degenerate elliptic equations and the strong maximum principle. arXiv:1911.08204.
[13]	Birindelli I, Galise G, Leoni F (2017) Liouville theorems for a family of very degenerate elliptic nonlinear operators. Nonlinear Anal 161: 198-211. doi: 10.1016/j.na.2017.06.002
[14]	Birindelli I, Lanconelli E (2003) A negative answer to a one-dimensional symmetry problem in the Heisenberg group. Calc Var Part Dif 18: 357-372. doi: 10.1007/s00526-003-0194-0
[15]	Birindelli I, Mazzeo R (2009) Symmetry for solutions of two-phase semilinear elliptic equations on hyperbolic space. Indiana Univ Math J 58: 2347-2368. doi: 10.1512/iumj.2009.58.3714
[16]	Birindelli I, Prajapat J (2002) Monotonicity and symmetry results for degenerate elliptic equations on nilpotent Lie groups. Pacific J Math 201: 1-17.
[17]	Del Pino M, Kowalczyk M, Wei J (2011) On De Giorgi's conjecture in dimension N ≥ 9. Ann Math 174: 1485-1569. doi: 10.4007/annals.2011.174.3.3
[18]	Farina A (1999) Symmetry for solutions of semilinear elliptic equations in $\mathbb{R}^N$ and related conjectures. Ricerche Mat 48: 129-154.
[19]	Farina A (2003) Rigidity and one-dimensional symmetry for semilinear elliptic equations in the whole of $\mathbb{R}^N$ and in half spaces. Adv Math Sci Appl 13: 65-82.
[20]	Farina A, Valdinoci E (2011) Rigidity results for elliptic PDEs with uniform limits: An abstract framework with applications. Indiana Univ Math J 60: 121-141. doi: 10.1512/iumj.2011.60.4433
[21]	Ghoussoub N, Gui C (1998) On a conjecture of De Giorgi and some related problems. Math Ann 311: 481-491. doi: 10.1007/s002080050196
[22]	Parini E, Rossi J, Salort A (2020) Reverse Faber-Krahn inequality for a truncated laplacian operator. arXiv:2003.12107.
[23]	Polacik P (2017) Propagating terraces in a proof of the Gibbons conjecture and related results. J Fixed Point Theory Appl 19: 113-128. doi: 10.1007/s11784-016-0343-7
[24]	Savin O (2003) Phase transitions: Regularity of flat level sets, PhD Thesis of UT Austin.

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