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January  2015, 14(1): 127-132. doi: 10.3934/cpaa.2015.14.127

Phragmén--Lindelöf theorem for infinity harmonic functions

Seppo Granlund 1, and  Niko Marola 1,

1. 

University of Helsinki, Department of Mathematics and Statistics, P.O. Box 68, FI-00014 University of Helsinki, Finland, Finland

Received  October 2013 Revised  January 2014 Published  September 2014

We investigate a version of the Phragmén--Lindelöf theorem for solutions of the equation $\Delta_\infty u=0$ in unbounded convex domains. The method of proof is to consider this infinity harmonic equation as the limit of the $p$-harmonic equation when $p$ tends to $\infty$.
Keywords: $p$-Laplace equation, $p$-harmonic equation, Growth estimate, infinity harmonic, Phragmén--Lindelöf principle, $p$-Laplacian..
Mathematics Subject Classification: Primary: 35B40; Secondary: 35J7.
Citation: Seppo Granlund, Niko Marola. Phragmén--Lindelöf theorem for infinity harmonic functions. Communications on Pure & Applied Analysis, 2015, 14 (1) : 127-132. doi: 10.3934/cpaa.2015.14.127
References:
[1]

G. Aronsson, On the partial differential equation $u_x^{2}u_{xx} +2u_xu_yu_{xy}+u_y^{2}u_{yy}=0$, Ark. Mat., 7 (1968), 395-425.  Google Scholar

[2]

G. Aronsson, M. G. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc. (N.S.), 41 (2004), 439-505. doi: 10.1090/S0273-0979-04-01035-3.  Google Scholar

[3]

T. Bhattacharya, On the behaviour of $\infty$-harmonic functions on some special unbounded domains, Pacific J. Math., 219 (2005), 237-253. doi: 10.2140/pjm.2005.219.237.  Google Scholar

[4]

T. Bhattacharya, A note on non-negative singular infinity-harmonic functions in the half-space, Rev. Mat. Complut., 18 (2005), 377-385.  Google Scholar

[5]

T. Bhattacharya, E. DiBenedetto and J. Manfredi, Limits as $p\to\infty$ of $\Delta_pu_p=f$ and related extremal problems, Rend. Sem. Mat. Univ. Politec. Torino, (1989), 15-68.  Google Scholar

[6]

I. Capuzzo Dolcetta and A. Vitolo, A qualitative Phragmén-Lindelöf theorem for fully nonlinear elliptic equations, J. Differential Equations, 243 (2007), 578-592. doi: 10.1016/j.jde.2007.08.001.  Google Scholar

[7]

S. Granlund, P. Lindqvist and O. Martio, Phragmén-Lindelöf's and Lindelöf's theorems, Ark. Mat., 23 (1985), 103-128. doi: 10.1007/BF02384420.  Google Scholar

[8]

R. Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient, Arch. Rational Mech. Anal., 123 (1993), 51-74. doi: 10.1007/BF00386368.  Google Scholar

[9]

Z. Jin and K. Lancaster, Theorems of Phragmén-Lindelöf type for quasilinear elliptic equations, J. Reine Angew. Math., 514 (1999), 165-197. doi: 10.1515/crll.1999.070.  Google Scholar

[10]

P. Lindqvist, On the growth of the solutions of the differential equation div$(|\nabla u|^{p-2}\nabla u)=0$ in $n$-dimensional space, J. Differential Equations, 58 (1985), 307-317. doi: 10.1016/0022-0396(85)90002-6.  Google Scholar

[11]

P. Lindqvist and J. Manfredi, The Harnack inequality for $\infty$-harmonic functions, Electron. J. Differential Equations, 4 (1995), 1-5.  Google Scholar

[12]

E. Phragmén and E. Lindelöf, Sur une extension d'un principe classique de l'analyse et sur quelques propriétés des fonctions monogénes dans le voisinage d'un point singulier, Acta Math., 31 (1908), 381-406. doi: 10.1007/BF02415450.  Google Scholar

show all references

References:
[1]

G. Aronsson, On the partial differential equation $u_x^{2}u_{xx} +2u_xu_yu_{xy}+u_y^{2}u_{yy}=0$, Ark. Mat., 7 (1968), 395-425.  Google Scholar

[2]

G. Aronsson, M. G. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc. (N.S.), 41 (2004), 439-505. doi: 10.1090/S0273-0979-04-01035-3.  Google Scholar

[3]

T. Bhattacharya, On the behaviour of $\infty$-harmonic functions on some special unbounded domains, Pacific J. Math., 219 (2005), 237-253. doi: 10.2140/pjm.2005.219.237.  Google Scholar

[4]

T. Bhattacharya, A note on non-negative singular infinity-harmonic functions in the half-space, Rev. Mat. Complut., 18 (2005), 377-385.  Google Scholar

[5]

T. Bhattacharya, E. DiBenedetto and J. Manfredi, Limits as $p\to\infty$ of $\Delta_pu_p=f$ and related extremal problems, Rend. Sem. Mat. Univ. Politec. Torino, (1989), 15-68.  Google Scholar

[6]

I. Capuzzo Dolcetta and A. Vitolo, A qualitative Phragmén-Lindelöf theorem for fully nonlinear elliptic equations, J. Differential Equations, 243 (2007), 578-592. doi: 10.1016/j.jde.2007.08.001.  Google Scholar

[7]

S. Granlund, P. Lindqvist and O. Martio, Phragmén-Lindelöf's and Lindelöf's theorems, Ark. Mat., 23 (1985), 103-128. doi: 10.1007/BF02384420.  Google Scholar

[8]

R. Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient, Arch. Rational Mech. Anal., 123 (1993), 51-74. doi: 10.1007/BF00386368.  Google Scholar

[9]

Z. Jin and K. Lancaster, Theorems of Phragmén-Lindelöf type for quasilinear elliptic equations, J. Reine Angew. Math., 514 (1999), 165-197. doi: 10.1515/crll.1999.070.  Google Scholar

[10]

P. Lindqvist, On the growth of the solutions of the differential equation div$(|\nabla u|^{p-2}\nabla u)=0$ in $n$-dimensional space, J. Differential Equations, 58 (1985), 307-317. doi: 10.1016/0022-0396(85)90002-6.  Google Scholar

[11]

P. Lindqvist and J. Manfredi, The Harnack inequality for $\infty$-harmonic functions, Electron. J. Differential Equations, 4 (1995), 1-5.  Google Scholar

[12]

E. Phragmén and E. Lindelöf, Sur une extension d'un principe classique de l'analyse et sur quelques propriétés des fonctions monogénes dans le voisinage d'un point singulier, Acta Math., 31 (1908), 381-406. doi: 10.1007/BF02415450.  Google Scholar

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