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August  2019, 39(8): 4731-4746. doi: 10.3934/dcds.2019192

Infinity-harmonic potentials and their streamlines

Erik Lindgren 1,, and  Peter Lindqvist 2,

1. 

Department of Mathematics, Uppsala University, Box 480, 751 06 Uppsala, Sweden
2. 

Department of Mathematical Sciences, Norwegian University of Science and Technology, N–7491, Trondheim, Norway

* Corresponding author: Erik Lindgren

Received  October 2018 Revised  February 2019 Published  May 2019

We consider certain solutions of the Infinity-Laplace Equation in planar convex rings. Their ascending streamlines are unique while the descending ones may bifurcate. We prove that bifurcation occurs in the generic situation and as a consequence, the solutions cannot have Lipschitz continuous gradients.

Keywords: Infinity-Laplace equation, streamlines, convex rings, infinity-potential function.
Mathematics Subject Classification: Primary: 49N60, 35J15, 35J60, 35J65, 35J70.
Citation: Erik Lindgren, Peter Lindqvist. Infinity-harmonic potentials and their streamlines. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4731-4746. doi: 10.3934/dcds.2019192
References:
[1]

G. Aronsson, Extension of functions satisfying Lipschitz conditions, Arkiv För Matematik, 6 (1967), 551-561.  doi: 10.1007/BF02591928.

[2]

G. Aronsson, On the partial differential equation $u_x^2u_xx+2u_xu_yu_xy+u_y^2u_yy = 0, $, Arkiv för Matematik, 7 (1968), 397-425.  doi: 10.1007/BF02590989.

[3]

V. Caselles, J.-M. Morel and C. Sbert, An axiomatic approach to image interpolation, IEEE Transactions on Image Processing, 7 (1998), 376-386.  doi: 10.1109/83.661188.

[4]

M. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bulletin of the American Mathematical Society, 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.

[5]

G. Crasta and I. Fragalà, On the characterization of some classes of proximally smooth sets, ESAIM: Control, Optimisation and Calculus of Variations, 22 (2016), 710-727.  doi: 10.1051/cocv/2015022.

[6]

G. Crasta and I. Fragalà, On the Dirichlet and Serrin problems for the inhomogeneous infinity Laplacian in convex domains: regularity and geometric results, Archive for Rational Mechanics and Analysis, 218 (2015), 1577-1607.  doi: 10.1007/s00205-015-0888-4.

[7] L. Evans and R. Gariepy, Measure Theory and Fine Propeties of Functions, CRC Press, Boca Raton, 1992. 
[8]

L. Evans and O. Savin, $C^{1, \alpha}$regularity of infinite harmonic functios in two dimensions, Calculus of Variations and Partial Differential Equations, 32 (2008), 325-347.  doi: 10.1007/s00526-007-0143-4.

[9]

L. Evans and Ch. Smart, Everywhere differentiability of infinity harmonic functions, Calculus of Variations and Partial Differential Equations, 42 (2011), 289-299.  doi: 10.1007/s00526-010-0388-1.

[10]

U. Janfalk, Behaviour in the limit, as $p \to \infty$, of minimizers of functionals involving $p$-Dirichlet integrals, SIAM Journal on Mathematical Analysis, 27 (1996), 341-360.  doi: 10.1137/S0036141093252619.

[11]

R. Jensen, Uniqueness of Lipschitz extension: Minimizing the sup norm of the gradient, Archive for Rational Mechanics and Analysis, 123 (1993), 51-74.  doi: 10.1007/BF00386368.

[12]

V. Julin and P. Juutinen, A new proof for the equivalence of weak and viscosity solutions for the $p$-Laplace equation, Communications on Partial Differential Equations, 37 (2012), 934-946.  doi: 10.1080/03605302.2011.615878.

[13]

P. Juutinen, P. Lindqvist and J. Manfredi., On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation, SIAM Journal on Mathematical Analysis, 33 (2001), 699-717.  doi: 10.1137/S0036141000372179.

[14]

P. Juutinen, P. Lindqvist and J. Manfredi., The infinity Laplacian: examples and observations, Papers on analysis, Rep. Univ. Jyväskylä Dep. Math. Stat. Univ. Jyväskylä, 83 (2001), 207-217. 

[15]

P. Juutinen, P. Lindqvist and J. Manfredi, The $\infty$-eigenvalue problem, Archive for Rational Mechanics and Analysis, 148 (1999), 89-105.  doi: 10.1007/s002050050157.

[16]

H. Koch, Y. Zhang and Y. Zhou, An asymptotic sharp Sobolev regularity for planar infinity harmonic functions, Journal de Mathématiques Pures et Appliquées, 2019, arXiv: 1806.01982 doi: 10.1016/j.matpur.2019.02.008.

[17]

S. Koike, A Beginner's Guide to the Theory of Viscosity Solutions, MSJ Memoirs, Mathematical Society of Japan, 2004.

[18]

J. Lewis, Capacitory functions in convex rings, Archive for Rational Mechanics and Analysis, 66 (1977), 201-224.  doi: 10.1007/BF00250671.

[19]

J. Manfredi, A. Petrosyan and H. Shahgholian, A free boundary problem for $\infty$-Laplace equation, Calculus of Variations and Partial Differential Equations, 14 (2002), 359-384.  doi: 10.1007/s005260100107.

[20]

Y. Peres, O. Schramm, S. Sheffield and D. Wilson, Tug-of-war and the infinity Laplacian, Journal of the American Mathematical Society, 22 (2009), 167-210.  doi: 10.1090/S0894-0347-08-00606-1.

[21] R. Rockafeller, Convex Analysis, Princeton University Press, USA, 1970. 
[22]

O. Savin, $C^1$ regularity for infinity harmonic functions in two dimensions, Archive for Rational Mechanics and Analysis, 176 (2005), 351-361.  doi: 10.1007/s00205-005-0355-8.

[23]

O. Savin, C. Wang and Y. Yu, Asymptotic behaviour of infinity harmonic functions near an isolated singularity, International Mathematical Research Notes, 6 (2008), Art. ID rnm163, 23 pp. doi: 10.1093/imrn/rnm163.

show all references

References:
[1]

G. Aronsson, Extension of functions satisfying Lipschitz conditions, Arkiv För Matematik, 6 (1967), 551-561.  doi: 10.1007/BF02591928.

[2]

G. Aronsson, On the partial differential equation $u_x^2u_xx+2u_xu_yu_xy+u_y^2u_yy = 0, $, Arkiv för Matematik, 7 (1968), 397-425.  doi: 10.1007/BF02590989.

[3]

V. Caselles, J.-M. Morel and C. Sbert, An axiomatic approach to image interpolation, IEEE Transactions on Image Processing, 7 (1998), 376-386.  doi: 10.1109/83.661188.

[4]

M. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bulletin of the American Mathematical Society, 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.

[5]

G. Crasta and I. Fragalà, On the characterization of some classes of proximally smooth sets, ESAIM: Control, Optimisation and Calculus of Variations, 22 (2016), 710-727.  doi: 10.1051/cocv/2015022.

[6]

G. Crasta and I. Fragalà, On the Dirichlet and Serrin problems for the inhomogeneous infinity Laplacian in convex domains: regularity and geometric results, Archive for Rational Mechanics and Analysis, 218 (2015), 1577-1607.  doi: 10.1007/s00205-015-0888-4.

[7] L. Evans and R. Gariepy, Measure Theory and Fine Propeties of Functions, CRC Press, Boca Raton, 1992. 
[8]

L. Evans and O. Savin, $C^{1, \alpha}$regularity of infinite harmonic functios in two dimensions, Calculus of Variations and Partial Differential Equations, 32 (2008), 325-347.  doi: 10.1007/s00526-007-0143-4.

[9]

L. Evans and Ch. Smart, Everywhere differentiability of infinity harmonic functions, Calculus of Variations and Partial Differential Equations, 42 (2011), 289-299.  doi: 10.1007/s00526-010-0388-1.

[10]

U. Janfalk, Behaviour in the limit, as $p \to \infty$, of minimizers of functionals involving $p$-Dirichlet integrals, SIAM Journal on Mathematical Analysis, 27 (1996), 341-360.  doi: 10.1137/S0036141093252619.

[11]

R. Jensen, Uniqueness of Lipschitz extension: Minimizing the sup norm of the gradient, Archive for Rational Mechanics and Analysis, 123 (1993), 51-74.  doi: 10.1007/BF00386368.

[12]

V. Julin and P. Juutinen, A new proof for the equivalence of weak and viscosity solutions for the $p$-Laplace equation, Communications on Partial Differential Equations, 37 (2012), 934-946.  doi: 10.1080/03605302.2011.615878.

[13]

P. Juutinen, P. Lindqvist and J. Manfredi., On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation, SIAM Journal on Mathematical Analysis, 33 (2001), 699-717.  doi: 10.1137/S0036141000372179.

[14]

P. Juutinen, P. Lindqvist and J. Manfredi., The infinity Laplacian: examples and observations, Papers on analysis, Rep. Univ. Jyväskylä Dep. Math. Stat. Univ. Jyväskylä, 83 (2001), 207-217. 

[15]

P. Juutinen, P. Lindqvist and J. Manfredi, The $\infty$-eigenvalue problem, Archive for Rational Mechanics and Analysis, 148 (1999), 89-105.  doi: 10.1007/s002050050157.

[16]

H. Koch, Y. Zhang and Y. Zhou, An asymptotic sharp Sobolev regularity for planar infinity harmonic functions, Journal de Mathématiques Pures et Appliquées, 2019, arXiv: 1806.01982 doi: 10.1016/j.matpur.2019.02.008.

[17]

S. Koike, A Beginner's Guide to the Theory of Viscosity Solutions, MSJ Memoirs, Mathematical Society of Japan, 2004.

[18]

J. Lewis, Capacitory functions in convex rings, Archive for Rational Mechanics and Analysis, 66 (1977), 201-224.  doi: 10.1007/BF00250671.

[19]

J. Manfredi, A. Petrosyan and H. Shahgholian, A free boundary problem for $\infty$-Laplace equation, Calculus of Variations and Partial Differential Equations, 14 (2002), 359-384.  doi: 10.1007/s005260100107.

[20]

Y. Peres, O. Schramm, S. Sheffield and D. Wilson, Tug-of-war and the infinity Laplacian, Journal of the American Mathematical Society, 22 (2009), 167-210.  doi: 10.1090/S0894-0347-08-00606-1.

[21] R. Rockafeller, Convex Analysis, Princeton University Press, USA, 1970. 
[22]

O. Savin, $C^1$ regularity for infinity harmonic functions in two dimensions, Archive for Rational Mechanics and Analysis, 176 (2005), 351-361.  doi: 10.1007/s00205-005-0355-8.

[23]

O. Savin, C. Wang and Y. Yu, Asymptotic behaviour of infinity harmonic functions near an isolated singularity, International Mathematical Research Notes, 6 (2008), Art. ID rnm163, 23 pp. doi: 10.1093/imrn/rnm163.

Figure 1.  The streamlines of $ V_\infty $ when $ \Omega $ is the square $ -1<x_1<1,\,\,-1<x_2<1 $
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