# American Institute of Mathematical Sciences

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

## Infinity-harmonic potentials and their streamlines

 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.

Citation: Erik Lindgren, Peter Lindqvist. Infinity-harmonic potentials and their streamlines. Discrete & Continuous Dynamical Systems - A, 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [7] L. Evans and R. Gariepy, Measure Theory and Fine Propeties of Functions, CRC Press, Boca Raton, 1992. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [17] S. Koike, A Beginner's Guide to the Theory of Viscosity Solutions, MSJ Memoirs, Mathematical Society of Japan, 2004. Google Scholar [18] J. Lewis, Capacitory functions in convex rings, Archive for Rational Mechanics and Analysis, 66 (1977), 201-224. doi: 10.1007/BF00250671. Google Scholar [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. Google Scholar [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. Google Scholar [21] R. Rockafeller, Convex Analysis, Princeton University Press, USA, 1970. Google Scholar [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. Google Scholar [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. Google Scholar

show all references

##### References:
 [1] G. Aronsson, Extension of functions satisfying Lipschitz conditions, Arkiv För Matematik, 6 (1967), 551-561. doi: 10.1007/BF02591928. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [7] L. Evans and R. Gariepy, Measure Theory and Fine Propeties of Functions, CRC Press, Boca Raton, 1992. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [17] S. Koike, A Beginner's Guide to the Theory of Viscosity Solutions, MSJ Memoirs, Mathematical Society of Japan, 2004. Google Scholar [18] J. Lewis, Capacitory functions in convex rings, Archive for Rational Mechanics and Analysis, 66 (1977), 201-224. doi: 10.1007/BF00250671. Google Scholar [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. Google Scholar [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. Google Scholar [21] R. Rockafeller, Convex Analysis, Princeton University Press, USA, 1970. Google Scholar [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. Google Scholar [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. Google Scholar
The streamlines of $V_\infty$ when $\Omega$ is the square $-1<x_1<1,\,\,-1<x_2<1$
 [1] Yinbin Deng, Yi Li, Wei Shuai. Existence of solutions for a class of p-Laplacian type equation with critical growth and potential vanishing at infinity. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 683-699. doi: 10.3934/dcds.2016.36.683 [2] Giovanna Cerami, Riccardo Molle. On some Schrödinger equations with non regular potential at infinity. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 827-844. doi: 10.3934/dcds.2010.28.827 [3] Iryna Pankratova, Andrey Piatnitski. On the behaviour at infinity of solutions to stationary convection-diffusion equation in a cylinder. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 935-970. doi: 10.3934/dcdsb.2009.11.935 [4] Fang Liu. An inhomogeneous evolution equation involving the normalized infinity Laplacian with a transport term. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2395-2421. doi: 10.3934/cpaa.2018114 [5] Mohameden Ahmedou, Mohamed Ben Ayed, Marcello Lucia. On a resonant mean field type equation: A "critical point at Infinity" approach. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1789-1818. doi: 10.3934/dcds.2017075 [6] José Caicedo, Alfonso Castro, Arturo Sanjuán. Bifurcation at infinity for a semilinear wave equation with non-monotone nonlinearity. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1857-1865. doi: 10.3934/dcds.2017078 [7] Goro Akagi, Kazumasa Suzuki. On a certain degenerate parabolic equation associated with the infinity-laplacian. Conference Publications, 2007, 2007 (Special) : 18-27. doi: 10.3934/proc.2007.2007.18 [8] Luis Barreira, Claudia Valls. Topological conjugacies and behavior at infinity. Communications on Pure & Applied Analysis, 2014, 13 (2) : 687-701. doi: 10.3934/cpaa.2014.13.687 [9] Yinbin Deng, Wei Shuai. Positive solutions for quasilinear Schrödinger equations with critical growth and potential vanishing at infinity. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2273-2287. doi: 10.3934/cpaa.2014.13.2273 [10] Victor S. Kozyakin, Alexander M. Krasnosel’skii, Dmitrii I. Rachinskii. Arnold tongues for bifurcation from infinity. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 107-116. doi: 10.3934/dcdss.2008.1.107 [11] Victor Kozyakin, Alexander M. Krasnosel’skii, Dmitrii Rachinskii. Asymptotics of the Arnold tongues in problems at infinity. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 989-1011. doi: 10.3934/dcds.2008.20.989 [12] Shota Sato. Blow-up at space infinity of a solution with a moving singularity for a semilinear parabolic equation. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1225-1237. doi: 10.3934/cpaa.2011.10.1225 [13] Alexander Krasnosel'skii, Jean Mawhin. The index at infinity for some vector fields with oscillating nonlinearities. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 165-174. doi: 10.3934/dcds.2000.6.165 [14] Michel Chipot, Aleksandar Mojsic, Prosenjit Roy. On some variational problems set on domains tending to infinity. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3603-3621. doi: 10.3934/dcds.2016.36.3603 [15] Guillaume James, Dmitry Pelinovsky. Breather continuation from infinity in nonlinear oscillator chains. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1775-1799. doi: 10.3934/dcds.2012.32.1775 [16] Yukihiro Seki. A remark on blow-up at space infinity. Conference Publications, 2009, 2009 (Special) : 691-696. doi: 10.3934/proc.2009.2009.691 [17] Francesco Della Pietra, Ireneo Peral. Breaking of resonance for elliptic problems with strong degeneration at infinity. Communications on Pure & Applied Analysis, 2011, 10 (2) : 593-612. doi: 10.3934/cpaa.2011.10.593 [18] Begoña Alarcón, Víctor Guíñez, Carlos Gutierrez. Hopf bifurcation at infinity for planar vector fields. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 247-258. doi: 10.3934/dcds.2007.17.247 [19] A.M. Krasnosel'skii, Jean Mawhin. The index at infinity of some twice degenerate compact vector fields. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 207-216. doi: 10.3934/dcds.1995.1.207 [20] 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

2018 Impact Factor: 1.143