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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 |
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.
Mathematics Subject Classification:
Primary: 49N60, 35J15, 35J60, 35J65, 35J70.
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
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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. |
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