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August 2017, 10(4): 799-813. doi: 10.3934/dcdss.2017040

## On the geometry of the p-Laplacian operator

 1 Mathematisches Institut, Universität zu Köln, 50923 Köln, Germany 2 Fakultät Maschinenbau, TH Ingolstadt, Postfach 21 04 54,85019 Ingolstadt, Germany

* Corresponding author: Bernd Kawohl

Received  April 2016 Revised  August 2016 Published  April 2017

The
 $p$
-Laplacian operator
 $\Delta_pu={\rm div }\left(|\nabla u|^{p-2}\nabla u\right)$
is not uniformly elliptic for any
 $p\in(1,2)\cup(2,\infty)$
and degenerates even more when
 $p\to \infty$
or
 $p\to 1$
. In those two cases the Dirichlet and eigenvalue problems associated with the
 $p$
-Laplacian lead to intriguing geometric questions, because their limits for
 $p\to\infty$
or
 $p\to 1$
can be characterized by the geometry of
 $\Omega$
. In this little survey we recall some well-known results on eigenfunctions of the classical 2-Laplacian and elaborate on their extensions to general
 $p\in[1,\infty]$
. We report also on results concerning the normalized or game-theoretic
 $p$
-Laplacian
 $\Delta_p^Nu:=\tfrac{1}{p}|\nabla u|^{2-p}\Delta_pu=\tfrac{1}{p}\Delta_1^Nu+\tfrac{p-1}{p}\Delta_\infty^Nu$
and its parabolic counterpart
 $u_t-\Delta_p^N u=0$
. These equations are homogeneous of degree 1 and
 $\Delta_p^N$
is uniformly elliptic for any
 $p\in (1,\infty)$
. In this respect it is more benign than the
 $p$
-Laplacian, but it is not of divergence type.
Citation: Bernd Kawohl, Jiří Horák. On the geometry of the p-Laplacian operator. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 799-813. doi: 10.3934/dcdss.2017040
##### References:
 [1] T. N. Anoop, P. Drábek and S. Sarath, On the structure of the second eigenfunctions of the p-Laplacian on a ball, Proc. Amer. Math. Soc., 144 (2016), 2503-2512. doi: 10.1090/proc/12902. [2] S. N. Armstrong and C. K. Smart, A finite difference approach to the infinity Laplace equation and tug-of-war games, Trans. Amer. Math. Soc., 364 (2012), 595-636. doi: 10.1090/S0002-9947-2011-05289-X. [3] G. Aronsson, Extension of functions satisfying Lipschitz conditions, Ark. Mat., 6 (1967), 551-561. doi: 10.1007/BF02591928. [4] A. Attouch, M. Parviainen and E. Ruosteenoja, C1, α regularity for the normalized p-Poisson problem, preprint, arXiv: 1603.06391, to appear in J. Math. Pures Appl. [5] A. Banerjee and N. Garofalo, On the Dirichlet boundary value problem for the normalized p-Laplacian evolution, Commun. Pure Appl. Anal., 14 (2015), 1-21. doi: 10.3934/cpaa.2015.14.1. [6] T. Bhattacharya, E. DiBenedetto and J. Manfredi, Limits as p → ∞ of ∆pup = f and related extremal problems, Some topics in nonlinear PDEs (Turin, 1989), Rend. Sem. Mat. Univ. Politec. , Torino, Special Issue, (1991), 15-68. [7] I. Birindelli and F. Demengel, Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators, Comm. Pure Applied Anal., 6 (2007), 335-366. [8] I. Birindelli and F. Demengel, Regularity and uniqueness of the first eigenfunction for singular fully nonlinear operators, J. Differential Equations, 249 (2010), 1089-1110. doi: 10.1016/j.jde.2010.03.015. [9] L. Brasco, C. Nitsch and C. Trombetti, An inequality á la Szegö-Weinberger for the p-Laplacian on convex sets, Communications in Contemporary Mathematics, 18 (2016), 1550086, 23pp. doi: 10.1142/S0219199715500868. [10] J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, in Problems in Analysis, A Symposium in Honor of Salomon Bochner, (ed. R. C. Gunning), Princeton Univ. Press, (2015), 195-200. doi: 10.1515/9781400869312-013. [11] M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67. [12] G. Crasta and I. Fragalá, A C1 regularity result for the inhomogeneous normalized infinity Laplacian, Proc. Amer. Math. Soc., 144 (2016), 2547-2558. doi: 10.1090/proc/12916. [13] K. Does, An evolution equation involving the normalized p-Laplacian, Commun. Pure Appl. Anal., 10 (2011), 361-396. doi: 10.3934/cpaa.2011.10.361. [14] L. Esposito, B. Kawohl, C. Nitsch and C. Trombetti, The Neumann eigenvalue problem for the ∞-Laplacian, Rend. Lincei Mat.Appl., 26 (2015), 119-134. doi: 10.4171/RLM/697. [15] L. Esposito, V. Ferone, B. Kawohl, C. Nitsch and C. Trombetti, The longest shortest fence and sharp Poincaré Sobolev inequalities, Arch. Ration. Mech. Anal., 206 (2012), 821-851. doi: 10.1007/s00205-012-0545-0. [16] L. C. Evans and J. Spruck, Motion of level sets by mean curvature Ⅰ, Chapter: Fundamental Contributions to the Continuum Theory of Evolving Phase Interfaces in Solids, 33 (1991), 328-374. doi: 10.1007/978-3-642-59938-5_13. [17] H. Gajewski and K. Gärtner, Domain separation by means of sign changing eigenfunctions of p-Laplacians, Appl. Anal., 79 (2001), 483-501. doi: 10.1080/00036810108840974. [18] J. Horák, Numerical investigation of the smallest eigenvalues of the p-Laplace operator on planar domains, Electron. J. Differential Equations, 2011 (2011), 1-30. [19] R. Hynd, C. K. Smart and Y. Yu, Nonuniqueness of infinity ground states, Calc. Var. Partial Differential Equations, 48 (2013), 545-554. doi: 10.1007/s00526-012-0561-9. [20] 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. [21] T. Jin and L. Silvestre, Hölder gradient estimates for parabolic homogeneous p-Laplacian equations, preprint, arXiv: 1505.05525 doi: 10.1016/j.matpur.2016.10.010. [22] V. Julin and P. Juutinen, A new proof for the equivalence of weak and viscosity solutions for the p-Laplace equation, Comm. Partial Differential Equations, 37 (2012), 934-946. doi: 10.1080/03605302.2011.615878. [23] P. Juutinen, P. Lindqvist and J. Manfredi, The ∞-eigenvalue problem, Arch. Ration. Mech. Anal., 148 (1999), 89-105. [24] P. Juutinen, p-Harmonic approximation of functions of least gradient, Indiana Univ. Math. J., 54 (2005), 1015-1029. doi: 10.1512/iumj.2005.54.2658. [25] P. Juutinen and B. Kawohl, On the evolution governed by the infinity Laplacian, Math. Ann., 335 (2006), 819-851. doi: 10.1007/s00208-006-0766-3. [26] P. Juutinen, Principal eigenvalue of a very badly degenerate operator and applications, J. Differential Equations, 236 (2007), 532-550. doi: 10.1016/j.jde.2007.01.020. [27] P. Juutinen and P. Lindqvist, On the higher eigenvalues for the ∞-eigenvalue problem, Calc. Var. Partial Differential Equations, 23 (2005), 169-192. doi: 10.1007/s00526-004-0295-4. [28] B. Kawohl, On a family of torsional creep problems, J. Reine Angew. Math., 410 (1990), 1-22. [29] B. Kawohl and N. Kutev, Maximum and comparison principle for one-dimensional anisotropic diffusion, Math. Ann., 311 (1989), 107-123. doi: 10.1007/s002080050179. [30] B. Kawohl and V. Fridman, Isoperimetric estimates for the first eigenvalue of the p-Laplace operator and the Cheeger constant, Comment. Math. Univ. Carolinae, 44 (2003), 659-667. [31] B. Kawohl and H. Shahgholian, Gamma limits in some Bernoulli free boundary problems, Archiv D. Math., 84 (2005), 79-87. doi: 10.1007/s00013-004-1334-2. [32] B. Kawohl and Th. Lachand-Robert, Characterization of Cheeger sets for convex subsets of the plane, Pacific J.Math., 225 (2006), 103-118. doi: 10.2140/pjm.2006.225.103. [33] B. Kawohl and P. Lindqvist, Positive eigenfunctions for the p-Laplace operator revisited, Analysis, (Munich), 26 (2006), 539-544. doi: 10.1524/anly.2006.26.4.545. [34] B. Kawohl and N. Kutev, Global behaviour of solutions to a parabolic mean curvature equation, Differential Integral Equations, 8 (1995), 1923-1946. [35] B. Kawohl, Variational versus PDE-based Approaches in Mathematical Image Processing, CRM Proceedings and Lecture Notes, 44 (2008), 113-126. [36] B. Kawohl, Variations on the p-Laplacian in Nonlinear Elliptic Partial Differential Equations, (eds. Bonheure D., P. Takač et al.), Contemporary Mathematics, 540 (2011), 35-46. doi: 10.1090/conm/540/10657. [37] B. Kawohl, J. Manfredi and M. Parviainen, Solutions of nonlinear PDEs in the sense of averages, J. Math. Pures Appl., 97 (2012), 173-188. doi: 10.1016/j.matpur.2011.07.001. [38] B. Kawohl, S. Krömer and J. Kurtz, Radial eigenfunctions for the game-theoretic p-Laplacian on a ball, Differential and Integral Equations, 27 (2014), 659-670. [39] B. Kawohl and F. Schuricht, First eigenfunctions of the 1-Laplacian are viscosity solutions, Commun. Pure Appl. Anal., 14 (2015), 329-339. doi: 10.3934/cpaa.2015.14.329. [40] G. Lu and P. Wang, Inhomogeneous infinity Laplace equation, Advances in Mathematics, 217 (2008), 1838-1868. doi: 10.1016/j.aim.2007.11.020. [41] G. Lu and P. Wang, A uniqueness theorem for degenerate elliptic equations, Lecture Notes of Seminario Interdisciplinare di Matematica, Conference on Geometric Methods in PDEs, On the Occasion of 65th Birthday of Ermanno Lanconelli, Bologna, May 27-30,2008, (eds. Giovanna Citti, Annamaria Montanari, Andrea Pascucci, Sergio Polidoro), 207-222. [42] Th. Lachand-Robert and É. Oudet, Minimizing within convex bodies using a convex hull method, SIAM J. Optim., 16 (2005), 368-379. [43] P. J. Martínez-Aparicio, M. Pérez-Llanos and J. D. Rossi, The limit as p → ∞ for the eigenvalue problem of the 1-homogeneous p-Laplacian, Rev. Mat. Complut., 27 (2014), 241-258. [44] E. Parini, The second eigenvalue of the p-Laplacian as p goes to 1, Int. J. Diff. Eq. , (2010), Art. ID 984671, 23 pp. [45] E. Parini, An introduction to the Cheeger problem, Surv. Math. Appl., 6 (2011), 9-22. [46] J. D. Rossi and N. Saintier, On the first nontrivial eigenvalue of the ∞-Laplacian with Neumann boundary conditions, Houston J. Math., 42 (2016), 613-635. [47] R. P. Sperb, Maximum Principles and Their Applications, Academic Press, New York-London, 1981. [48] Y. Yu, Some properties of the ground states of the infinity Laplacian, Indiana Univ. Math. J., 56 (2007), 947-964. doi: 10.1512/iumj.2007.56.2935.

show all references

##### References:
 [1] T. N. Anoop, P. Drábek and S. Sarath, On the structure of the second eigenfunctions of the p-Laplacian on a ball, Proc. Amer. Math. Soc., 144 (2016), 2503-2512. doi: 10.1090/proc/12902. [2] S. N. Armstrong and C. K. Smart, A finite difference approach to the infinity Laplace equation and tug-of-war games, Trans. Amer. Math. Soc., 364 (2012), 595-636. doi: 10.1090/S0002-9947-2011-05289-X. [3] G. Aronsson, Extension of functions satisfying Lipschitz conditions, Ark. Mat., 6 (1967), 551-561. doi: 10.1007/BF02591928. [4] A. Attouch, M. Parviainen and E. Ruosteenoja, C1, α regularity for the normalized p-Poisson problem, preprint, arXiv: 1603.06391, to appear in J. Math. Pures Appl. [5] A. Banerjee and N. Garofalo, On the Dirichlet boundary value problem for the normalized p-Laplacian evolution, Commun. Pure Appl. Anal., 14 (2015), 1-21. doi: 10.3934/cpaa.2015.14.1. [6] T. Bhattacharya, E. DiBenedetto and J. Manfredi, Limits as p → ∞ of ∆pup = f and related extremal problems, Some topics in nonlinear PDEs (Turin, 1989), Rend. Sem. Mat. Univ. Politec. , Torino, Special Issue, (1991), 15-68. [7] I. Birindelli and F. Demengel, Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators, Comm. Pure Applied Anal., 6 (2007), 335-366. [8] I. Birindelli and F. Demengel, Regularity and uniqueness of the first eigenfunction for singular fully nonlinear operators, J. Differential Equations, 249 (2010), 1089-1110. doi: 10.1016/j.jde.2010.03.015. [9] L. Brasco, C. Nitsch and C. Trombetti, An inequality á la Szegö-Weinberger for the p-Laplacian on convex sets, Communications in Contemporary Mathematics, 18 (2016), 1550086, 23pp. doi: 10.1142/S0219199715500868. [10] J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, in Problems in Analysis, A Symposium in Honor of Salomon Bochner, (ed. R. C. Gunning), Princeton Univ. Press, (2015), 195-200. doi: 10.1515/9781400869312-013. [11] M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67. [12] G. Crasta and I. Fragalá, A C1 regularity result for the inhomogeneous normalized infinity Laplacian, Proc. Amer. Math. Soc., 144 (2016), 2547-2558. doi: 10.1090/proc/12916. [13] K. Does, An evolution equation involving the normalized p-Laplacian, Commun. Pure Appl. Anal., 10 (2011), 361-396. doi: 10.3934/cpaa.2011.10.361. [14] L. Esposito, B. Kawohl, C. Nitsch and C. Trombetti, The Neumann eigenvalue problem for the ∞-Laplacian, Rend. Lincei Mat.Appl., 26 (2015), 119-134. doi: 10.4171/RLM/697. [15] L. Esposito, V. Ferone, B. Kawohl, C. Nitsch and C. Trombetti, The longest shortest fence and sharp Poincaré Sobolev inequalities, Arch. Ration. Mech. Anal., 206 (2012), 821-851. doi: 10.1007/s00205-012-0545-0. [16] L. C. Evans and J. Spruck, Motion of level sets by mean curvature Ⅰ, Chapter: Fundamental Contributions to the Continuum Theory of Evolving Phase Interfaces in Solids, 33 (1991), 328-374. doi: 10.1007/978-3-642-59938-5_13. [17] H. Gajewski and K. Gärtner, Domain separation by means of sign changing eigenfunctions of p-Laplacians, Appl. Anal., 79 (2001), 483-501. doi: 10.1080/00036810108840974. [18] J. Horák, Numerical investigation of the smallest eigenvalues of the p-Laplace operator on planar domains, Electron. J. Differential Equations, 2011 (2011), 1-30. [19] R. Hynd, C. K. Smart and Y. Yu, Nonuniqueness of infinity ground states, Calc. Var. Partial Differential Equations, 48 (2013), 545-554. doi: 10.1007/s00526-012-0561-9. [20] 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. [21] T. Jin and L. Silvestre, Hölder gradient estimates for parabolic homogeneous p-Laplacian equations, preprint, arXiv: 1505.05525 doi: 10.1016/j.matpur.2016.10.010. [22] V. Julin and P. Juutinen, A new proof for the equivalence of weak and viscosity solutions for the p-Laplace equation, Comm. Partial Differential Equations, 37 (2012), 934-946. doi: 10.1080/03605302.2011.615878. [23] P. Juutinen, P. Lindqvist and J. Manfredi, The ∞-eigenvalue problem, Arch. Ration. Mech. Anal., 148 (1999), 89-105. [24] P. Juutinen, p-Harmonic approximation of functions of least gradient, Indiana Univ. Math. J., 54 (2005), 1015-1029. doi: 10.1512/iumj.2005.54.2658. [25] P. Juutinen and B. Kawohl, On the evolution governed by the infinity Laplacian, Math. Ann., 335 (2006), 819-851. doi: 10.1007/s00208-006-0766-3. [26] P. Juutinen, Principal eigenvalue of a very badly degenerate operator and applications, J. Differential Equations, 236 (2007), 532-550. doi: 10.1016/j.jde.2007.01.020. [27] P. Juutinen and P. Lindqvist, On the higher eigenvalues for the ∞-eigenvalue problem, Calc. Var. Partial Differential Equations, 23 (2005), 169-192. doi: 10.1007/s00526-004-0295-4. [28] B. Kawohl, On a family of torsional creep problems, J. Reine Angew. Math., 410 (1990), 1-22. [29] B. Kawohl and N. Kutev, Maximum and comparison principle for one-dimensional anisotropic diffusion, Math. Ann., 311 (1989), 107-123. doi: 10.1007/s002080050179. [30] B. Kawohl and V. Fridman, Isoperimetric estimates for the first eigenvalue of the p-Laplace operator and the Cheeger constant, Comment. Math. Univ. Carolinae, 44 (2003), 659-667. [31] B. Kawohl and H. Shahgholian, Gamma limits in some Bernoulli free boundary problems, Archiv D. Math., 84 (2005), 79-87. doi: 10.1007/s00013-004-1334-2. [32] B. Kawohl and Th. Lachand-Robert, Characterization of Cheeger sets for convex subsets of the plane, Pacific J.Math., 225 (2006), 103-118. doi: 10.2140/pjm.2006.225.103. [33] B. Kawohl and P. Lindqvist, Positive eigenfunctions for the p-Laplace operator revisited, Analysis, (Munich), 26 (2006), 539-544. doi: 10.1524/anly.2006.26.4.545. [34] B. Kawohl and N. Kutev, Global behaviour of solutions to a parabolic mean curvature equation, Differential Integral Equations, 8 (1995), 1923-1946. [35] B. Kawohl, Variational versus PDE-based Approaches in Mathematical Image Processing, CRM Proceedings and Lecture Notes, 44 (2008), 113-126. [36] B. Kawohl, Variations on the p-Laplacian in Nonlinear Elliptic Partial Differential Equations, (eds. Bonheure D., P. Takač et al.), Contemporary Mathematics, 540 (2011), 35-46. doi: 10.1090/conm/540/10657. [37] B. Kawohl, J. Manfredi and M. Parviainen, Solutions of nonlinear PDEs in the sense of averages, J. Math. Pures Appl., 97 (2012), 173-188. doi: 10.1016/j.matpur.2011.07.001. [38] B. Kawohl, S. Krömer and J. Kurtz, Radial eigenfunctions for the game-theoretic p-Laplacian on a ball, Differential and Integral Equations, 27 (2014), 659-670. [39] B. Kawohl and F. Schuricht, First eigenfunctions of the 1-Laplacian are viscosity solutions, Commun. Pure Appl. Anal., 14 (2015), 329-339. doi: 10.3934/cpaa.2015.14.329. [40] G. Lu and P. Wang, Inhomogeneous infinity Laplace equation, Advances in Mathematics, 217 (2008), 1838-1868. doi: 10.1016/j.aim.2007.11.020. [41] G. Lu and P. Wang, A uniqueness theorem for degenerate elliptic equations, Lecture Notes of Seminario Interdisciplinare di Matematica, Conference on Geometric Methods in PDEs, On the Occasion of 65th Birthday of Ermanno Lanconelli, Bologna, May 27-30,2008, (eds. Giovanna Citti, Annamaria Montanari, Andrea Pascucci, Sergio Polidoro), 207-222. [42] Th. Lachand-Robert and É. Oudet, Minimizing within convex bodies using a convex hull method, SIAM J. Optim., 16 (2005), 368-379. [43] P. J. Martínez-Aparicio, M. Pérez-Llanos and J. D. Rossi, The limit as p → ∞ for the eigenvalue problem of the 1-homogeneous p-Laplacian, Rev. Mat. Complut., 27 (2014), 241-258. [44] E. Parini, The second eigenvalue of the p-Laplacian as p goes to 1, Int. J. Diff. Eq. , (2010), Art. ID 984671, 23 pp. [45] E. Parini, An introduction to the Cheeger problem, Surv. Math. Appl., 6 (2011), 9-22. [46] J. D. Rossi and N. Saintier, On the first nontrivial eigenvalue of the ∞-Laplacian with Neumann boundary conditions, Houston J. Math., 42 (2016), 613-635. [47] R. P. Sperb, Maximum Principles and Their Applications, Academic Press, New York-London, 1981. [48] Y. Yu, Some properties of the ground states of the infinity Laplacian, Indiana Univ. Math. J., 56 (2007), 947-964. doi: 10.1512/iumj.2007.56.2935.
The positive viscosity solution of (4.4)
Conceivable nodal lines of the second eigenfunction for $p=\infty$ in the disc
Illustration of (5.4) and (5.5)
Numerical simulation of $u_{15}$ and side view in diagonal direction
Numerical simulation of $u_p$: normalized values along half of the diagonal for $p=2, 3, 4, 6, 8, 10, 15$ (left), and for $p=15$ compared to the line $y=x$ (right)
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