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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. AnoopP. 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. Google Scholar

[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. Google Scholar

[3]

G. Aronsson, Extension of functions satisfying Lipschitz conditions, Ark. Mat., 6 (1967), 551-561. doi: 10.1007/BF02591928. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

L. BrascoC. 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. Google Scholar

[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. Google Scholar

[11]

M. G. CrandallH. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

L. EspositoB. KawohlC. Nitsch and C. Trombetti, The Neumann eigenvalue problem for the ∞-Laplacian, Rend. Lincei Mat.Appl., 26 (2015), 119-134. doi: 10.4171/RLM/697. Google Scholar

[15]

L. EspositoV. FeroneB. KawohlC. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

R. HyndC. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[23]

P. JuutinenP. Lindqvist and J. Manfredi, The ∞-eigenvalue problem, Arch. Ration. Mech. Anal., 148 (1999), 89-105. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[28]

B. Kawohl, On a family of torsional creep problems, J. Reine Angew. Math., 410 (1990), 1-22. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[34]

B. Kawohl and N. Kutev, Global behaviour of solutions to a parabolic mean curvature equation, Differential Integral Equations, 8 (1995), 1923-1946. Google Scholar

[35]

B. Kawohl, Variational versus PDE-based Approaches in Mathematical Image Processing, CRM Proceedings and Lecture Notes, 44 (2008), 113-126. Google Scholar

[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. Google Scholar

[37]

B. KawohlJ. 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. Google Scholar

[38]

B. KawohlS. Krömer and J. Kurtz, Radial eigenfunctions for the game-theoretic p-Laplacian on a ball, Differential and Integral Equations, 27 (2014), 659-670. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[42]

Th. Lachand-Robert and É. Oudet, Minimizing within convex bodies using a convex hull method, SIAM J. Optim., 16 (2005), 368-379. Google Scholar

[43]

P. J. Martínez-AparicioM. 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. Google Scholar

[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.Google Scholar

[45]

E. Parini, An introduction to the Cheeger problem, Surv. Math. Appl., 6 (2011), 9-22. Google Scholar

[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. Google Scholar

[47]

R. P. Sperb, Maximum Principles and Their Applications, Academic Press, New York-London, 1981.Google Scholar

[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. Google Scholar

show all references

References:
[1]

T. N. AnoopP. 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. Google Scholar

[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. Google Scholar

[3]

G. Aronsson, Extension of functions satisfying Lipschitz conditions, Ark. Mat., 6 (1967), 551-561. doi: 10.1007/BF02591928. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

L. BrascoC. 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. Google Scholar

[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. Google Scholar

[11]

M. G. CrandallH. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

L. EspositoB. KawohlC. Nitsch and C. Trombetti, The Neumann eigenvalue problem for the ∞-Laplacian, Rend. Lincei Mat.Appl., 26 (2015), 119-134. doi: 10.4171/RLM/697. Google Scholar

[15]

L. EspositoV. FeroneB. KawohlC. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

R. HyndC. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[23]

P. JuutinenP. Lindqvist and J. Manfredi, The ∞-eigenvalue problem, Arch. Ration. Mech. Anal., 148 (1999), 89-105. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[28]

B. Kawohl, On a family of torsional creep problems, J. Reine Angew. Math., 410 (1990), 1-22. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[34]

B. Kawohl and N. Kutev, Global behaviour of solutions to a parabolic mean curvature equation, Differential Integral Equations, 8 (1995), 1923-1946. Google Scholar

[35]

B. Kawohl, Variational versus PDE-based Approaches in Mathematical Image Processing, CRM Proceedings and Lecture Notes, 44 (2008), 113-126. Google Scholar

[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. Google Scholar

[37]

B. KawohlJ. 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. Google Scholar

[38]

B. KawohlS. Krömer and J. Kurtz, Radial eigenfunctions for the game-theoretic p-Laplacian on a ball, Differential and Integral Equations, 27 (2014), 659-670. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[42]

Th. Lachand-Robert and É. Oudet, Minimizing within convex bodies using a convex hull method, SIAM J. Optim., 16 (2005), 368-379. Google Scholar

[43]

P. J. Martínez-AparicioM. 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. Google Scholar

[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.Google Scholar

[45]

E. Parini, An introduction to the Cheeger problem, Surv. Math. Appl., 6 (2011), 9-22. Google Scholar

[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. Google Scholar

[47]

R. P. Sperb, Maximum Principles and Their Applications, Academic Press, New York-London, 1981.Google Scholar

[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. Google Scholar

Figure 1.  The positive viscosity solution of (4.4)
Figure 2.  Conceivable nodal lines of the second eigenfunction for $p=\infty$ in the disc
Figure 3.  Illustration of (5.4) and (5.5)
Figure 4.  Numerical simulation of $u_{15}$ and side view in diagonal direction
Figure 5.  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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