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January  2015, 14(1): 329-339. doi: 10.3934/cpaa.2015.14.329

First eigenfunctions of the 1-Laplacian are viscosity solutions

Bernd Kawohl 1, and  Friedemann Schuricht 2,

1. 

Mathematisches Institut, Universität zu Köln, 50923 Köln
2. 

Fachrichtung Mathematik, Technische Universität Dresden,01062 Dresden, Germany

Received  February 2014 Revised  March 2014 Published  September 2014

We address the question if eigenfunctions of the 1-Laplacian, which are obtained through a variational argument, are also viscosity solutions of the associated strongly degenerate formal Euler equation. The answer is positive, but examples show also that there are many more viscosity solutions than expected.
Keywords: Discontinuous viscosity solution, Cheeger set, 1-Laplacian.
Mathematics Subject Classification: Primary: 35D40, 35J70, 49G05, 49A5.
Citation: Bernd Kawohl, Friedemann Schuricht. First eigenfunctions of the 1-Laplacian are viscosity solutions. Communications on Pure and Applied Analysis, 2015, 14 (1) : 329-339. doi: 10.3934/cpaa.2015.14.329
References:
[1]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, Oxford, 2000.

[2]

G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl., 135 (1983), 193-318. doi: 10.1007/BF01781073.

[3]

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

[4]

M. Degiovanni and P. Margone, Linking solutions for quasilinear equations at critical growth involving the 1-Laplace operator, Calc. Var., 36 (2009), 591-609. doi: 10.1007/s00526-009-0246-1.

[5]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992.

[6]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed. Springer-Verlag, Heidelberg, 1983. doi: 10.1007/978-3-642-61798-0.

[7]

E. Giusti, On the equation of surfaces of prescribed mean curvature, existence and uniqueness without boundary conditions, Invent. Math., 46 (1978), 111-137.

[8]

J. Horák, Numerical investigation of the smallest eigenvalues of the p-Laplace operator on planar domains, Electron. J. Differential Equations, 132 (2011).

[9]

P. Juutinen, P. Lindqvist and J. J. Manfredi, The $\infty$-Eigenvalue Problem, Arch Ration Mech. Anal., 148 (1999), 89-105. doi: 10.1007/s002050050157.

[10]

B. Kawohl and N. Kutev, Global behaviour of solutions to a parabolic mean curvature equation, Diff. int. Eqs., 8 (1995), 1923-1946.

[11]

B. Kawohl and N. Kutev, Viscosity solutions for degenerate and nonmonotone elliptic equations, in Applied Nonlinear Analysis (B. da Vega, A. Sequeira and J. Videman eds), Plenum Press, New York u. London, (1999), 185-210.

[12]

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.

[13]

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.

[14]

B. Kawohl and F. Schuricht, Dirichlet problems for the 1-Laplace operator, including the eigenvalue problem, Communications in Contemporary Mathematics, 9 (2007), 515-543. doi: 10.1142/S0219199707002514.

[15]

B. Kawohl, Variations on the $p$-Laplacian, in Nonlinear Elliptic Partial Differential Equations (D. Bonheure, P. Takač et al. eds), Contemporary Mathematics, 540 (2011), 35-46. doi: 10.1090/conm/540/10657.

[16]

Z. Milbers and F. Schuricht, Existence of a sequence of eigensolutions for the 1-Laplace operator, J. Lond. Math. Soc., 82 (2010), 74-88. doi: 10.1112/jlms/jdq012.

[17]

Z. Milbers and F. Schuricht, Some special aspects related to the 1-Laplace operator, Adv. Calc. Var., 4 (2011), 101-126. doi: 10.1515/ACV.2010.021.

[18]

Z. Milbers and F. Schuricht, Necessary condition for eigensolutions of the 1-Laplace operator by means of inner variations, Math. Ann., 356 (2013), 147-177.

[19]

E. Parini, An Introduction to the Cheeger problem, Surveys in Mathematics and its Applications, 6 (2011), 9-22.

show all references

References:
[1]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, Oxford, 2000.

[2]

G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl., 135 (1983), 193-318. doi: 10.1007/BF01781073.

[3]

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

[4]

M. Degiovanni and P. Margone, Linking solutions for quasilinear equations at critical growth involving the 1-Laplace operator, Calc. Var., 36 (2009), 591-609. doi: 10.1007/s00526-009-0246-1.

[5]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992.

[6]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed. Springer-Verlag, Heidelberg, 1983. doi: 10.1007/978-3-642-61798-0.

[7]

E. Giusti, On the equation of surfaces of prescribed mean curvature, existence and uniqueness without boundary conditions, Invent. Math., 46 (1978), 111-137.

[8]

J. Horák, Numerical investigation of the smallest eigenvalues of the p-Laplace operator on planar domains, Electron. J. Differential Equations, 132 (2011).

[9]

P. Juutinen, P. Lindqvist and J. J. Manfredi, The $\infty$-Eigenvalue Problem, Arch Ration Mech. Anal., 148 (1999), 89-105. doi: 10.1007/s002050050157.

[10]

B. Kawohl and N. Kutev, Global behaviour of solutions to a parabolic mean curvature equation, Diff. int. Eqs., 8 (1995), 1923-1946.

[11]

B. Kawohl and N. Kutev, Viscosity solutions for degenerate and nonmonotone elliptic equations, in Applied Nonlinear Analysis (B. da Vega, A. Sequeira and J. Videman eds), Plenum Press, New York u. London, (1999), 185-210.

[12]

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.

[13]

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.

[14]

B. Kawohl and F. Schuricht, Dirichlet problems for the 1-Laplace operator, including the eigenvalue problem, Communications in Contemporary Mathematics, 9 (2007), 515-543. doi: 10.1142/S0219199707002514.

[15]

B. Kawohl, Variations on the $p$-Laplacian, in Nonlinear Elliptic Partial Differential Equations (D. Bonheure, P. Takač et al. eds), Contemporary Mathematics, 540 (2011), 35-46. doi: 10.1090/conm/540/10657.

[16]

Z. Milbers and F. Schuricht, Existence of a sequence of eigensolutions for the 1-Laplace operator, J. Lond. Math. Soc., 82 (2010), 74-88. doi: 10.1112/jlms/jdq012.

[17]

Z. Milbers and F. Schuricht, Some special aspects related to the 1-Laplace operator, Adv. Calc. Var., 4 (2011), 101-126. doi: 10.1515/ACV.2010.021.

[18]

Z. Milbers and F. Schuricht, Necessary condition for eigensolutions of the 1-Laplace operator by means of inner variations, Math. Ann., 356 (2013), 147-177.

[19]

E. Parini, An Introduction to the Cheeger problem, Surveys in Mathematics and its Applications, 6 (2011), 9-22.

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