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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 & 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, (2000).   Google Scholar

[2]

G. Anzellotti, Pairings between measures and bounded functions and compensated compactness,, \emph{Ann. Mat. Pura Appl.}, 135 (1983), 193.  doi: 10.1007/BF01781073.  Google Scholar

[3]

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

[4]

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

[5]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, CRC Press, (1992).   Google Scholar

[6]

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

[7]

E. Giusti, On the equation of surfaces of prescribed mean curvature, existence and uniqueness without boundary conditions,, \emph{Invent. Math.}, 46 (1978), 111.   Google Scholar

[8]

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

[9]

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

[10]

B. Kawohl and N. Kutev, Global behaviour of solutions to a parabolic mean curvature equation,, \emph{Diff. int. Eqs.}, 8 (1995), 1923.   Google Scholar

[11]

B. Kawohl and N. Kutev, Viscosity solutions for degenerate and nonmonotone elliptic equations,, in \emph{Applied Nonlinear Analysis} (B. da Vega, (1999), 185.   Google Scholar

[12]

B. Kawohl and V. Fridman, Isoperimetric estimates for the first eigenvalue of the $p$-Laplace operator and the Cheeger constant,, \emph{Comment. Math. Univ. Carolinae}, 44 (2003), 659.   Google Scholar

[13]

B. Kawohl and Th. Lachand-Robert, Characterization of Cheeger sets for convex subsets of the plane,, \emph{Pacific J. Math.}, 225 (2006), 103.  doi: 10.2140/pjm.2006.225.103.  Google Scholar

[14]

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

[15]

B. Kawohl, Variations on the $p$-Laplacian,, in \emph{Nonlinear Elliptic Partial Differential Equations} (D. Bonheure, 540 (2011), 35.  doi: 10.1090/conm/540/10657.  Google Scholar

[16]

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

[17]

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

[18]

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

[19]

E. Parini, An Introduction to the Cheeger problem,, \emph{Surveys in Mathematics and its Applications}, 6 (2011), 9.   Google Scholar

show all references

References:
[1]

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

[2]

G. Anzellotti, Pairings between measures and bounded functions and compensated compactness,, \emph{Ann. Mat. Pura Appl.}, 135 (1983), 193.  doi: 10.1007/BF01781073.  Google Scholar

[3]

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

[4]

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

[5]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, CRC Press, (1992).   Google Scholar

[6]

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

[7]

E. Giusti, On the equation of surfaces of prescribed mean curvature, existence and uniqueness without boundary conditions,, \emph{Invent. Math.}, 46 (1978), 111.   Google Scholar

[8]

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

[9]

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

[10]

B. Kawohl and N. Kutev, Global behaviour of solutions to a parabolic mean curvature equation,, \emph{Diff. int. Eqs.}, 8 (1995), 1923.   Google Scholar

[11]

B. Kawohl and N. Kutev, Viscosity solutions for degenerate and nonmonotone elliptic equations,, in \emph{Applied Nonlinear Analysis} (B. da Vega, (1999), 185.   Google Scholar

[12]

B. Kawohl and V. Fridman, Isoperimetric estimates for the first eigenvalue of the $p$-Laplace operator and the Cheeger constant,, \emph{Comment. Math. Univ. Carolinae}, 44 (2003), 659.   Google Scholar

[13]

B. Kawohl and Th. Lachand-Robert, Characterization of Cheeger sets for convex subsets of the plane,, \emph{Pacific J. Math.}, 225 (2006), 103.  doi: 10.2140/pjm.2006.225.103.  Google Scholar

[14]

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

[15]

B. Kawohl, Variations on the $p$-Laplacian,, in \emph{Nonlinear Elliptic Partial Differential Equations} (D. Bonheure, 540 (2011), 35.  doi: 10.1090/conm/540/10657.  Google Scholar

[16]

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

[17]

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

[18]

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

[19]

E. Parini, An Introduction to the Cheeger problem,, \emph{Surveys in Mathematics and its Applications}, 6 (2011), 9.   Google Scholar

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