• Previous Article
    Some properties of minimizers of a variational problem involving the total variation functional
  • CPAA Home
  • This Issue
  • Next Article
    Nonuniqueness in vector-valued calculus of variations in $L^\infty$ and some Linear elliptic systems
January  2015, 14(1): 329-339. doi: 10.3934/cpaa.2015.14.329

First eigenfunctions of the 1-Laplacian are viscosity solutions

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.
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.

[1]

Boris Andreianov, Kenneth H. Karlsen, Nils H. Risebro. On vanishing viscosity approximation of conservation laws with discontinuous flux. Networks and Heterogeneous Media, 2010, 5 (3) : 617-633. doi: 10.3934/nhm.2010.5.617

[2]

Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Existence and applications to the level-set approach. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1047-1069. doi: 10.3934/dcds.2008.21.1047

[3]

Sarah Constantin, Robert S. Strichartz, Miles Wheeler. Analysis of the Laplacian and spectral operators on the Vicsek set. Communications on Pure and Applied Analysis, 2011, 10 (1) : 1-44. doi: 10.3934/cpaa.2011.10.1

[4]

Lluís Alsedà, Sylvie Ruette. On the set of periods of sigma maps of degree 1. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 4683-4734. doi: 10.3934/dcds.2015.35.4683

[5]

Diogo A. Gomes. Viscosity solution methods and the discrete Aubry-Mather problem. Discrete and Continuous Dynamical Systems, 2005, 13 (1) : 103-116. doi: 10.3934/dcds.2005.13.103

[6]

Yoshikazu Giga, Jürgen Saal. $L^1$ maximal regularity for the laplacian and applications. Conference Publications, 2011, 2011 (Special) : 495-504. doi: 10.3934/proc.2011.2011.495

[7]

David E. Bernholdt, Mark R. Cianciosa, Clement Etienam, David L. Green, Kody J. H. Law, Jin M. Park. Corrigendum to "Cluster, classify, regress: A general method for learning discontinuous functions [1]". Foundations of Data Science, 2020, 2 (1) : 81-81. doi: 10.3934/fods.2020005

[8]

C. R. Chen, S. J. Li. Semicontinuity of the solution set map to a set-valued weak vector variational inequality. Journal of Industrial and Management Optimization, 2007, 3 (3) : 519-528. doi: 10.3934/jimo.2007.3.519

[9]

Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1801-1816. doi: 10.3934/dcdsb.2017107

[10]

Xi Wang, Zuhan Liu, Ling Zhou. Asymptotic decay for the classical solution of the chemotaxis system with fractional Laplacian in high dimensions. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 4003-4020. doi: 10.3934/dcdsb.2018121

[11]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521

[12]

J. Huang, Marius Paicu. Decay estimates of global solution to 2D incompressible Navier-Stokes equations with variable viscosity. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4647-4669. doi: 10.3934/dcds.2014.34.4647

[13]

Fei Chen, Boling Guo, Xiaoping Zhai. Global solution to the 3-D inhomogeneous incompressible MHD system with discontinuous density. Kinetic and Related Models, 2019, 12 (1) : 37-58. doi: 10.3934/krm.2019002

[14]

Victor A. Kovtunenko, Anna V. Zubkova. Mathematical modeling of a discontinuous solution of the generalized Poisson-Nernst-Planck problem in a two-phase medium. Kinetic and Related Models, 2018, 11 (1) : 119-135. doi: 10.3934/krm.2018007

[15]

Guangwu Wang, Boling Guo. Global weak solution to the quantum Navier-Stokes-Landau-Lifshitz equations with density-dependent viscosity. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 6141-6166. doi: 10.3934/dcdsb.2019133

[16]

Wenjie Li, Lihong Huang, Jinchen Ji. Globally exponentially stable periodic solution in a general delayed predator-prey model under discontinuous prey control strategy. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2639-2664. doi: 10.3934/dcdsb.2020026

[17]

Mei Wang, Zilai Li, Zhenhua Guo. Global weak solution to 3D compressible flows with density-dependent viscosity and free boundary. Communications on Pure and Applied Analysis, 2017, 16 (1) : 1-24. doi: 10.3934/cpaa.2017001

[18]

Na An, Chaobao Huang, Xijun Yu. Error analysis of discontinuous Galerkin method for the time fractional KdV equation with weak singularity solution. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 321-334. doi: 10.3934/dcdsb.2019185

[19]

Baojun Bian, Shuntai Hu, Quan Yuan, Harry Zheng. Constrained viscosity solution to the HJB equation arising in perpetual American employee stock options pricing. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5413-5433. doi: 10.3934/dcds.2015.35.5413

[20]

Xiao Wu, Mingkang Ni. Solution of contrast structure type for a reaction-diffusion equation with discontinuous reactive term. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3249-3266. doi: 10.3934/dcdss.2020341

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (139)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]