# American Institute of Mathematical Sciences

• 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 & 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
 [1] Boris Andreianov, Kenneth H. Karlsen, Nils H. Risebro. On vanishing viscosity approximation of conservation laws with discontinuous flux. Networks & 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 & Continuous Dynamical Systems - A, 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 & 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 & Continuous Dynamical Systems - A, 2015, 35 (10) : 4683-4734. doi: 10.3934/dcds.2015.35.4683 [5] 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 [6] Diogo A. Gomes. Viscosity solution methods and the discrete Aubry-Mather problem. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 103-116. doi: 10.3934/dcds.2005.13.103 [7] C. R. Chen, S. J. Li. Semicontinuity of the solution set map to a set-valued weak vector variational inequality. Journal of Industrial & Management Optimization, 2007, 3 (3) : 519-528. doi: 10.3934/jimo.2007.3.519 [8] Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1801-1816. doi: 10.3934/dcdsb.2017107 [9] Xi Wang, Zuhan Liu, Ling Zhou. Asymptotic decay for the classical solution of the chemotaxis system with fractional Laplacian in high dimensions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 4003-4020. doi: 10.3934/dcdsb.2018121 [10] Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521 [11] J. Huang, Marius Paicu. Decay estimates of global solution to 2D incompressible Navier-Stokes equations with variable viscosity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4647-4669. doi: 10.3934/dcds.2014.34.4647 [12] Mei Wang, Zilai Li, Zhenhua Guo. Global weak solution to 3D compressible flows with density-dependent viscosity and free boundary. Communications on Pure & Applied Analysis, 2017, 16 (1) : 1-24. doi: 10.3934/cpaa.2017001 [13] Baojun Bian, Shuntai Hu, Quan Yuan, Harry Zheng. Constrained viscosity solution to the HJB equation arising in perpetual American employee stock options pricing. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5413-5433. doi: 10.3934/dcds.2015.35.5413 [14] Guangwu Wang, Boling Guo. Global weak solution to the quantum Navier-Stokes-Landau-Lifshitz equations with density-dependent viscosity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6141-6166. doi: 10.3934/dcdsb.2019133 [15] Dung Le. On the regular set of BMO weak solutions to $p$-Laplacian strongly coupled nonregular elliptic systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3245-3265. doi: 10.3934/dcdsb.2014.19.3245 [16] Fei Chen, Boling Guo, Xiaoping Zhai. Global solution to the 3-D inhomogeneous incompressible MHD system with discontinuous density. Kinetic & Related Models, 2019, 12 (1) : 37-58. doi: 10.3934/krm.2019002 [17] 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 & Related Models, 2018, 11 (1) : 119-135. doi: 10.3934/krm.2018007 [18] Na An, Chaobao Huang, Xijun Yu. Error analysis of discontinuous Galerkin method for the time fractional KdV equation with weak singularity solution. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 321-334. doi: 10.3934/dcdsb.2019185 [19] Quansen Jiu, Zhouping Xin. The Cauchy problem for 1D compressible flows with density-dependent viscosity coefficients. Kinetic & Related Models, 2008, 1 (2) : 313-330. doi: 10.3934/krm.2008.1.313 [20] Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 763-800. doi: 10.3934/dcds.2008.21.763

2018 Impact Factor: 0.925