American Institute of Mathematical Sciences

Advanced
  • 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

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

[1]

Kien Trung Nguyen, Vo Nguyen Minh Hieu, Van Huy Pham. Inverse group 1-median problem on trees. Journal of Industrial & Management Optimization, 2021, 17 (1) : 221-232. doi: 10.3934/jimo.2019108
[2]

Evan Greif, Daniel Kaplan, Robert S. Strichartz, Samuel C. Wiese. Spectrum of the Laplacian on regular polyhedra. Communications on Pure & Applied Analysis, 2021, 20 (1) : 193-214. doi: 10.3934/cpaa.2020263
[3]

Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020169
[4]

Zexuan Liu, Zhiyuan Sun, Jerry Zhijian Yang. A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction. Electronic Research Archive, 2020, 28 (4) : 1487-1501. doi: 10.3934/era.2020078
[5]

Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, , () : -. doi: 10.3934/era.2020120
[6]

Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 537-552. doi: 10.3934/dcds.2020268
[7]

Chandra Shekhar, Amit Kumar, Shreekant Varshney, Sherif Ibrahim Ammar. $ \bf{M/G/1} $ fault-tolerant machining system with imperfection. Journal of Industrial & Management Optimization, 2021, 17 (1) : 1-28. doi: 10.3934/jimo.2019096
[8]

Yi An, Bo Li, Lei Wang, Chao Zhang, Xiaoli Zhou. Calibration of a 3D laser rangefinder and a camera based on optimization solution. Journal of Industrial & Management Optimization, 2021, 17 (1) : 427-445. doi: 10.3934/jimo.2019119
[9]

Yasmine Cherfaoui, Mustapha Moulaï. Biobjective optimization over the efficient set of multiobjective integer programming problem. Journal of Industrial & Management Optimization, 2021, 17 (1) : 117-131. doi: 10.3934/jimo.2019102
[10]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319
[11]

Dorothee Knees, Chiara Zanini. Existence of parameterized BV-solutions for rate-independent systems with discontinuous loads. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 121-149. doi: 10.3934/dcdss.2020332
[12]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264
[13]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261
[14]

Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020164
[15]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348
[16]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440
[17]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118
[18]

Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020073
[19]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070
[20]

Sihem Guerarra. Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 75-86. doi: 10.3934/naco.2020016

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (81)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences