-
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
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 |
References:
[1] |
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems,, Oxford University Press, (2000).
|
[2] |
G. Anzellotti, Pairings between measures and bounded functions and compensated compactness,, \emph{Ann. Mat. Pura Appl.}, 135 (1983), 193.
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,, \emph{Bull. Amer. Math. Soc.}, 27 (1992), 1.
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,, \emph{Calc. Var.}, 36 (2009), 591.
doi: 10.1007/s00526-009-0246-1. |
[5] |
L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, CRC Press, (1992).
|
[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. |
[7] |
E. Giusti, On the equation of surfaces of prescribed mean curvature, existence and uniqueness without boundary conditions,, \emph{Invent. Math.}, 46 (1978), 111.
|
[8] |
J. Horák, Numerical investigation of the smallest eigenvalues of the p-Laplace operator on planar domains,, \emph{Electron. J. Differential Equations}, 132 (2011).
|
[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. |
[10] |
B. Kawohl and N. Kutev, Global behaviour of solutions to a parabolic mean curvature equation,, \emph{Diff. int. Eqs.}, 8 (1995), 1923.
|
[11] |
B. Kawohl and N. Kutev, Viscosity solutions for degenerate and nonmonotone elliptic equations,, in \emph{Applied Nonlinear Analysis} (B. da Vega, (1999), 185.
|
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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).
|
[2] |
G. Anzellotti, Pairings between measures and bounded functions and compensated compactness,, \emph{Ann. Mat. Pura Appl.}, 135 (1983), 193.
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,, \emph{Bull. Amer. Math. Soc.}, 27 (1992), 1.
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,, \emph{Calc. Var.}, 36 (2009), 591.
doi: 10.1007/s00526-009-0246-1. |
[5] |
L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, CRC Press, (1992).
|
[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. |
[7] |
E. Giusti, On the equation of surfaces of prescribed mean curvature, existence and uniqueness without boundary conditions,, \emph{Invent. Math.}, 46 (1978), 111.
|
[8] |
J. Horák, Numerical investigation of the smallest eigenvalues of the p-Laplace operator on planar domains,, \emph{Electron. J. Differential Equations}, 132 (2011).
|
[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. |
[10] |
B. Kawohl and N. Kutev, Global behaviour of solutions to a parabolic mean curvature equation,, \emph{Diff. int. Eqs.}, 8 (1995), 1923.
|
[11] |
B. Kawohl and N. Kutev, Viscosity solutions for degenerate and nonmonotone elliptic equations,, in \emph{Applied Nonlinear Analysis} (B. da Vega, (1999), 185.
|
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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
Tools
Metrics
Other articles
by authors
[Back to Top]