- Volume 26 Issue 4
In this paper, we deal with the discrete p-Laplacian operators with a potential term having the smallest nonnegative eigenvalue. Such operators are classified as its smallest eigenvalue is positive or zero. We discuss differences between them such as an existence of solutions of p-Laplacian equations on networks and properties of the energy functional. Also, we give some examples of Poisson equations which suggest a difference between linear types and nonlinear types. Finally, we study characteristics of the set of a potential those involving operator has the smallest positive eigenvalue.
S.-Y. Chung and C. A. Berenstein,
$\omega$-harmonic functions and inverse conductivity problems on networks, SIAM J. Appl. Math. 65 (2005), no. 4, 1200-1226. https://doi.org/10.1137/S0036139903432743
- S.-Y. Chung and J. H. Kim, The p-Laplacian and the uniqueness of inverse problems on nonlinear networks, preprint.
- E. B. Curtis, D. Ingerman, and J. A. Morrow, Circular planar graphs and resistor networks, Linear Algebra Appl. 283 (1998), no. 1-3, 115-150. https://doi.org/10.1016/S0024-3795(98)10087-3
- E. B. Curtis, E. Mooers, and J. A. Morrow, Finding the conductors in circular networks from boundary measurements, RAIRO Model. Math. Anal. Numer. 28 (1994), no. 7, 781-814. https://doi.org/10.1051/m2an/1994280707811
- E. B. Curtis and J. A. Morrow, Determining the resistors in a network, SIAM J. Appl. Math. 50 (1990), no. 3, 918-930. https://doi.org/10.1137/0150055
- E. B. Curtis and J. A. Morrow, The Dirichlet to Neumann map for a resistor network, SIAM J. Appl. Math. 51 (1991), no. 4, 1011-1029. https://doi.org/10.1137/0151051
- D. Ingerman and J. A. Morrow, On a characterization of the kernel of the Dirichlet-to- Neumann map for a planar region, SIAM J. Math. Anal. 29 (1998), no. 1, 106-115. https://doi.org/10.1137/S0036141096300483
- J.-H. Park, J.-H. Kim, and S.-Y. Chung, The p-Schrodinger equations on finite networks, Publ. Res. Inst. Math. Sci. 45 (2009), 363-381. https://doi.org/10.2977/prims/1241553123