First eigenfunctions of the 1-Laplacian are viscosity solutions. (English) Zbl 1322.35106
In this very interesting paper, the authors consider a nonlinear eigenvalue problem arising from the 1-Laplacian in the space BV over a bounded domain with homogeneous Dirichlet boundary conditions. One of the main problems in the area is that the Euler-Lagrange equation arising from the variational formulation of the constrained eigenvalue problem is highly degenerate. Herein, the authors apply a suitable modification of the theory of viscosity solutions to this context in the space BV. The main result of this paper is that BV functions which are minimisers of the variational constrained eigenvalue problem are viscosity solutions of the equation. Hence, the authors prove necessity of the equation for the variational eigenvalue problem. However, as they show by examples, sufficiency is not true.
Reviewer: Nikos Katzourakis (Reading)
MSC:
| 35P30 | Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs |
| 35D40 | Viscosity solutions to PDEs |
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