Some special aspects related to the 1-Laplace operator

Abstract

The eigenvalue problem for the 1-Laplace operator, which is considered to be the EulerLagrange equation for an associated variational problem in BV (Ω), is formally given byHowever the undetermined expressions Du/|Du| and u/|u| have to be replaced with an appropriate vector field z related to u and a measurable selection s of the set-valued sign function Sgn (u(·)), respectively, such that −Div z = λs. For the special case of a square Ω ⊂ R 2 and the known minimizer u = χ C of the related variational problem, the paper presents a somehow explicit construction of corresponding vector fields z. In particular it is shown that, for a fixed selection s, the field z is not determined by means of the differential equation and its coupling conditions with u, but there are infinitely many continuous vector fields z that even differ on the boundary of Ω.

Perturbation results involving the 1-Laplace operator

“…was verified in Milbers & Schuricht [16] by minimax methods. While in [16] the classes S α k are defined by means of category as topological index, we know from Littig & Schuricht [14] that these eigenvalues λ k,1 coincide with that using S α k := {S ⊆ L 1 (Ω) compact, symmetric ; G 1 = 1 on S, gen L 1 S ≥ k} with genus gen L 1 S as topological index in (1.11). Investigating bifurcation for the formal problems (1.6) and (1.7) we are confronted with the question how to define solutions.…”

“…was verified in Milbers & Schuricht [16] by minimax methods. While in [16] the classes S α k are defined by means of category as topological index, we know from Littig & Schuricht [14] that these eigenvalues λ k,1 coincide with that using…”

The Cheeger N-problem in terms of BV functions

“…Note that neither E 1 nor G 1 are differentiable, such that the concept of the weak slope is applied to define critical points in that context (cf. [KS07], [MS10], [Cha09], [MS12]). It has been shown in [KS07] that eigenfunctions satisfy the following Euler-Lagrange equation.…”