Doubly nonlinear equations for the 1-Laplacian

• Published in: Journal of evolution equations Vol. 23; no. 4
• Main Authors: Mazón, J. M., Molino, A., Toledo, J.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.12.2023; Springer Nature B.V
• Subjects: Analysis; Mathematical analysis; Mathematics; Mathematics and Statistics; Neumann problem; Nonlinear equations; Functions of bounded variations; 35D35; 47H20; Doubly nonlinear equations; 35D30; 1-Laplacian; Nonlinear Semigroup theory; 35K65
• ISSN: 1424-3199; 1424-3202
• DOI: 10.1007/s00028-023-00917-8

This paper is concerned with the Neumann problem for a class of doubly nonlinear equations for the 1-Laplacian, ∂ v ∂ t - Δ 1 u ∋ 0 in ( 0 , ∞ ) × Ω , v ∈ γ ( u ) , and initial data in  L 1 ( Ω ) , where Ω is a bounded smooth domain in  R N and  γ is a maximal monotone graph in R × R . We prove that, under certain assumptions on the graph  γ , there is existence and uniqueness of solutions. Moreover, we proof that these solutions coincide with the ones of the Neumann problem for the total variational flow. We show that such assumptions are necessary.