Potential theory for the sum of the 1-Laplacian and p-Laplacian. (January 2015)

• Record Type: Journal Article
• Authors: Krügel, Florian
• Is Part Of: Nonlinear analysis. Volume 112(2015)
• Page Start: 165
• Page End: 180
• Publication Date: 2015-01
• Subjects: 1-Laplacian -- Nonlinear potential theory -- Comparison principle -- Cheeger constant; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2014.08.019
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: The variational problem of minimizing the functional u ↦ ∫ Ω | D u | + 1 p ∫ Ω | D u | p − ∫ Ω a u on a domain Ω ⊂ R n under zero boundary values, which appears for example in the theory of Bingham fluids, shows interesting phenomena such as the formation of a subset of Ω with positive measure where the minimizer is constant (a "plateau"). The corresponding Euler–Lagrange equation should be − Δ 1 u − Δ p u = a, but the 1-Laplacian is not defined in points where D u = 0 . We show that it is nevertheless possible to define subsolutions and supersolutions for this equation. We show that the comparison principle is valid, and we begin developing a potential theory analogous to what is known about the p -Laplacian. In addition we apply those notions to rotationally invariant solutions, and we discuss properties of the plateau and a possible connection to the theory.