Explicit examples of Lipschitz, one-homogeneous solutions of log-singular planar elliptic systems. (September 2015)

• Record Type: Journal Article
• Authors: Bevan, Jonathan J.
• Is Part Of: Nonlinear analysis. Volume 125(2015)
• Page Start: 659
• Page End: 680
• Publication Date: 2015-09
• Subjects: 49K20 -- 49N60; One-homogeneous -- Singular solutions -- Variational; 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.2015.06.012
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We give examples of systems of Partial Differential Equations that admit non-trivial, Lipschitz and one-homogeneous solutions in the form u ( R, θ ) = R g ( θ ), where ( R, θ ) are plane polar coordinates and g takes values in R m, m ≥ 2 . The systems are singular in the sense that they arise as the Euler–Lagrange equations of the functionals I ( u ) = ∫ B W ( x, ∇ u ( x ) ) d x, where D F W ( x, F ) behaves like 1 | x | as | x | → 0 and W satisfies an ellipticity condition. Such solutions cannot exist when | x | D F W ( x, F ) → 0 as | x | → 0, so the condition is optimal. The associated analysis exploits the well-known Fefferman–Stein duality (Fefferman and Stein, 1972). We also discuss conditions for the uniqueness of these one-homogeneous solutions and demonstrate that they are minimizers of certain variational functionals.