The fractional unstable obstacle problem. (April 2020)

• Record Type: Journal Article
• Authors: Allen, Mark; Smit Vega Garcia, Mariana
• Is Part Of: Nonlinear analysis. Volume 193(2020)
• Publication Date: 2020-04
• Subjects: Obstacle problem -- Unstable -- Free boundary; 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.2019.02.012
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study a model for combustion on a boundary. Specifically, we study certain generalized solutions of the equation ( − Δ ) s u = χ { u > c } for 0 < s < 1 and an arbitrary constant c . Our main object of study is the free boundary ∂ { u > c } . We study the behavior of the free boundary and prove an upper bound for the Hausdorff dimension of the singular set. We also show that when s ≤ 1 ∕ 2 certain symmetric solutions are stable; however, when s > 1 ∕ 2 these solutions are not stable and therefore not minimizers of the corresponding functional.