The fractional unstable obstacle problem

2018/11/29 by Allen, Mark, Garcia, Mariana Smit Vega
#34A08 #35R35 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1811.12497

Abstract

We study a model for combustion on a boundary. Specifically, we study certain generalized solutions of the equation (-Δ)^s u = χ_\ugt;c\ for 0c\. 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.

Related

• The Singular Set of Higher Dimensional Unstable Obstacle Type Problems
• Structure and regularity of the singular set in the obstacle problem for the fractional Laplacian
• Fractional optimal maximization problem and the unstable fractional obstacle problem
• The local structure of the free boundary in the fractional obstacle problem
• The two-phase fractional obstacle problem