A convexity problem for a semi-linear PDE. Issue 8 (11th June 2021)

• Record Type: Journal Article
• Authors: El Hajj, Layan
• Is Part Of: Applicable analysis. Volume 100:Issue 8(2021)
• Page Start: 1816
• Page End: 1826
• Publication Date: 2021-06-11
• Subjects: Convexity -- star-shapedness -- uniqueness -- free boundary; 35R35; Mathematical analysis -- Periodicals; 515
• DOI: 10.1080/00036811.2019.1667981
• Languages: English
• ISSNs: 0003-6811

Abstract

ABSTRACT: In this paper, we prove convexity of super-level sets of a semi-linear PDE with a non-monotone right-hand side, and with a free boundary (1) Δ u = χ { 0 < u < 1 } i n R n ∖ D, u = 2 o n ∂ D . Here D is assumed to be convex, and n ≥ 2 . The main difficulty of this problem is that the right-hand side is non-monotone and no a-priori regularity is known about the boundary ∂ { u > 0 } .