A convexity problem for a semi-linear PDE. Issue 8 (11th June 2021) - El Hajj, Layan
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 } .