id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-625	Mebrate, Benyam; Porru, Giovanni	Convexity of solutions to elliptic PDE's	2024	11	.pdf	application/pdf	4607	322	84	To show this, we shall prove that the corresponding concavity function c(x, y) satisfies c(x, y) ≥ 0 in Ω×Ω. Let us show first that c(x, y) cannot have a minimum at a point (x, y) with y ∈ ∂Ω. Indeed, suppose a minimum occurs at a point (x, y) with y ∈ ∂Ω. If we take the derivative of c(x, y) with respect to ν (external normal) and compute it for x ∈ Ω and y ∈ ∂Ω, we find cν(x, y) = a, u(y) = b, and choosing r = φ1/2(a) and s = φ1/2(b), we find that Lc(x, y) > 1 2 ( φ1/2(a) + φ1/2(b) )2 f (a+ b 2 ) − φ(a)f(a)− φ(b)f(b) + k2|Du(z)|2 [1 2 ( φ1/2(a) + φ1/2(b) )2 g (a+ b 2 )	cache/ejde-625.pdf	txt/ejde-625.txt