Convex Configurations for Solutions to Semilinear Elliptic Problems in Convex Rings

A free boundary problem for a discontinuous semilinear elliptic equations in convex ring

“…For many generalizations in the literature, we refer to [1], [15], [17], [19] and [24] for further results.…”

“…This phenomenon arise in several different context like fluid dynamics, potential flow in fluid mechanics, combustion theory. See for instance [1], [2], [11], [15], [20], [21], [22] and the references therein.…”

Convexity of level sets for elliptic problems in convex domains or convex rings: Two counterexamples

“…Caffarelli and Spruck [17] proved the quasiconcavity of u in any dimension N ≥ 2 when f (0) = 0 and f is nonincreasing. We refer to [16,36,37] for further positive results using properties of the curvatures of the level sets of u or the rank of the Hessian matrix of g(u) for some increasing g, to [17,20,24,28] for positive results using properties of minimal points of the quasiconcavity function (x, y) → u((x + y)/2) − min(u(x), u(y)) in Ω × Ω, to [13,18,19] for positive results using the maximum principle for the quasiconcave envelope of the function u and to [4,38] for further existence results of quasiconcave solutions to some equations of the type (1.3). Lastly, if the open sets Ω 1 and Ω 2 are just assumed to be starshaped with respect to a point x 0 ∈ Ω 2 , then the superlevel sets of the solutions u of (1.3) are known to be starshaped with respect to x 0 when f (0) = 0 and f is nonincreasing, since (x − x 0 ) • ∇u(x) < 0 in Ω from the maximum principle and Hopf lemma, see [2,20,26,28,45] for further results in this direction.…”