Geometric properties of the Gelfand problem through a parabolic approach

We consider the asymptotic profiles of the nonlinear parabolic flows(eu)t=Δu+λeu to show the geometric properties of minimal solutions of the following elliptic nonlinear eigenvalue problems known as the Gelfand problem:Δφ+λeφ=0,φ>0in Ωφ=0on Ω posed in a strictly convex domain Ω⊂Rn. In this work, we show that there is a strictly increasing function f(s) such that f−1(φ(x)) is convex for 0<λ⩽λ⁎, i.e., we prove that level set of φ is convex. Moreover, we also present the boundary condition of φ which guarantees the f-convexity of solution φ.