Geometric Properties of Gelfand's Problems with Parabolic Approach

TL;DR

This paper investigates the geometric convexity properties of solutions to Gelfand's elliptic problem using a parabolic approach, establishing conditions under which solutions have convex level sets in convex domains.

Contribution

It introduces a parabolic method to analyze the convexity of solutions to Gelfand's problem and identifies conditions ensuring the convexity of level sets.

Findings

Existence of a strictly increasing function f making f^{-1}(ϕ) convex

Convexity of level sets of ϕ in convex domains for certain λ values

Boundary conditions that guarantee f-convexity of solutions

Abstract

We consider the asymptotic profiles of the nonlinear parabolic flows $$(e^{u})_{t}= \La u+\lambda e^u$$ to show the geometric properties of the following elliptic nonlinear eigenvalue problems known as a Gelfand's problem: \begin{equation*} \begin{split} \La \vp &+ \lambda e^{\vp}=0, \quad \vp>0\quad\text{in $\Omega$}\\ \vp&=0\quad\text{on $\Omega$} \end{split} \end{equation*} posed in a strictly convex domain $\Omega\subset\re^n$. In this work, we show that there is a strictly increasing function $f(s)$ such that $f^{-1}(\vp(x))$ is convex for $0<\lambda\leq\lambda^{\ast}$, i.e., we prove that level set of $\vp$ is convex. Moreover, we also present the boundary condition of $\vp$ which guarantee the $f$-convexity of solution $\vp$.