Geometric Properties of Gelfand's Problems with Parabolic Approach (1305.1065v2)
Abstract: We consider the asymptotic profiles of the nonlinear parabolic flows $$(e{u})_{t}= \La u+\lambda eu$$ 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 $\Omega\subset\ren$. 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 $\vp$.
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