Geometric property of the Ground State Eigenfunction for Cauchy Process

Abstract

We consider the asymptotic behavior of nonlinear nonlocal flows $u_t+(-\La)^{1/2}u=0$ to find the geometric property of the solutions in nonlinear eigenvalue problem: (-\La)^{1/2}\vp=λ\vp posed in a strictly convex domain $Ω\subset\R^n$ with $\vp>0$ in $Ω$ and $\vp=0$ on $\R^n\bsΩ$. This is corresponding to an eigenvalue problem for Cauchy process. The concavity of $\vp$ is well known for the dimension $n=1$. In this paper, we will show $\vp^{-\frac{2}{n+1}}$ is convex. Moreover, the eventual power-convexity of the parabolic flows is also proved. In the final section, We extend geometric results to Cauchy problem for the fractional Heat operator.

25 pages. This paper has been withdrawn by the author. The reason is as follows. We've checked our paper for the purpose of reviewing our work. Unfortunately, we find that the proof for the main result is not perfect, i.e., there are some gaps and mistakes