Geometric property of the Ground State Eigenfunction for Cauchy Process

Details Geometric property of the Ground State Eigenfunction for Cauchy Process Geometric property of the Ground State Eigenfunction for Cauchy Process

2011-05-17

arxiv.org/abs/1105.3283v1

Mathematics

Analysis of PDEs

25 pages

Scientific paper

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=\lambda\vp posed in a strictly convex domain $\Omega\subset\R^n$ with $\vp>0$ in $\Omega$ and $\vp=0$ on $\R^n\bs\Omega$. This is corresponding to an eigenvalue problem for Chauchy 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 when $1

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