{"paper":{"title":"Geometric property of the Ground State Eigenfunction for Cauchy Process","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ki-Ahm Lee, SungHoon Kim","submitted_at":"2011-05-17T06:10:25Z","abstract_excerpt":"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 Cauchy process.\nThe 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 e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.3283","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}