2023/12/01 by Gerassimos Barbatis, Barbatis, Gerassimos, Achilles Tertikas +1
Engineering · Mathematics · #35A23 #35G05 #35G20 (Primary) 31B30 #35J30 #46E35 (Secondary) #Analysis of PDEs (math.AP) #FOS: Mathematics #Fatigue and fracture mechanics #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2312.00433
openalex publication_date 2023/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
There are two Rellich inequalities for the bilaplacian, that is for ∫ (Δu)2dx, the one involving |∇ u| and the other involving |u| at the RHS. In this article we consider these inequalities with sharp constants and obtain sharp Sobolev-type improvements. More precisely, in our first result we improve the Rellich inequality with |∇ u| obtained recently by Cazacu in dimensions n=3,4 by a sharp Sobolev term thus complementing existing results for the case n≥ 5. In the second theorem the sharp constant of the Sobolev improvement for the Rellich inequality with |u| is obtained.