Regularity of Extremal Solutions in Fourth Order Nonlinear Eigenvalue Problems on General Domains

2010/03/19 by Craig Cowan, Cowan, Craig, Pierpaolo Esposito +3 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics #math.AP

paper · pdf · doi:10.48550/arxiv.1003.3862

19 pages. Updated versions - if any - can be downloaded at http://www.birs.ca/~nassif/

arxiv created 2010/03/19 · openalex publication_date 2010/03/19 · arxiv updated 2010/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We examine the regularity of the extremal solution of the nonlinear eigenvalue problem Δ² u = λf(u) on a general bounded domain Ω in \IR^N, with the Navier boundary condition u=Δu =0 on \pOm. Here λ is a positive parameter and f is a non-decreasing nonlinearity with f(0)=1. We give general pointwise bounds and energy estimates which show that for any convex and superlinear nonlinearity f, the extremal solution u^* is smooth provided N≤ 5.

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