Elliptic fourth-order operators with Wentzell boundary conditions on Lipschitz domains

2024/05/03 by David Ploß, Ploß, David
Computer Science · Mathematics · #35B65 (secondary) #35K35 (primary) #47A07 #47D06 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2405.02064

openalex publication_date 2024/05/03 · openalex created_date 2024/05/10 · openalex updated_date 2026/07/28

Abstract

For bounded domains Ω with Lipschitz boundary Γ, we investigate boundary value problems for elliptic operators with variable coefficients of fourth order subject to Wentzell (or dynamic) boundary conditions. Using form methods, we begin by showing general results for an even wider class of operators defined via two (intertwined) quadratic forms by defining very abstract concepts of weak traces. Even in this general setting, we prove generation of an analytic semigroup on the product space L²(Ω) × L²(Γ). Using recent results concerning weak co-normal traces, we apply our abstract theory to the elliptic fourth-order case and are able to fully characterize the domain in terms of Sobolev regularity, also obtaining Hölder-regularity of solutions. Finally, we also discuss asymptotic behavior and (eventual) positivity.

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