A nonlinear Korn inequality on a surface with an explicit estimate of the constant

Abstract

A nonlinear Korn inequality on a surface estimates a distance between a surface θ(ω) and another surface φ(ω) in terms of distances between their fundamental forms in the space Lp (ω), 1 < p <∞. We establish a new inequality of this type. The novelty is that the immersion θ belongs to a specific set of mappings of class C 1 from ω into R3 with a unit vector field also of class C 1 over ω. Résumé. Une inégalité de Korn non linéaire sur une surface estime une distance entre une surfaceθ(ω) et une autre surfaceφ(ω) en fonction des distances entre leur formes fondamentales dans l’espace Lp (ω), 1 < p <∞. Nous établissons une nouvelle inégalité de ce type. La nouveauté réside dans l’appartenance de l’immersion θ à un ensemble particulier d’applications de classe C 1 de ω dans R3 avec un champ de vecteurs normaux unitaires aussi de classe C 1 dans ω. Funding. The work of the second author was substantially supported by a grant from City University of Hong Kong (Project No. 7005495). Manuscript received 18th September 2020, accepted 23rd September 2020. ∗Corresponding author. ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/ 106 Maria Malin and Cristinel Mardare 1. Notation and definitions Vector and matrix fields are denoted by boldface letters. Given any open set Ω ⊂ Rn , n > 1, any subset V ⊂ Y of a finite-dimensional vector space Y , and any integer `> 0, the notation C (Ω;V ) designates the set of all fields v = (vi ) :Ω→ Y such that v (x) ∈ V for all x ∈ Ω and vi ∈ C (Ω). Likewise, given any real number p > 1, the notation Lp (Ω;V ), resp. W `, p (Ω;V ), designates the set of all fields v = (vi ) :Ω→ Y such that v (x) ∈ V for almost all x ∈Ω and vi ∈ Lp (Ω), resp. vi ∈W `, p (Ω). The space of all real matrices with k rows and ` columns is denotedMk×`. We also let M :=Mk×k ,S := { A ∈Mk ; A = A } , S> := { A ∈Sk ; A is positive-definite } , and O+ := { A ∈Mk ; A A = I and det A = 1 } . A k × ` matrix whose column vectors are the vectors v 1, . . . , v` ∈ Rk is denoted (v 1| . . . |v`). If A ∈S>, there exists a unique matrix U ∈S> such that U 2 = A; this being the case, we let A1/2 :=U . The Euclidean norm in R3 is denoted | · |. Spaces of matrices are equipped with the Frobenius norm, also denoted | · |. The spaces Lp (Ω), Lp (Ω;Rk ), and Lp (Ω;Mk×`), are respectively equipped with the norms denoted and defined by