On multiwell Liouville theorems in higher dimension

2008/02/06 by Robert L. Jerrard, Jerrard, Robert L., Andrew Lorent +1
Mathematics · #26B99 #30C70 #Advanced Operator Algebra Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Holomorphic and Operator Theory #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.0802.0850

openalex publication_date 2008/02/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider certain subsets of the space of n× n matrices of the form K = ∪_i=1^m SO(n)A_i, and we prove that for p>1, q ≥ 1 and for connected Ω'⊂⊂Ω⊂ \Rⁿ, there exists positive constant a<1 depending on n,p,q, Ω, Ω' such that for \veps=‖ dist(Du, K)‖_L^p(Ω)^p we have inf_{R∈ K}‖Du-R‖^p_L^p(Ω')≤ M\veps^1/p provided u satisfies the inequality ‖ D² u‖_L^q(Ω)^q≤ a\veps^1-q. Our main result holds whenever m=2, and also for \em generic m≤ n in every dimension n≥ 3, as long as the wells SO(n)A₁,..., SO(n)A_m satisfy a certain connectivity condition. These conclusions are mostly known when n=2, and they are new for n≥ 3.

Citations

• A theorem on geometric rigidity and the derivation of nonlinear plate theory from three‐dimensional elasticity
• A Hierarchy of Plate Models Derived from Nonlinear Elasticity by Gamma-Convergence

Related

• Bounds on Dimension Reduction in the Nuclear Norm
• Lines of full rank matrices in large subspaces
• A generalization of the boundedness of certain integral operators in variable Lebesgue spaces
• Inequalities involving upper bounds for certain matrix operators
• New methods for determining speciality of linear systems based at fat points in Pⁿ