New projection and Korn estimates for a class of constant-rank operators on domains

2021/09/29 by Adolfo Arroyo-Rabasa, Arroyo-Rabasa, Adolfo
Mathematics · #Nonlinear Partial Differential Equations #Advanced Harmonic Analysis Research #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2109.14602

Abstract

Let 1 < p < ∞ and let Ω be an open and bounded set of \mathbb Rⁿ. We establish classical Korn inequalities inf_\substackv ∈ L^p(Ω)
\mathcal A v = 0 ‖u - v‖_W^k,p(Ω) ≤ C ‖ \mathcal A u‖_L^p(Ω) for all kth order operators \mathcal A satisfying the maximal-rank condition. This new condition is satisfied by the divergence, Laplacian, Laplace-Beltrami, and Wirtinger operators, among others. As such, our estimates generalize Fuchs' estimates for the del-bar operator to maximal-rank operators and to arbitrary domains. For domains with sufficiently regular boundary ∂ Ω, we are able to construct an L^p(Ω)-bounded projection P, onto the kernel of the operator. This projection is shown to satisfy a classical Fonseca-Müller projection estimate ‖u - Pu‖_L^p(Ω) ≤ C ‖ \mathcal A u‖_W^-k,p(Ω) as well as analogous estimates for higher-order derivatives. As a particular application of our results, we are able to establish a weak Korn inequality for general constant-rank operators (by taking the infimum over all \mathcal A-harmonic maps instead of taking it over all \mathcal A-free maps). Several examples are discussed.

Related

• Constant rank operators in Korn-Maxwell-Sobolev inequalities
• L¹-estimates for constant rank operators
• On the complex constant rank condition and inequalities for differential operators
• Korn and Poincaré-Korn inequalities: A different perspective
• On the growth rate of powers of a strongly Kreiss bounded operator on L^p-spaces