L^p-trace-free version of the generalized Korn inequality for incompatible tensor fields in arbitrary dimensions

Abstract

For nge 3 and 1<p<infty , we prove an L^p-version of the generalized trace-free Korn-type inequality for incompatible, p-integrable tensor fields P:Omega rightarrow mathbb {R}^{ntimes n} having p-integrable generalized {text {Curl}}_{n} and generalized vanishing tangential trace P,tau _l=0 on partial Omega , denoting by {tau _l}_{l=1,ldots , n-1} a moving tangent frame on partial Omega . More precisely, there exists a constant c=c(n,p,Omega ) such that ‖P‖Lp(Ω,Rn×n)≤c‖devnsymP‖Lp(Ω,Rn×n)+‖CurlnP‖LpΩ,Rn×n(n-1)2,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\Vert P \\Vert _{L^p(\\Omega ,\\mathbb {R}^{n\\times n})}\\le c\\,\\left( \\Vert {\\text {dev}}_n {\\text {sym}}P \\Vert _{L^p(\\Omega ,\\mathbb {R}^{n \\times n})}+ \\Vert {\\text {Curl}}_{n} P \\Vert _{L^p\\left( \\Omega ,\\mathbb {R}^{n\\times \\frac{n(n-1)}{2}}\\right) }\\right) , \\end{aligned}$$\\end{document}where the generalized {text {Curl}}_{n} is given by ({text {Curl}}_{n} P)_{ijk} :=partial _i P_{kj}-partial _j P_{ki} and denotes the deviatoric (trace-free) part of the square matrix X. The improvement towards the three-dimensional case comes from a novel matrix representation of the generalized cross product.