Korn inequalities for incompatible tensor fields in three space dimensions with conformally invariant dislocation energy

Abstract

Let Omega subset mathbb {R}^3 be an open and bounded set with Lipschitz boundary and outward unit normal nu . For 1<p<infty we establish an improved version of the generalized L^p-Korn inequality for incompatible tensor fields P in the new Banach space W01,p,r(devsymCurl;Ω,R3×3)={P∈Lp(Ω;R3×3)∣devsymCurlP∈Lr(Ω;R3×3),devsym(P×ν)=0on∂Ω}\\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}$$\\begin{aligned}&W^{1,\\,p,\\,r}_0({{\\,\\mathrm{dev}\\,}}{{\\,\\mathrm{sym}\\,}}{{\\,\\mathrm{Curl}\\,}}; \\Omega ,\\mathbb {R}^{3\\times 3}) \\\\&\\quad = \\{ P \\in L^p(\\Omega ; \\mathbb {R}^{3 \\times 3}) \\mid {{\\,\\mathrm{dev}\\,}}{{\\,\\mathrm{sym}\\,}}{{\\,\\mathrm{Curl}\\,}}P \\in L^r(\\Omega ; \\mathbb {R}^{3 \\times 3}),\\ {{\\,\\mathrm{dev}\\,}}{{\\,\\mathrm{sym}\\,}}(P \\times \\nu ) = 0 \\text { on }\\partial \\Omega \\} \\end{aligned}$$\\end{document}where r∈[1,∞),1r≤1p+13,r>1ifp=32.\\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}$$\\begin{aligned} r \\in [1, \\infty ), \\qquad \\frac{1}{r} \\le \\frac{1}{p} + \\frac{1}{3}, \\qquad r >1 \\quad \\text {if }p = \\frac{3}{2}. \\end{aligned}$$\\end{document}Specifically, there exists a constant c=c(p,Omega ,r)>0 such that the inequality ‖P‖Lp(Ω,R3×3)≤c‖symP‖Lp(Ω,R3×3)+‖devsymCurlP‖Lr(Ω,R3×3)\\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}$$\\begin{aligned} \\Vert P \\Vert _{L^p(\\Omega ,\\mathbb {R}^{3\\times 3})}\\le c\\,\\left( \\Vert {{\\,\\mathrm{sym}\\,}}P \\Vert _{L^p(\\Omega ,\\mathbb {R}^{3\\times 3})} + \\Vert {{\\,\\mathrm{dev}\\,}}{{\\,\\mathrm{sym}\\,}}{{\\,\\mathrm{Curl}\\,}}P \\Vert _{L^{r}(\\Omega ,\\mathbb {R}^{3\\times 3})}\\right) \\end{aligned}$$\\end{document}holds for all tensor fields Pin W^{1,,p, , r}_0({{,mathrm{dev},}}{{,mathrm{sym},}}{{,mathrm{Curl},}}; Omega ,mathbb {R}^{3times 3}). Here, {{,mathrm{dev},}}X :=X -frac{1}{3} {{,mathrm{tr},}}(X),{mathbb {1}} denotes the deviatoric (trace-free) part of a 3 times 3 matrix X and the boundary condition is understood in a suitable weak sense. This estimate also holds true if the boundary condition is only satisfied on a relatively open, non-empty subset Gamma subset partial Omega . If no boundary conditions are imposed then the estimate holds after taking the quotient with the finite-dimensional space K_{S,dSC} which is determined by the conditions {{,mathrm{sym},}}P =0 and {{,mathrm{dev},}}{{,mathrm{sym},}}{{,mathrm{Curl},}}P = 0. In that case one can replace Vert {{,mathrm{dev},}}{{,mathrm{sym},}}{{,mathrm{Curl},}}P Vert _{L^r(Omega ,mathbb {R}^{3times 3})} by Vert {{,mathrm{dev},}}{{,mathrm{sym},}}{{,mathrm{Curl},}}P Vert _{W^{-1,p}(Omega ,mathbb {R}^{3times 3})}. The new L^p-estimate implies a classical Korn’s inequality with weak boundary conditions by choosing P=mathrm {D}u and a deviatoric-symmetric generalization of Poincaré’s inequality by choosing P=Ain {{,mathrm{mathfrak {so}},}}(3). The proof relies on a representation of the third derivatives mathrm {D}^3 P in terms of mathrm {D}^2 {{,mathrm{dev},}}{{,mathrm{sym},}}{{,mathrm{Curl},}}P combined with the Lions lemma and the Nečas estimate. We also discuss applications of the new inequality to the relaxed micromorphic model, to Cosserat models with the weakest form of the curvature energy, to gradient plasticity with plastic spin and to incompatible linear elasticity.