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Lewintan, Peter; Neff, Patrizio
Mathematical methods in the applied sciences, 30 September 2021, Letnik: 44, Številka: 14Journal Article
For 1 < p < ∞, we prove an Lp‐version of the generalized Korn inequality for incompatible tensor fields P in W01,p(Curl;Ω,ℝ3×3). More precisely, let Ω⊂ℝ3 be a bounded Lipschitz domain. Then there exists a constant c = c(p, Ω) > 0 such that ‖P‖Lp(Ω,ℝ3×3)≤c‖symP‖Lp(Ω,ℝ3×3)+‖CurlP‖Lp(Ω,ℝ3×3) holds for all tensor fields P∈W01,p(Curl;Ω,ℝ3×3), that is, for all P∈W1,p(Curl;Ω,ℝ3×3) with vanishing tangential trace P×ν=0 on ∂Ω where ν denotes the outward unit normal vector field to ∂Ω. For compatible P=Du, this recovers an Lp‐version of the classical Korn's first inequality and for skew‐symmetric P=A an Lp‐version of the Poincaré inequality.
