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

where

Specifically, there exists a constant \(c=c(p,\Omega ,r)>0\) such that the inequality

holds for all tensor fields \(P\in W^{1,\,p, \, r}_0({{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}; \Omega ,\mathbb {R}^{3\times 3})\). 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 \(\Vert {{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}P \Vert _{L^r(\Omega ,\mathbb {R}^{3\times 3})} \) by \(\Vert {{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}P \Vert _{W^{-1,p}(\Omega ,\mathbb {R}^{3\times 3})}\). 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 \(P=A\in {{\,\mathrm{\mathfrak {so}}\,}}(3)\). 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.

令\(\Omega \subset \mathbb {R}^3\)是一个开有界集，具有 Lipschitz 边界和外向单位法向\(\nu \)。对于\(1<p<\infty \)，我们为新 Banach 空间中的不相容张量场P建立了广义\(L^p\) -Korn 不等式的改进版本

在哪里

具体来说，存在一个常数\(c=c(p,\Omega ,r)>0\)使得不等式

对于所有张量场\(P\in W^{1,\,p, \, r}_0({{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,} }{{\,\mathrm{Curl}\,}}; \Omega ,\mathbb {R}^{3\times 3})\)。这里，\({{\,\mathrm{dev}\,}}X :=X -\frac{1}{3} {{\,\mathrm{tr}\,}}(X)\,{\ mathbb {1}}\)表示\(3 \times 3\)矩阵X的偏（无迹）部分，边界条件在适当的弱意义上被理解。如果边界条件仅在相对开放的非空子集\(\Gamma \subset \partial \Omega \)上满足，则该估计也适用。如果没有施加边界条件，则在对有限维空间\(K_{S,dSC}\)取商后估计成立这是由条件\({{\,\mathrm{sym}\,}}P =0\)和\({{\,\mathrm{dev}\,}}{{\,\mathrm{sym }\,}}{{\,\mathrm{Curl}\,}}P = 0\)。在这种情况下，可以替换\(\Vert {{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}P \Vert _{L^r(\Omega ,\mathbb {R}^{3\times 3})} \)由\(\Vert {{\,\mathrm{dev}\,}}{{\,\ mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}P \Vert _{W^{-1,p}(\Omega ,\mathbb {R}^{3\times 3} )}\)。新的\(L^p\)估计通过选择\(P=\mathrm {D}u\)隐含了具有弱边界条件的经典 Korn 不等式，并通过选择\(P=A \in {{\,\mathrm{\mathfrak {so}}\,}}(3)\). 该证明依赖于对\(\mathrm {D}^2 {{\,\mathrm{dev}\,}}{{\ )的三阶导数\(\mathrm {D}^3 P\)的表示,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}P\)结合 Lions 引理和 Nečas 估计。我们还讨论了新的不等式在松弛微晶模型、曲率能量最弱形式的 Cosserat 模型、具有塑性自旋的梯度塑性和不相容的线性弹性中的应用。