Dev-Div- and DevSym-DevCurl-inequalities for incompatible square tensor fields with mixed boundary conditions

Bauer, Sebastian; Neff, Patrizio; Pauly, Dirk; Starke, Gerhard

ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 1, pp. 112-133.

Let $\Omega \subset {ℝ}^n,n\ge 2$, be a bounded Lipschitz domain and $1<q<\infty$. We prove the inequality

$$\begin{array}{c}\hfill {\parallel T\parallel }_{L^q\left(\Omega \right)}\le C_{DD}\left({\parallel \mathrm{dev}T\parallel }_{L^q\left(\Omega \right)}+{\parallel \mathrm{Div}T\parallel }_{L^q\left(\Omega \right)}\right)\end{array}$$

being valid for tensor fields $T:\Omega \to {ℝ}^{n\times n}$ with a normal boundary condition on some open and non-empty part $\Gamma {}_{\nu }$ of the boundary $\partial \Omega$. Here $\mathrm{dev}T=T-\frac{1}{n}tr\left(T\right)·$ It denotes the deviatoric part of the tensor $T$ and Div is the divergence row-wise. Furthermore, we prove

$$\begin{array}{ccccc}\hfill {\parallel T\parallel }_{L^2\left(\Omega \right)}& & \le C_{DSC}\left({\parallel \mathrm{dev}\mathrm{sym}T\parallel }_{L^2\left(\Omega \right)}+{\parallel \mathrm{Curl}T\parallel }_{L^2\left(\Omega \right)}\right)\hfill & & \mathrm{if}n\ge 3,\\ \hfill {\parallel T\parallel }_{L^2\left(\Omega \right)}& & \le C_{DSDC}\left({\parallel \mathrm{dev}\mathrm{sym}T\parallel }_{L^2\left(\Omega \right)}+{\parallel \mathrm{dev}\mathrm{Curl}T\parallel }_{L^2\left(\Omega \right)}\right)\hfill & & \mathrm{if}n=3,\end{array}$$

being valid for tensor fields $T$ with a tangential boundary condition on some open and non-empty part $\Gamma {}_{\tau }$ of $\partial \Omega$. Here, $\mathrm{sym}T=\frac{1}{2}(T+T^{\top })$denotes the symmetric part of denotes the symmetric part of $T$ and Curl is the rotation row-wise.

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