论文标题
最佳不兼容的Korn-Maxwell-Sobolev在所有维
Optimal incompatible Korn-Maxwell-Sobolev inequalities in all dimensions
论文作者
论文摘要
我们表征所有线性映射$ \ mathcal {a} \ colon \ mathbb {r}^{n \ times n} \ to \ mathbb {r}^{n \ times n} $ \ | p \ | _ {l^{p^{*}}（\ Mathbb {r}^{n}）} \ leq c \，\ big（\ | \ Mathcal {a} [p] \ | _ {l^{p^{*}}}（\ MathBb {r} \ end {align*}保留在c_ {c}^{\ infty}（\ Mathbb {r}^{n}; \ \ m mathbb {r}^{n \ times n}）$，where $ \ \ mathrm {curl curlix purl curl curl cull curl curl curl curlix curlix curlix curlix curlix curlix也适用于不兼容的，即非梯度矩阵字段，这种不平等概括了通常使用的korn型不平等现象，例如在线性弹性中。与以前的贡献不同，本文收集的结果适用于所有维度和最佳。这尤其需要在$ \ Mathcal {a} $，集成性$ p $和基础空间尺寸$ n $之间进行不同星座的区别，尤其是在二维情况下需要进行更精细的分析。
We characterise all linear maps $\mathcal{A}\colon\mathbb{R}^{n\times n}\to\mathbb{R}^{n\times n}$ such that, for $1\leq p<n$, \begin{align*} \|P\|_{L^{p^{*}}(\mathbb{R}^{n})}\leq c\,\Big(\|\mathcal{A}[P]\|_{L^{p^{*}}(\mathbb{R}^{n})}+\|\mathrm{Curl} P\|_{L^{p}(\mathbb{R}^{n})} \Big) \end{align*} holds for all compactly supported $P\in C_{c}^{\infty}(\mathbb{R}^{n};\mathbb{R}^{n\times n})$, where $\mathrm{Curl} P$ displays the matrix curl. Being applicable to incompatible, that is, non-gradient matrix fields as well, such inequalities generalise the usual Korn-type inequalities used e.g. in linear elasticity. Different from previous contributions, the results gathered in this paper are applicable to all dimensions and optimal. This particularly necessitates the distinction of different constellations between the ellipticities of $\mathcal{A}$, the integrability $p$ and the underlying space dimensions $n$, especially requiring a finer analysis in the two-dimensional situation.
