Constant rank operators in Korn-Maxwell-Sobolev inequalities
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| Format: | Preprint |
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2024
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| _version_ | 1866910752918470656 |
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| author | Lewintan, Peter Stephan, Paul |
| author_facet | Lewintan, Peter Stephan, Paul |
| contents | We focus on Korn-Maxwell-Sobolev inequalities for operators of reduced constant rank. These inequalities take the form \[ \|P - Π_{\mathbb{B}} Π_{\ker\mathscr{A}} P\|_{\dot{\mathrm{W}}^{k-1, p^*}(\mathbb{R}^n)} \le c \, (\|\mathscr{A}[P]\|_{\dot{\mathrm{W}}^{k-1, p^*}(\mathbb{R}^n)} + \|\mathbb{B} P\|_{\mathrm{L}^p(\mathbb{R}^n)}) \] for all $ P \in \mathrm{C}_c^\infty(\mathbb{R}^n; V) $, where $ V $ is a finite-dimensional vector space, $ \mathscr{A} $ is a linear mapping, and $ \mathbb{B} $ is a constant coefficient homogeneous differential operator of order $ k $. In particular, we can treat the combination $(p,\mathscr{A},\mathbb{B},k)=(1,\operatorname{tr},\operatorname{Curl},1)$. Our results generalize the techniques from Gmeineder et al. (Math.Mod.Met.Appl.Sci,34:03,2024; arXiv:2405.10349), which exclusively dealt with reduced elliptic operators. In contrast to the reduced ellipticity case, however, the reduced constant rank case necessitates to introduce a correction, namely the projection $Π_\mathbb{B}$ on the left-hand side of the inequality. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_14866 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Constant rank operators in Korn-Maxwell-Sobolev inequalities Lewintan, Peter Stephan, Paul Analysis of PDEs We focus on Korn-Maxwell-Sobolev inequalities for operators of reduced constant rank. These inequalities take the form \[ \|P - Π_{\mathbb{B}} Π_{\ker\mathscr{A}} P\|_{\dot{\mathrm{W}}^{k-1, p^*}(\mathbb{R}^n)} \le c \, (\|\mathscr{A}[P]\|_{\dot{\mathrm{W}}^{k-1, p^*}(\mathbb{R}^n)} + \|\mathbb{B} P\|_{\mathrm{L}^p(\mathbb{R}^n)}) \] for all $ P \in \mathrm{C}_c^\infty(\mathbb{R}^n; V) $, where $ V $ is a finite-dimensional vector space, $ \mathscr{A} $ is a linear mapping, and $ \mathbb{B} $ is a constant coefficient homogeneous differential operator of order $ k $. In particular, we can treat the combination $(p,\mathscr{A},\mathbb{B},k)=(1,\operatorname{tr},\operatorname{Curl},1)$. Our results generalize the techniques from Gmeineder et al. (Math.Mod.Met.Appl.Sci,34:03,2024; arXiv:2405.10349), which exclusively dealt with reduced elliptic operators. In contrast to the reduced ellipticity case, however, the reduced constant rank case necessitates to introduce a correction, namely the projection $Π_\mathbb{B}$ on the left-hand side of the inequality. |
| title | Constant rank operators in Korn-Maxwell-Sobolev inequalities |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2412.14866 |