Boundary ellipticity and limiting $L^1$-estimates on halfspaces
Fuente:
arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866914650922156032 |
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| author | Gmeineder, Franz Raiţă, Bogdan Van Schaftingen, Jean |
| author_facet | Gmeineder, Franz Raiţă, Bogdan Van Schaftingen, Jean |
| contents | We identify necessary and sufficient conditions on $k$th order differential operators $\mathbb{A}$ in terms of a fixed halfspace $H^+\subset\mathbb{R}^n$ such that the Gagliardo--Nirenberg--Sobolev inequality
$$
\|D^{k-1}u\|_{\mathrm{L}^{\frac{n}{n-1}}(H^+)}\leq c\|\mathbb{A} u\|_{\mathrm{L}^1(H^+)}\quad\text{for }u\in\mathrm{C}^\infty_c (\mathbb{R}^{n},V)
$$
holds. This comes as a consequence of sharp trace theorems on $H=\partial H^+$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_08167 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Boundary ellipticity and limiting $L^1$-estimates on halfspaces Gmeineder, Franz Raiţă, Bogdan Van Schaftingen, Jean Analysis of PDEs Functional Analysis 26D10, 35E05, 35G15, 35G35 We identify necessary and sufficient conditions on $k$th order differential operators $\mathbb{A}$ in terms of a fixed halfspace $H^+\subset\mathbb{R}^n$ such that the Gagliardo--Nirenberg--Sobolev inequality $$ \|D^{k-1}u\|_{\mathrm{L}^{\frac{n}{n-1}}(H^+)}\leq c\|\mathbb{A} u\|_{\mathrm{L}^1(H^+)}\quad\text{for }u\in\mathrm{C}^\infty_c (\mathbb{R}^{n},V) $$ holds. This comes as a consequence of sharp trace theorems on $H=\partial H^+$. |
| title | Boundary ellipticity and limiting $L^1$-estimates on halfspaces |
| topic | Analysis of PDEs Functional Analysis 26D10, 35E05, 35G15, 35G35 |
| url | https://arxiv.org/abs/2211.08167 |