Boundary ellipticity and limiting \(\mathrm{L}^1\)-estimates on halfspaces.(English)Zbl 07807342
Summary: 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\[\| D^{k - 1} u \|_{\mathrm{L}^{\frac{n}{n - 1}} (H^+)} \leqslant c \| \mathbb{A} u \|_{\mathrm{L}^1 (H^+)} \quad \text{for } u \in \mathrm{C}_c^\infty(\mathbb{R}^n, V)\]holds. This comes as a consequence of sharp trace theorems on \(H = \partial H^+\).
MSC:
| 26D10 | Inequalities involving derivatives and differential and integral operators |
| 35E05 | Fundamental solutions to PDEs and systems of PDEs with constant coefficients |
| 35G15 | Boundary value problems for linear higher-order PDEs |
| 35G35 | Systems of linear higher-order PDEs |
Keywords:
boundary ellipticity;canceling operators;\(\mathbb{C}\)-ellipticity;trace theorem;Uspenskiĭ’s theorem;Aronszajn’s coercive inequalitiesCite
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