|
| | | | |
| | | |
| Stat |
Members: 3657
Articles: 2'599'751
Articles rated: 2609
15 October 2024 |
|
| | | | |
|
Article overview
| |
|
|
The $mathrm{CMO}$-Dirichlet problem for elliptic systems in the upper half-space | | Mingming Cao
; | | Date: |
1 Jun 2022 | | Abstract: | We prove that for any second-order, homogeneous, $N imes N$ elliptic system
$L$ with constant complex coefficients in $mathbb{R}^n$, the Dirichlet problem
in $mathbb{R}^n_+$ with boundary data in $mathrm{CMO}(mathbb{R}^{n-1},
mathbb{C}^N)$ is well-posed under the assumption that $dmu(x’, t) := |
abla
u(x)|^2, t , dx’ dt$ is a strong vanishing Carleson measure in
$mathbb{R}^n_+$ in some sense. This solves an open question posed by Martell
et al. The proof relies on a quantitative Fatou-type theorem, which not only
guarantees the existence of the pointwise nontangential boundary trace for
smooth null-solutions satisfying a strong vanishing Carleson measure condition,
but also includes a Poisson integral representation formula of solutions along
with a characterization of $mathrm{CMO}(mathbb{R}^{n-1}, mathbb{C}^N)$ in
terms of the traces of solutions of elliptic systems. Moreover, we are able to
establish the well-posedness of the Dirichlet problem in $mathbb{R}^n_+$ for a
system $L$ as above in the case when the boundary data belongs to
$mathrm{XMO}(mathbb{R}^{n-1}, mathbb{C}^N)$, which lines in between
$mathrm{CMO}(mathbb{R}^{n-1}, mathbb{C}^N)$ and
$mathrm{VMO}(mathbb{R}^{n-1}, mathbb{C}^N)$. Analogously, we formulate a new
brand of strong Carleson measure conditions and a characterization of
$mathrm{XMO}(mathbb{R}^{n-1}, mathbb{C}^N)$ in terms of the traces of
solutions of elliptic systems. | | Source: | arXiv, 2206.00318 | | Services: | Forum | Review | PDF | Favorites | |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
|
| |
|
|
|
REGISTER