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The $mathrm{CMO}$Dirichlet problem for elliptic systems in the upper halfspace  Mingming Cao
;  Date: 
1 Jun 2022  Abstract:  We prove that for any secondorder, 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 $mathrm{CMO}(mathbb{R}^{n1},
mathbb{C}^N)$ is wellposed 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 Fatoutype theorem, which not only
guarantees the existence of the pointwise nontangential boundary trace for
smooth nullsolutions satisfying a strong vanishing Carleson measure condition,
but also includes a Poisson integral representation formula of solutions along
with a characterization of $mathrm{CMO}(mathbb{R}^{n1}, mathbb{C}^N)$ in
terms of the traces of solutions of elliptic systems. Moreover, we are able to
establish the wellposedness of the Dirichlet problem in $mathbb{R}^n_+$ for a
system $L$ as above in the case when the boundary data belongs to
$mathrm{XMO}(mathbb{R}^{n1}, mathbb{C}^N)$, which lines in between
$mathrm{CMO}(mathbb{R}^{n1}, mathbb{C}^N)$ and
$mathrm{VMO}(mathbb{R}^{n1}, mathbb{C}^N)$. Analogously, we formulate a new
brand of strong Carleson measure conditions and a characterization of
$mathrm{XMO}(mathbb{R}^{n1}, mathbb{C}^N)$ in terms of the traces of
solutions of elliptic systems.  Source:  arXiv, 2206.00318  Services:  Forum  Review  PDF  Favorites 


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