- journal article metadata
European Mathematical Society Publishing House
2016-10-06 23:45:01
Revista Matemática Iberoamericana
Rev. Mat. Iberoamericana
RMI
0213-2230
2235-0616
General
10.4171/RMI
http://www.ems-ph.org/doi/10.4171/RMI
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2012)
32
2016
3
The Dirichlet problem for elliptic systems with data in Köthe function spaces
José María
Martell
Universidad Autónoma de Madrid, MADRID, SPAIN
Dorina
Mitrea
University of Missouri, COLUMBIA, UNITED STATES
Irina
Mitrea
Temple University, PHILADELPHIA, UNITED STATES
Marius
Mitrea
University of Missouri, COLUMBIA, UNITED STATES
Dirichlet problem, second-order elliptic system, nontangential maximal function, Hardy–Littlewood maximal operator, Poisson kernel, Green function, Köthe function space, Muckenhoupt weight, Lebesgue space, variable exponent Lebesgue space, Lorentz space, Zygmund space, Orlicz space, Hardy space, Beurling algebra, Hardy–Beurling space, semigroup, Fatou type theorem
We show that the boundedness of the Hardy–Littlewood maximal operator on a Köthe function space $\mathbb X$ and on its Köthe dual $\mathbb X$' is equivalent to the well-posedness of the $\mathbb X$-Dirichlet and $\mathbb X$'-Dirichlet problems in$\mathbb R^n_+$ in the class of all second-order, homogeneous, elliptic systems, with constant complex coefficients. As a consequence, we obtain that the Dirichlet problem for such systems is well-posed for boundary data in Lebesgue spaces, variable exponent Lebesgue spaces, Lorentz spaces, Zygmund spaces, as well as their weighted versions. We also discuss a version of the aforementioned result which contains, as a particular case, the Dirichlet problem for elliptic systems with data in the classical Hardy space $H^1$, and the Beurling-Hardy space HA$^p$ for $p \in (1,\infty)$. Based on the well-posedness of the $L^p$-Dirichlet problem we then prove the uniqueness of the Poisson kernel associated with such systems, as well as the fact that they generate a strongly continuous semigroup in natural settings. Finally, we establish a general Fatou type theorem guaranteeing the existence of the pointwise nontangential boundary trace for null-solutions of such systems.
Partial differential equations
Fourier analysis
Functional analysis
Mechanics of deformable solids
913
970
10.4171/RMI/903
http://www.ems-ph.org/doi/10.4171/RMI/903