The Dirichlet problem for elliptic systems with data in Köthe function spaces

Abstract

We show that the boundedness of the Hardy-Littlewood maximal operator on a Köthe function space $\mathbb{X}$ and on its Köthe dual ${\mathbb{X}}^{\prime }$ is equivalent to the well-posedness of the $\mathbb{X}$-Dirichlet and ${\mathbb{X}}^{\prime }$-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 ${\mathrm{HA}}^p$ 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.

References in corpus (1)

• The method of layer potentials in $L^p$ and endpoint spaces for elliptic operators with $L^{\infty }$ coefficients

Cited by in corpus (8)

• The BMO-Dirichlet problem for elliptic systems in the upper-half space and quantitative characterizations of VMO
• The generalized Hölder and Morrey-Campanato Dirichlet problems for elliptic systems in the upper-half space
• On the $L^p$-Poisson semigroup associated with elliptic systems
• The $L^p$ Dirichlet boundary problem for second order Elliptic Systems with rough coefficients
• The $L^p$ Dirichlet and Regularity problems for second order Elliptic Systems with application to the Lamé system
• The $\mathrm{CMO}$-Dirichlet problem for elliptic systems in the upper half-space
• The ${\dot{W}}^{-1,p}$ Neumann problem for higher order elliptic equations
• A fractional Korn-type inequality