The higher order regularity Dirichlet problem for elliptic systems in the upper-half space

We identify a large class of constant (complex) coefficient, second order elliptic systems for which the Dirichlet problem in the upper-half space with data in $L^p$-based Sobolev spaces, $1<p<\infty$, of arbitrary smoothness $\ell$, is well-posed in the class of functions whose nontangential maximal operator of their derivatives up to, and including, order $\ell$ is $L^p$-integrable. This class includes all scalar, complex coefficient elliptic operators of second order, as well as the Lam\'e system of elasticity, among others

A Characterization of Rationally Convex Immersions

Let $S$ be a smooth, totally real, compact immersion in $\mathbb{C}^n$ of real dimension $m \leq n$, which is locally polynomially convex and it has finitely many points where it self-intersects finitely many times, transversely or non-transversely. We prove that $S$ is rationally convex if and only if it is isotropic with respect to a "degenerate" K\"ahler form in $\mathbb{C}^n$.Comment: In this second version of the paper, we strengthen the statement of the main theorem, address some typos that the first version contains and enhance the clarity of some parts of the proof of the main resul

Maxwell meets Korn: A New Coercive Inequality for Tensor Fields with Square-Integrable Exterior Derivative

Maxwell meets Korn: A New Coercive Inequality for Tensor Fields with Square-Integrable Exterior DerivativeComment: Key Words: Korn's inequality, theory of generalized Maxwell equations, Helmholtz decomposition, Poincare/Friedrichs-type estimates, incompatible tensor