Regularity results for elliptic and parabolic systems of partial differential equations

TL;DR

This paper establishes regularity and pointwise derivative estimates for solutions to elliptic and parabolic systems of PDEs, enhancing understanding of their smoothness in various functional spaces.

Contribution

It provides new pointwise estimates and regularity results for elliptic and parabolic systems with vector-valued functions, including coupled first-order operators.

Findings

Pointwise derivative estimates for semigroups

Regularity results in Hölder, Zygmund, Sobolev, and Besov spaces

Enhanced understanding of PDE solution smoothness

Abstract

We study Cauchy problems associated to elliptic operators acting on vector-valued functions and coupled up to the first-order. We prove pointwise estimates for the spatial derivatives of the semigroup associated to these problems in the space of bounded and continuous functions over $\R^d$. Consequently, we deduce relevant regularity results both in H\"older and Zygmund spaces and in Sobolev and Besov spaces.