Log-type ultra-analyticity of elliptic equations with gradient terms

TL;DR

This paper investigates the ultra-analyticity properties of solutions to elliptic equations with entire coefficients of exponential type, establishing sharp logarithmic bounds that highlight limitations in their ultra-analytic regularity.

Contribution

It provides the first quantitative ultra-analytic bounds for solutions of elliptic equations with exponential type entire coefficients, showing these bounds are sharp.

Findings

Solutions satisfy a sharp logarithmic ultra-analytic bound.

The ultra-analyticity of solutions is weaker than that of the coefficients.

The bounds demonstrate fundamental limitations in the ultra-analytic regularity of solutions.

Abstract

It is well known that every solution of an elliptic equation is analytic if its coefficients are analytic. However, less is known about the ultra-analyticity of such solutions. This work addresses the problem of elliptic equations with lower-order terms, where the coefficients are entire functions of exponential type. We prove that every solution satisfies a quantitative logarithmic ultra-analytic bound and demonstrate that this bound is sharp. The results suggest that the ultra-analyticity of solutions to elliptic equations cannot be expected to achieve the same level of ultra-analyticity as the coefficients.