Summable solutions of the Goursat problem for some partial differential equations with constant coefficients

TL;DR

This paper investigates the Goursat problem for certain linear PDEs with constant coefficients in two complex variables, focusing on conditions for summable solutions when the Newton polygon has a specific geometric property.

Contribution

It provides new criteria for the summability of solutions to the Goursat problem under geometric constraints on the Newton polygon.

Findings

Derived conditions for summable solutions when the Newton polygon has one positively sloped side.

Established a link between the geometry of the Newton polygon and solution summability.

Extended understanding of solution behavior for linear PDEs with constant coefficients.

Abstract

We consider the Goursat problem for linear partial differential equations with constant coefficients in two complex variables. We find the conditions for summable solutions of the Goursat problem in the case when the Newton polygon has exactly one side with a positive slope.