Multi-dimensional Weiss operators

TL;DR

This paper generalizes Weiss operators to multi-dimensional spaces, providing a new theorem and methods to identify null solutions for complex PDEs, with practical examples demonstrating its effectiveness.

Contribution

It introduces a d-dimensional Weiss operator framework and a corresponding theorem, enabling the analysis of PDEs in higher-dimensional spaces.

Findings

Derived a d-dimensional Weiss Theorem

Identified null class functions for PDEs

Validated approach with examples of linear and nonlinear PDEs

Abstract

We present a solution of the Weiss operator family generalized for the case of $\mathbb{R}^{d}$ and formulate a d-dimensional analogue of the Weiss Theorem. Most importantly, the generalization of the Weiss Theorem allows us to find a sub-set of null class functions for a partial differential equation with the generalized Weiss operators. We illustrate the significance of our approach through several examples of both linear and non-linear partial differential equations.