Integral Formulas for Differential Forms on Weighted Manifolds and Applications

TL;DR

This paper develops integral formulas for differential forms on weighted manifolds with boundary, leading to new inequalities, boundary value problem solutions, and eigenvalue estimates that extend existing results.

Contribution

It introduces a Reilly formula for differential forms on weighted manifolds and applies it to derive inequalities and eigenvalue bounds, extending prior work.

Findings

Derived a Reilly formula for differential forms on weighted manifolds.

Proved a Poincaré-type inequality in this context.

Obtained new eigenvalue estimates extending known results.

Abstract

In this paper, we derive a Reilly formula for differential forms on weighted manifolds with nonempty boundary. As an application of this formula, we prove a Poincar\'e-type inequality in the same context and explore several of its consequences. We also present weighted versions of some boundary value problems and obtain new eigenvalue estimates that extend previously known results.