Conformal Killing vector fields and Rellich type identities on Riemannian Manifolds, II

TL;DR

This paper develops a general Noetherian approach to derive Rellich integral identities on Riemannian manifolds, including higher order and biharmonic identities, and proves a nonexistence result for certain biharmonic semilinear systems.

Contribution

It introduces a novel Noetherian method to obtain Rellich identities on Riemannian manifolds with homothetic transformations, extending previous results to higher order and biharmonic cases.

Findings

Derived higher order Rellich identities involving polyharmonic operators.

Established a biharmonic Rellich identity in a general setting.

Proved a nonexistence result for semilinear biharmonic systems.

Abstract

We propose a general Noetherian approach to Rellich integral identities. Using this method we obtain a higher order Rellich type identity involving the polyharmonic operator on Riemannian manifolds admitting homothetic transformations. Then we prove a biharmonic Rellich identity in a more general context. We also establish a nonexistence result for semilinear systems involving biharmonic operators.