Pohozaev-type identities for differential operators driven by homogeneous vector fields

TL;DR

This paper establishes Pohozaev-type identities for differential operators driven by homogeneous Hörmander vector fields, enabling non-existence results for certain boundary value problems involving these operators.

Contribution

It introduces new integral identities for solutions of PDEs associated with homogeneous Hörmander vector fields, extending classical Pohozaev identities to this setting.

Findings

Proved Pohozaev-type identities for second and fourth order equations

Derived non-existence results for boundary value problems

Extended classical identities to Hörmander vector field-driven operators

Abstract

We prove Pohozaev-type identities for smooth solutions of Euler-Lagrange equations of second and fourth order that arise from functional depending on homogeneous H\"{o}rmander vector fields. We then exploit such integral identities to prove non-existence results for the associated boundary value problems.