Improved Friedrichs inequality for a subhomogeneous embedding

TL;DR

This paper enhances the Friedrichs inequality for certain function spaces, providing quantified versions and applying them to establish existence results for a class of nonlinear PDEs involving the p-Laplacian.

Contribution

It introduces quantified Friedrichs inequalities for subhomogeneous embeddings and applies these results to prove the existence of solutions to a resonant p-Laplacian equation.

Findings

Established quantified Friedrichs inequalities for p ≥ q ≥ 2.

Proved existence of weak solutions for a resonant p-Laplacian equation.

Demonstrated the application of inequalities to nonlinear PDEs.

Abstract

For a smooth bounded domain $\Omega$ and $p \geq q \geq 2$, we establish quantified versions of the classical Friedrichs inequality $\|\nabla u\|_p^p - \lambda_1 \|u\|_q^p \geq 0$, $u \in W_0^{1,p}(\Omega)$, where $\lambda_1$ is a generalized least frequency. We apply one of the obtained quantifications to show that the resonant equation $-\Delta_p u = \lambda_1 \|u\|_q^{p-q} |u|^{q-2} u + f$ coupled with zero Dirichlet boundary conditions possesses a weak solution provided $f$ is orthogonal to the minimizer of $\lambda_1$.