Quadratic estimates for degenerate elliptic systems on manifolds with lower Ricci curvature bounds and boundary value problems

TL;DR

This paper establishes weighted quadratic estimates and solvability results for degenerate elliptic systems on manifolds with lower Ricci bounds, extending previous work to more general geometric and coefficient conditions.

Contribution

It proves new quadratic estimates and boundary value problem solvability for degenerate elliptic systems on manifolds with weaker curvature and weight assumptions.

Findings

Proved weighted quadratic estimates for certain differential operators.

Established Kato square root estimate under weaker assumptions.

Demonstrated solvability of boundary value problems on Lipschitz manifolds.

Abstract

Weighted quadratic estimates are proved for certain bisectorial firstorder differential operators with bounded measurable coefficients which are (not necessarily pointwise) accretive, on complete manifolds with positive injectivity radius. As compared to earlier results, Ricci curvature is only assumed to be bounded from below, and the weight is only assumed to be locally in A^2. The Kato square root estimate is proved under this weaker assumption. On compact Lipschitz manifolds we prove solvability estimates for solutions to degenerate elliptic systems with not necessarily self-adjoint coefficients, and with Dirichlet, Neumann and Atiyah-Patodi-Singer boundary conditions.