Boundary value problems for first order elliptic wedge operators

TL;DR

This paper develops an $L^2$ elliptic boundary value problem theory for first order wedge differential operators, extending classical results to singular geometries with minimal spectral assumptions.

Contribution

It introduces a new elliptic theory for wedge operators that does not require constant indicial roots, addressing a long-standing open problem in analysis on singular spaces.

Findings

Established a robust $L^2$ elliptic boundary value theory for wedge operators.

Unified classical boundary value problem theory with analysis on manifolds with edge singularities.

Provided a framework applicable to a broad class of singular geometric problems.

Abstract

We develop an elliptic theory based in $L^2$ of boundary value problems for general wedge differential operators of first order under only mild assumptions on the boundary spectrum. In particular, we do not require the indicial roots to be constant along the base of the boundary fibration. Our theory includes as a special case the classical theory of elliptic boundary value problems for first order operators with and without the Shapiro-Lopatinskii condition, and can be thought of as a natural extension of that theory to the geometrically and analytically relevant class of wedge operators. Wedge operators arise in the global analysis on manifolds with incomplete edge singularities. Our theory settles, in the first order case, the long-standing open problem to develop a robust elliptic theory of boundary value problems for such operators.