Characterization of intermediate boundary-weight positivity conditions

Determine the precise conditions on the domain, conductivity q^\dagger, source f, and state u^\dagger solving the elliptic conductivity equation that ensure the positivity condition (fu^\dagger+q^\dagger|\nabla u^\dagger|^2)(x)\geq c\,\mathrm{dist}(x,\partial\Omega)^\beta almost everywhere in \Omega for an exponent \beta\in(0,2).

Background

The paper’s L^2(\Omega) error estimate for recovering the conductivity depends on a weighted stability quantity involving fu^\dagger+q^\dagger|\nabla u^\dagger|^2. Condition \ref{Cond:positivity} assumes that this quantity is bounded below by a positive constant times the distance to the boundary raised to a power \beta\geq0. The exponent \beta measures how rapidly the stability weight may degenerate near \partial\Omega.

The paper notes that existing results give the condition with \beta=2 under relatively weak assumptions and with \beta=0 under stronger smoothness assumptions. It explicitly leaves unresolved the precise assumptions on the domain, conductivity, and source that yield intermediate exponents \beta\in(0,2).