The Structure Condition Implies Degenerate Ellipticity

Throughout we work in the setting of Second-Order Equations on Euclidean Open Sets and of Bounded Open Domain in Euclidean Space, whose notation, including that of Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation on which the former rests, is in force in the dimension $n$, a natural number with $1\le n$. In addition $T=\{t\in\mathbb{R}:0\le t\}$.

Let $F$ be a second-order equation operator on $\Omega$ which is continuous, and let $\omega:T\to\mathbb{R}$ be a modulus of continuity such that $F$ and $\omega$ satisfy the structure condition of the comparison principle for the Dirichlet problem.

Then $F$ is degenerate elliptic.