 Solution

The missing hypothesis is uniform ellipticity up to the boundary. The principal coefficient matrix and its quadratic form areAlthough this is positive-definite at every interior point, no single positive uniform ellipticity constant works on $R$: taking $\xi =(0,1)$ requires $y^2\ge \lambda$ for arbitrarily small positive $y$. The normal second-derivative coefficient vanishes on the lower side. This degenerate ellipticity allows the equation and boundedness to force a boundary trace, instead of allowing arbitrary continuous Dirichlet data. The coefficient matrix is positive-definite at every interior point, but lacks the uniform lower bound used by the usual bounded-domain existence theorem.