  We prove weak and strong maximum principles, including a Hopf lemma, for
smooth subsolutions to equations defined by linear, second-order, partial
differential operators whose principal symbols vanish along a portion of the
domain boundary. The boundary regularity property of the smooth subsolutions
along this boundary vanishing locus ensures that these maximum principles hold
irrespective of the sign of the Fichera function. Boundary conditions need only
be prescribed on the complement in the domain boundary of the principal symbol
vanishing locus. We obtain uniqueness and a priori maximum principle estimates
for smooth solutions to boundary value and obstacle problems defined by these
boundary-degenerate elliptic operators for partial Dirichlet or Neumann
boundary conditions along the complement of the boundary vanishing locus. We
also prove weak maximum principles and uniqueness for solutions to the
corresponding variational equations and inequalities defined with the aide of
weighted Sobolev spaces. The domain is allowed to be unbounded when the
operator coefficients and solutions obey certain growth conditions.
