We investigate the boundary behavior of variational solutions of Dirichlet problems for prescribed mean curvature equations at smooth boundary points where certain boundary curvature conditions are satisfied (which preclude the existence of local barrier functions). We prove that if the Dirichlet boundary data $\phi$ is continuous at such a point (and possibly nowhere else), then the solution of the variational problem is continuous at this point.
"Boundary Continuity of Nonparametric Prescribed Mean Curvature Surfaces." Taiwanese J. Math. 24 (2) 483 - 499, April, 2020. https://doi.org/10.11650/tjm/190504