Global boundedness, interior gradient estimates, and boundary regularity for the mean curvature equation with boundary conditions

where ν is the outward pointing unit normal of ∂Ω, and where cosθ is a given function on ∂Ω. (Thus, in the capillarity problem, we are considering geometrically a function u in Ω̄ whose graph has the prescribed mean curvature H and which meets the boundary cylinder in the prescribed angle θ.) Here, H = H(x,t) is assumed to be a given locally Lipschitz function in Ω×R satisfying the structural conditions