For prescribed mean curvature equations with a small W^{1,p} forcing term and small boundary data, a W^{2,q} solution exists near any given graphical minimal surface.
Gerhardt, Existence, regularity, and boundary behaviour of generalized surfaces of prescribed mean curvature, Math
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The Dirichlet problem for a prescribed mean curvature equation
For prescribed mean curvature equations with a small W^{1,p} forcing term and small boundary data, a W^{2,q} solution exists near any given graphical minimal surface.