Variants of Bernstein’s theorem for variational integrals with linear and nearly linear growth

Using a Caccioppoli-type inequality involving negative exponents for a directional weight we establish variants of Bernstein’s theorem for variational integrals with linear and nearly linear growth. We give some mild conditions for entire solutions of the equation aligned div [Df(∇ u) ] = 0 \,, aligned div [ D f ( ∇ u ) ] = 0 , under which solutions have to be affine functions. Here f is a smooth energy density satisfying D² f>0 D 2 f > 0 together with a natural growth condition for D² f D 2 f .