Variants of Bernstein's theorem for variational integrals with linear and nearly linear growth
- Resource Type
- Working Paper
- Authors
- Bildhauer, Michael; Fuchs, Martin
- Source
- Subject
- Mathematics - Analysis of PDEs
49Q20, 49Q05, 53A10, 35J20
- Language
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 \[ {\rm div} \Big[Df(\nabla u)\Big] = 0 \, , \] under which solutions have to be affine functions. Here $f$ is a smooth energy density satisfying $D^2 f>0$ together with a natural growth condition for $D^2 f$.