Splitting-type variational problems with linear growth conditions

Abstract

Regularity properties of solutions to variational problems are established for a broad class of strictly convex splitting-type energy densities of the principal form $f$: ${\mathbb{R}}^2\to \mathbb{R}$, \[ f(ξ_1,ξ_2) = f_1\big( ξ_1 \big) + f_2\big( ξ_2 \big) \, , \] with linear growth. As a main result it is shown that, regardless of a corresponding property of $f_2$, the assumption ($t\in \mathbb{R}$) $c_1(1+|t|{)}^{-{\mu }_1}\le f_1^{\prime \prime }(t)\le c_2,1<{\mu }_1<2,$ is sufficient to obtain higher integrability of ${\partial }_{1}u$ for any finite exponent. We also inculde a series of variants of our main theorem. We finally note that similar results in the case $f$: ${\mathbb{R}}^n\to \mathbb{R}$ hold with the obvious changes in notation.