Splitting-type variational problems with linear growth conditions

Regularity properties of solutions to variational problems are established for a broad class of strictly convex splitting-type energy densities of the principal form $f$: ${\mathbb{R}}^2\to \mathbb{R}$, $$f({\xi }_1,{\xi }_2)=f_1\left({\xi }_1\right)+f_2\left({\xi }_2\right),$$ with linear growth. As a main result it is shown that, regardless of a corresponding property of $f_2$, the assumption ($t\in \mathbb{R}$) $c_1 (1+|t|)^{-\mu_{1}} \le f_1’'(t) \le c_2, ,\quad 1 < \mu_1 < 2, ,$issufficienttoobtainhigherintegrabilityof$\partial_1 u$ 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.