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 $f$: $\mathbb{R}^2 \to \mathbb{R}$, \[ f(\xi_1,\xi_2) = f_1\big( \xi_1 \big) + f_2\big( \xi_2 \big) \, , \] with linear growth. As a main result it is shown that, regardless of a corresponding property of $f_2$, the assumption ($t\in \mathbb{R}$) $c_1 (1+|t|)^{-\mu_{1}} \le f_1''(t) \le c_2\, ,\quad 1 < \mu_1 < 2\, ,$ 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.