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 : ℝ2 → ℝ, f(ξ1, ξ2) = f1(ξ1) + f2(ξ2), with linear growth. We show that, regardless of the corresponding property of f2, the assumption (t ∈ ℝ) $$ {c}_1{\left(1+\left|t\right|\right)}^{-{\upmu}_1}\le {f}_1^{\hbox{'}\hbox{'}}(t)\le {c}_2,\kern1em 1<{\upmu}_1<2, $$ is sufficient to obtain higher integrability of ∂1u for any finite exponent. We also include a series of variants of our main theorem. In the case f : ℝn → ℝ, similar results hold with the obvious changes in notation. Bibliography: 30 titles.