A Priori Gradient Estimates for Bounded Generalized Solutions of a Class of Variational Problems with Linear Growth

Given an integrand $f$ of linear growth and assuming an ellipticity condition of the form $$D^{2}f(Z)(Y,Y)\geq c \big(1+|Z|^{2}\big)^{-\frac{\mu}{2}} |Y|^{2},\quad 1< \mu \leq 3\,,$$ we consider the variational problem $J[w] = \int_{\Omega} f(\nabla w)\,dx\to\min$ among mappings $w$: $\mathbb{R}^{n}\supset \Omega\to \mathbb{R}^{N}$ with prescribed Dirichlet boundary data. If we impose some boundedness condition, then the existence of a generalized minimizer $u^{\ast}$ is proved such that $\int_{\Omega'} |\nabla u^{\ast}|\log^{2}(1+|\nabla u^{\ast}|^{2})\,dx \leq c(\Omega')$ for any $\Omega'\Subset \Omega$. Here the limit case $\mu =3$ is included and we obtain a clear interpretation of the particular solution $u^{\ast}$. Moreover, if $\mu <3$ and if $f(Z)=g(|Z|^{2})$ is assumed in the vector-valued case, then we show local $C^{1,\alpha}$-regularity and uniqueness up to a constant of generalized minimizers. These results substantially improve earlier contributions of the author and M. Fuchs [Rend. Mat. Appl., VII. Ser. 22 (2002) 249–274], where only the case of exponents $1 < \mu <1 +2/n$ could be considered.