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

Given an integrand

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\,, D 2 f ( Z ) ( Y , Y ) ≥ c ( 1 + ∣ Z ∣ 2 ) − μ 2 ∣ Y ∣ 2 , 1 < μ ≤ 3   ,

we consider the variational problem

J[w] = \int_{\Omega} f(\nabla w)\,dx\to\min J [ w ] = ∫ Ω f ( ∇ w )   d x → min ⁡

among mappings

w w

:

\mathbb{R}^{n}\supset \Omega\to \mathbb{R}^{N} R n ⊃ Ω → R N

with prescribed Dirichlet boundary data. If we impose some boundedness condition, then the existence of a generalized minimizer

u^{\ast} u ∗

is proved such that

\int_{\Omega'} |\nabla u^{\ast}|\log^{2}(1+|\nabla u^{\ast}|^{2})\,dx \leq c(\Omega') ∫ Ω ′ ∣ ∇ u ∗ ∣ log ⁡ 2 ( 1 + ∣ ∇ u ∗ ∣ 2 )   d x ≤ c ( Ω ′ )

for any

\Omega'\Subset \Omega Ω ′ ⋐ Ω

. Here the limit case

\mu =3 μ = 3

is included and we obtain a clear interpretation of the particular solution

u^{\ast} u ∗

. Moreover, if

\mu <3 μ < 3

and if

f(Z)=g(|Z|^{2}) f ( Z ) = g ( ∣ Z ∣ 2 )

is assumed in the vector-valued case, then we show local

C^{1,\alpha} C 1 , α

-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 1 < μ < 1 + 2 / n

could be considered.