Partial regularity for minimizers of convex integrals with \(L\log L\)-growth
For a domain \(\Omega\subset \mathbb{R}^n\) the authors discuss the regularity properties of local minimizers \(u\in W^1_{\text{loc}}(\Omega, \mathbb{R}^n)\) of the variational integral \[ {\mathcal F}(u)= \int_\Omega \ln(1+|\nabla u|)|\nabla u|dx\tag{1} \] which is of nearly linear growth. In [\textit{M. Fuchs} and \textit{G. Seregin}, Calc. Var. Partial Differ. Equ. 6, No. 2, 171-187 (1998; Zbl 0929.49022)] partial regularity was established up to dimension \(n=4\). In the present paper the blow-up procedure is refined by a very careful choice of the quantity defining the excess which leads to partial \(C^1\)-regularity without any restriction on the dimension of the domain \(\Omega\). It is worth remarking that in [\textit{G. Mingione} and \textit{F. Siepe}, Z. Anal. Anwend. 18, No. 4, 1083-1100 (1999; preceding review)] even full \(C^1\)-regularity for local minimizers of (1) was proved by using the special structure of (1). Further extensions concerning functionals of the type \[ {\mathcal G}(u)= \int_\Omega g(|\nabla u|) dx\tag{2} \] with \(g\) being of nearly linear growth were presented in [\textit{M. Fuchs} and \textit{G. Mingione}, Manuscr. Math. 102, No. 2, 227-250 (2000)], for example, full \(C^1\)-regularity is true for \[ g(t)= t\ln(1+ \ln..(1+ \ln(1+ t))), \] i.e., for any iterated logarithm.
scientific article; zbMATH DE number 1439285