Fonseca, Irene; Malý, Jan; Mingione, Giuseppe Scalar minimizers with fractal singular sets. (English) Zbl 1049.49015 Arch. Ration. Mech. Anal. 172, No. 2, 295-307 (2004). Summary: Lack of regularity of local minimizers for convex functionals with non-standard growth conditions is considered. It is shown that for every \(\varepsilon > 0\) there exists a function \(a\in C^\alpha(\Omega)\) such that the functional \[ {\mathcal F}:u\mapsto \int_\Omega (| Du|^p+a(x)| Du|^q)\,dx \] admits a local minimizer \(u\in W^{1,p}(\Omega)\) whose set of non-Lebesgue points is a closed set \(\Sigma\) with \(\dim_{\mathcal H}(\Sigma) > N- p-\varepsilon\), and where \(1 < p < N <N+\alpha < q < +\infty\). Cited in 73 Documents MSC: 49J45 Methods involving semicontinuity and convergence; relaxation 49N60 Regularity of solutions in optimal control 28A80 Fractals Keywords:convex functional; fractal singular set PDFBibTeX XMLCite \textit{I. Fonseca} et al., Arch. Ration. Mech. Anal. 172, No. 2, 295--307 (2004; Zbl 1049.49015) Full Text: DOI