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Scalar minimizers with fractal singular sets. (English) Zbl 1049.49015

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 \(1 < p < N <N+\alpha < q < +\infty\).

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
49N60 Regularity of solutions in optimal control
28A80 Fractals
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