Higher integrability of the gradient and dimension of the singular set for minimisers of the Mumford-Shah functional

Research Report MRR01-020

Higher integrability of the gradient and dimension of the singular set for minimisers of the Mumford-Shah functional

Luigi Ambrosio, Nicola Fusco and John E. Hutchinson

Abstract: We investigate the regularity properties of minimisers of the Mumford-Shah functional in Rⁿ. We show that the set of points in the singular set where the appropriately scaled Dirichlet energy tends to zero has Hausdorff dimension at most n-2. (In case n=2 the analysis in [19] suggests that such points correspond to "triple junctions".) As a consequence, we show that if the gradient of the minimiser is locally p-summable for some p>2 then the dimension of the singular set is bounded above by max{n-2, n-p/2}.

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