Singular Sets of Minimizers for the Mumford-Shah Functional

By G. David. 2005

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The Singular Set of Lipschitzian Minima of Multiple Integrals (2007, methodology): “Using porosity properties to get dimension bounds is a strategy followed also in other contexts, and our approach undoubtedly has similarities with some of the methods developed in the parametric setting [4,6,36].”
Regularity of almost minimizers with free boundary (2013, background): “otice that when Ω is Lipschitz and u ∈ L1 loc(Ω) is such that ∇u ∈ L2(Ω), we can deﬁne the trace of u almost everywhere on ∂Ω, which lies in a slightly better space than L2(∂Ω), so (1.4) makes sense ([D]). They also showed that the minimizers are Lipschitz-continuous up to the free boundary ∂{u >”
Decay of correlations and the central limit theorem for meromorphic maps (2004, background): “deduces that kϕk Lp ≤ |m(ϕ)|+ckdϕk L2 ≤ kϕk L1 +ckdϕk L2. (2) Hence, W1,2 ⊂ Lp. The associated inclusion map is bounded. We will prove the following proposition for the reader’s convenience, see also [2, 17]. Proposition3.1 Let I ⊂ X be a compact set whose Hausdorﬀ (m− 1)- dimensional measure Hm−1(I”